text:
\section{Introduction}
\label{sec-intro}

Let $\gamma \in (0,2]$, let $U\subset\mathbbm C$ be an open set, and let $h$ be some variant of the Gaussian free field (GFF) on $U$. The \emph{$\gamma$-Liouville quantum gravity (LQG)} surface associated with $(U,h)$ is, heuristically speaking, the random two-dimensional Riemannian manifold parametrized by $U$ with Riemannian metric tensor $e^{\gamma h} \, (dx^2 + dy^2)$, where $dx^2 + dy^2$ is the Euclidean metric tensor. LQG surfaces arise as the scaling limits of various discrete random geometries, such as random planar maps and Liouville first passage percolation, which we discuss just below.

The above definition of an LQG surface does not make literal sense since $h$ is a distribution, not a function.
Nevertheless, it is possible to make sense of the volume form associated with an LQG surface as a limit of regularized versions of $e^{\gamma h} \,dz$, where $dz$ denotes Lebesgue measure; see~\cite{kahane,shef-kpz,rhodes-vargas-review}.
One can also make sense of LQG as a random metric space. This was first done in the special case when $\gamma=\sqrt{8/3}$ by Miller and Sheffield~\cite{lqg-tbm1,lqg-tbm2,lqg-tbm3}, in which case the resulting metric space is isometric to the so-called \emph{Brownian map}~\cite{legall-uniqueness,miermont-brownian-map}.
Very recently, Gwynne and Miller~\cite{gm-uniqueness} constructed the $\gamma$-LQG metric for all $\gamma\in(0,2)$, building on the works~\cite{dddf-lfpp,lqg-metric-estimates,local-metrics,gm-confluence}.

Several important properties of the $\gamma$-LQG are not yet fully understood.
For example, for $\gamma \not=  \sqrt{8/3} $, the Hausdorff dimension of the $\gamma$-LQG metric is unknown (the dimension is 4 for $\gamma=\sqrt{8/3}$).
However, recent progress on related problems has been made in~\cite{ding-goswami-watabiki,ding-dunlap-lqg-fpp,ding-zhang-geodesic-dim,dzz-heat-kernel,ghs-dist-exponent,ghs-map-dist,dg-lqg-dim,df-lqg-metric,ding-dunlap-lgd}.
Particularly relevant to us are the articles~\cite{ghs-map-dist,dzz-heat-kernel,dg-lqg-dim} which establish for each $\gamma \in (0,2)$ the existence of an exponent $d_\gamma > 2$ which arises in a variety of different approximations of LQG distances and which is equal to the Hausdorff dimension of the LQG metric~\cite{gp-kpz}.
See~\eqref{eqn-lambda-gamma} below for the appearance of $d_\gamma$ in our paper.
We define $d_2 := \lim_{\gamma\rightarrow 2^-} d_\gamma$, which exists since $\gamma\mapsto d_\gamma$ is increasing~\cite{dg-lqg-dim}.
It is known that $d_{\sqrt{8/3}} = 4$~\cite[Theorem 1.2]{dg-lqg-dim}, but for other $\gamma$ the value of $d_\gamma$ is unknown. The best-known physics prediction, due to Watabiki~\cite{watabiki-lqg}, is
\begin{equation} \label{eqn-watabiki}
d_\gamma^{\operatorname{Wat}} = 1 + \frac{\gamma^2}{4} + \frac14\sqrt{(4+\gamma^2)^2 + 16\gamma^2} .
\end{equation}
But, it was proven by Ding and Goswami~\cite{ding-goswami-watabiki} that this prediction is false for small values of $\gamma$.

One of the most natural ways to study $\gamma$-LQG distances is to consider the random metric obtained by exponentiating a continuous approximation of the GFF  (as is done in several of the above-cited works).
Such approximate metrics are referred to as \emph{Liouville first passage percolation (LFPP)}.
In this article, we will prove several estimates for LFPP distances which in particular lead to new bounds for $d_\gamma$ for general $\gamma \in (0,2]$, improving on the previous best known upper (resp.\ lower) bound from~\cite{dg-lqg-dim} in the case when $\gamma > \sqrt{8/3}$ (resp.\ $\gamma \in (0.4981, \sqrt{8/3})$) (Corollary~\ref{cor-d-bound}). We also establish an upper bound for the Euclidean dimension of LFPP geodesics.

Our bounds are valid not only for discretizations of $\gamma$-LQG with $\gamma \in (0,2]$ but also for discreteizations of a certain extension of LQG beyond this phase: LQG with central charge in $(1,25)$, which corresponds to $\xi > 2/d_2$ in the model which we define just below.
In this extended regime, our bounds are inconsistent with the analytic continuation of Watabiki's prediction for a range of parameter values; see Corollary~\ref{cor-watabiki-contradict}.
\medskip

\noindent\textbf{Acknowledgments.} We thank an anonymous referee for helpful comments on an earlier version of this article. We thank Scott Sheffield for helpful discussions. Part of the project was carried out during J.P.'s visit to the Isaac Newton Institute in Cambridge, UK in Summer 2018. We thank the institute for its hospitality.  J.P.\ was partially supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374.

\subsection*{Definition of the model}

Let $h$ be a whole-plane GFF, normalized so that its circle average over $\partial\mathbbm D$ is zero. For $\varepsilon > 0$ and $z\in\mathbbm C$, we write $h_\varepsilon(z)$ for the average of $h$ over the circle of radius $\varepsilon$ centered at $z$ (see~\cite[Section 3.1]{shef-kpz} for more on the circle average process).
We write $\mathbbm S  := [0,1]^2$.
For $\varepsilon > 0$, we define $\mathbbm S^\varepsilon := (\varepsilon\mathbbm Z^2) \cap \mathbbm S$ and we equip $\mathbbm S^\varepsilon$ with its standard nearest-neighbor graph structure.

For $\varepsilon , \xi \geq 0$ and a lattice path $P :\{0,1,\ldots,N\}   \rightarrow  \mathbbm S^\varepsilon$ for some $N \in\mathbbm N$, we define the \emph{$\varepsilon$-LFPP length} of $P$, with parameter $\xi$, by
\begin{equation}
L_h^{\xi,\varepsilon}(P) := \sum_{j=0}^N \varepsilon e^{\xi h_\varepsilon(P(j))} .
\end{equation}
The reason for the factor of $\varepsilon$ is that edges of $\mathbbm Z^2$ have side length $\varepsilon$, so this factor makes it so that $L_h^{\xi,\varepsilon}$ approximates the integral of $e^{\xi h_\varepsilon}$ along a linearly interpolated version of $P$.
For $z,w\in \mathbbm S^\varepsilon$, we define the \emph{$\xi$-LFPP distance} by\footnote{\label{footnote-lfpp-variants} There are several other natural variants of LFPP besides the one we consider here.
For example, we can replace the circle average process by the white noise approximation or by the discrete GFF.
We can also define distances by integrating along continuous paths rather than by summing along paths in $\varepsilon\mathbbm Z^2$.
The arguments of this paper work for any of these approximations; all we need is that the variance of the approximating field is of order $\log\varepsilon^{-1}  + O_\varepsilon(1)$, uniformly over $\mathbbm S$.
It follows from~\cite[Lemma 3.1 and Proposition 3.16]{dg-lqg-dim} that with probability tending to 1 as $\varepsilon\rightarrow 0$, LFPP distances are scaled by a factor of at most $\varepsilon^{o_\varepsilon(1)}$ if we replace the circle average process by the white noise decomposition and/or we integrate along continuous paths instead of discrete paths.
\cite[Theorem 1.4]{ang-discrete-lfpp} gives a similar statement comparing LFPP defined using the discrete GFF instead of the circle average process of the continuum GFF.
In particular, the exponents for distances in the above variants of LFPP are all the same.
We will use this fact without comment when we cite results from~\cite{dg-lqg-dim}. }
\begin{equation} \label{eqn-lfpp-def}
D_h^{\xi,\varepsilon}(z,w  ) := \inf_{P : z\rightarrow w}  L_h^{\xi,\varepsilon}(P)  ,
\end{equation}
where the infimum is over all lattice paths in $\mathbbm S^\varepsilon$ from $z$ to $w$.
Let $\partial_{\operatorname{L}} \mathbbm S^\varepsilon$ (resp.\ $\partial_{\operatorname{R}} \mathbbm S^\varepsilon$) be the left (resp.\ right) boundary of $\mathbbm S^\varepsilon$, i.e., the set of vertices of $\mathbbm S^\varepsilon$ whose nearest neighbor to the left (resp.\ right) in $\varepsilon\mathbbm Z^2$ is not in $\mathbbm S^\varepsilon$.
For $\xi \geq 0$, we define the \emph{$\xi$-LFPP distance exponent}
\begin{equation} \label{eqn-lambda-def}
\lambda(\xi) := \sup\left\{ \alpha \in\mathbbm R  : \liminf_{\varepsilon\rightarrow 0} \mathbbm P\left[ D_h^{\xi,\varepsilon}\left( \partial_{\operatorname{L}} \mathbbm S^\varepsilon , \partial_{\operatorname{R}} \mathbbm S^\varepsilon \right) \leq \varepsilon^{ \alpha} \right] = 1 \right\} .
\end{equation}
We have the following \emph{a priori} bounds for $\lambda(\xi)$.

\begin{lem} \label{lem-lambda-basic}
$\lambda(\xi) \in [-1/2 , 1]$ for all $\xi \geq 0$. Furthermore, $\xi\mapsto \lambda(\xi)$ is 2-Lipschitz on $[0,\infty)$.
\end{lem}
\begin{proof}
Let $\zeta >0$ be a small exponent. By~\cite[Lemma 5.3]{ghpr-central-charge}, it holds with probability tending to 1 as $\varepsilon\rightarrow 0$ that there is a path $P$ from $\partial_{\operatorname{L}} \mathbbm S^\varepsilon$ to $\partial_{\operatorname{R}} \mathbbm S^\varepsilon$ whose total number of vertices satisfies $\# P \leq \varepsilon^{-3/2-\zeta}$ such that $|h_\varepsilon(P(j))| \leq \zeta \log \varepsilon^{-1} $ for each $j=0,\dots,\# P-1$. For this path, one has $L_h^{\xi,\varepsilon}(P) \leq \varepsilon^{-1/2- (\xi+1)\zeta}$, so $\lambda(\xi) \geq -1/2- (\xi+1) \zeta$.
Furthermore, any path between the upper and lower boundaries of $\mathbbm S^\varepsilon$ must cross $P$, so must have $L_h^{\xi,\varepsilon}$-length at least $\varepsilon^{1+\xi\zeta}$. By $\pi/2$-rotational symmetric this implies $\lambda(\xi) \leq 1+\xi \zeta$. Since $\zeta>0$ is arbitrary we get $\lambda(\xi) \in [-1/2,1]$.

For the Lipschitz continuity, we observe that $\sup_{z\in\mathbbm S} |h_\varepsilon(z)| \leq (2+\zeta) \log\varepsilon^{-1}$ with probability tending to 1 as $\varepsilon\rightarrow 0$ (see, e.g.,~\cite[Proposition 2.4]{lqg-tbm2}). On the event that this bound holds, for any path $P$ and any $0\leq \widetilde\xi\leq \xi$, we have $\varepsilon^{ (2+\zeta)(\xi-\widetilde\xi)} \leq L_h^{\xi,\varepsilon}(P) / L_h^{\widetilde\xi,\varepsilon}(P) \leq \varepsilon^{-(2+\zeta)(\xi-\widetilde\xi)}$. Therefore $|\lambda(\xi) - \lambda(\widetilde\xi)| \leq (2+\zeta)(\xi-\widetilde\xi)$. Again since $\zeta > 0$ is arbitrary, we get the 2-Lipschitz continuity.
\end{proof}

It is shown in~\cite[Theorem 1.5]{dg-lqg-dim} (c.f.\ Footnote~\ref{footnote-lfpp-variants}) that, with $d_\gamma$ the dimension exponent for $\gamma$-LQG, it holds with probability tending to 1 as $\varepsilon\rightarrow 0$ that $D_h^{\gamma/d_\gamma , \varepsilon}(\partial_{\operatorname{L}} \mathbbm S^\varepsilon , \partial_{\operatorname{R}}\mathbbm S^\varepsilon) = \varepsilon^{1 - \gamma Q/d_\gamma + o_\varepsilon(1)}$, where $Q = 2/\gamma + \gamma/2$. Moreover, the same is true for several other quantities related to LFPP distances, such as diameters and point-to-point distances.
In particular,
\begin{equation} \label{eqn-lambda-gamma}
\lambda(\gamma/d_\gamma) = 1 - \frac{\gamma}{d_\gamma} Q  = 1 - \frac{\gamma}{d_\gamma} \left(\frac{2}{\gamma} + \frac{\gamma}{2}\right) ,\quad\forall \gamma \in (0,2] .
\end{equation}
Note that~\eqref{eqn-lambda-gamma} for $\gamma=2$ follows from the case $\gamma <2$ and the continuity of $\lambda$. Since $d_{\sqrt{8/3}} = 4$,~\eqref{eqn-lambda-gamma} implies that $\lambda(1/\sqrt 6) =1/6 $.

As we will explain in Section~\ref{sec-cc}, we expect that LFPP with $\xi > 2/d_2$ is connected to Liouville quantum gravity with central charge in $(1,25]$ (note that $\gamma$-LQG for $\gamma\in (0,2]$ corresponds to central charge in $(-\infty,1]$).
In this regime, we do not know that $ D_h^{\xi,\varepsilon}\left( \partial_{\operatorname{L}} \mathbbm S^\varepsilon , \partial_{\operatorname{R}} \mathbbm S^\varepsilon \right) \geq \varepsilon^{\lambda(\xi) + o_\varepsilon(1)}$ with high probability.
However, we expect that this can be proven using arguments similar to those used to show the existence of an exponent for Liouville graph distance in~\cite{dzz-heat-kernel}.

By~\eqref{eqn-lambda-gamma}, Watabiki's prediction~\eqref{eqn-watabiki} is equivalent to $\lambda(\xi)  =  \xi^2 $, $\forall \xi \in [0,2/d_2]$: indeed, $d_\gamma^{\operatorname{Wat}}$ is the positive solution to $d_\gamma^2  - \gamma Q d_\gamma =  \gamma^2  $, and dividing this by $d_\gamma^2$ gives $1-\xi Q  = \xi^2  $.

One might guess that $\lambda(\xi)$ is an analytic function of $\xi$ at least up until the smallest $\xi > 0$ for which $\lambda(\xi) = 1$. Indeed, as explained in Section~\ref{sec-cc}, $\lambda(\xi) = 1$ corresponds to the critical point at which the ``background charge" $Q$ is zero and the ``central charge" $\mathbf{c}$ is 25.
We do not know if there is a finite value of $\xi_* > 0$ for which $\lambda(\xi_*) =1$, but if such a $\xi_*$ exists we expect that LFPP is in some sense degenerate for $\xi > \xi_*$. The vast majority of exponents associated with LQG depend analytically on their parameter values in the range for which the objects are non-degenerate.
For example, the KPZ formula extends analytically to the case when $\mathbf{c} \in (1,25)$~\cite[Theorem 1.5]{ghpr-central-charge}.\footnote{If one assumes that a metric on LQG with $\xi  > 2/d_2$ and $Q\in (0,2)$ exists and satisfies certain axioms (which should be satisfied if LFPP for $\xi > 2/d_2$ has a scaling limit), then one can show that the formulas for exponents / dimensions from~\cite{gp-kpz,lqg-metric-estimates} extend analytically to this regime, provided they still give finite positive answers. Examples of such formulas include the optimal H\"older exponent of the Euclidean metric w.r.t.\ the LQG metric~\cite[Theorem 1.7]{lqg-metric-estimates} and the LQG dimension of the $\alpha$-thick points of the field~\cite[Theorem 1.5]{gp-kpz}. The proofs of these formulas should be the same as the proofs in~\cite{gp-kpz,lqg-metric-estimates}, but such proofs have not been written down.}
We emphasize, though, that the analyticity of $\lambda(\xi)$ for $\xi < \xi_*$ is only a guess, and we would not be very surprised if this guess turns out to be false.

If $\lambda(\xi)$ were analytic for $\xi < \xi_*$, then we would have the following extension of Watabiki's prediction from the case when $\xi \in (0,2/d_2]$ to the case of general $\xi >0$:
\begin{equation} \label{eqn-watabiki-extend}
\lambda^{\operatorname{Wat}}(\xi)  =  \min\{\xi^2 ,1\}  ,\quad\forall \xi  \geq 0 .
\end{equation}
We will show in Corollary~\ref{cor-watabiki-contradict} that~\eqref{eqn-watabiki-extend} is false for a specific subset of $[2/d_2,\infty)$.

\section{Main results}
\label{sec-results}

The starting point of our main results is the following comparison of LFPP lengths of a path for different values of $\xi$, which will be proven (via a one-page argument) in Section~\ref{sec-proof}.

\begin{thm} \label{thm-length-compare}
Let $0 \leq \widetilde\xi \leq \xi  $ and fix a small parameter $\zeta > 0$.
With probability tending to 1 as $\varepsilon\rightarrow 0$, each simple path $P$ in $\mathbbm S^\varepsilon$ with $D_h^{\xi,\varepsilon}$-length $L_h^{\xi,\varepsilon}(P) \leq \varepsilon^{\lambda(\xi) -\zeta}$ satisfies
\begin{equation} \label{eqn-length-compare}
L_h^{\widetilde\xi,\varepsilon}(P) \leq  \varepsilon^{\lambda(\xi)   -  (\xi-\widetilde\xi) \left( \sqrt{2 + 2 \lambda(\xi) + \xi^2}  -\xi  \right) - \zeta } .
\end{equation}
\end{thm}

\begin{cor}[Upper differential inequality for $\lambda$] \label{cor-lambda-mono}
If $ 0 \leq \widetilde\xi < \xi$, then
\begin{equation} \label{eqn-lambda-mono}
\frac{\lambda(\xi) -  \lambda(\widetilde\xi) }{\xi - \widetilde \xi} \leq   \sqrt{2 + 2 \lambda(\xi) + \xi^2}  -\xi    .
\end{equation}
In particular, for Lebesgue-a.e.\ $\xi  \geq 0$,
\begin{equation} \label{eqn-lambda-deriv}
\lambda'(\xi) \leq  \sqrt{2 + 2 \lambda(\xi) + \xi^2}  -\xi  .
\end{equation}
\end{cor}
\begin{proof}
The relation~\eqref{eqn-lambda-mono} is immediate from the definition~\eqref{eqn-lambda-def} of $\lambda$ and Theorem~\ref{thm-length-compare} applied to a path from $\partial_{\operatorname{L}} \mathbbm S^\varepsilon$ to $\partial_{\operatorname{R}} \mathbbm S^\varepsilon$ with minimal $D_h^{\xi,\varepsilon}$-length.
This relation implies~\eqref{eqn-lambda-deriv} since $\xi\mapsto \lambda(\xi)$ is Lipschitz, hence differentiable a.e.
\end{proof}

Corollary~\ref{cor-lambda-mono} complements the monotonicity relation from~\cite[Lemma 2.5]{dg-lqg-dim} which shows that for $0 \leq \widetilde\xi \leq \xi$,
\begin{equation} \label{eqn-dg-mono}
\lambda(\widetilde\xi) + \frac{\widetilde\xi^2}{2} \leq \lambda(\xi) + \frac{\xi^2}{2} \quad \text{and hence} \quad \lambda'(\xi) \geq - \xi .
\end{equation}
By combining these two differential inequalities and the fact that $\lambda(0) = 0$ and $\lambda(1/\sqrt 6) = 1/6$ (see the discussion just below~\eqref{eqn-lambda-gamma}), we get the following theorem.

\begin{thm}[Bounds for $\lambda(\xi)$] \label{thm-lambda-bound}
For $\xi  \geq 0$,
\begin{equation} \label{eqn-lambda-bound}
\underline\lambda(\xi) \leq \lambda(\xi) \leq \overline\lambda(\xi)
\end{equation}
where
\begin{equation} \label{eqn-lambda-lower}
\underline\lambda(\xi) :=
\begin{dcases}
\max\left\{ \left(\sqrt{\frac{5}{2}} - \frac{1}{\sqrt 6} \right) \xi  - \frac{\sqrt{15}-2}{6}   ,   - \frac{\xi^2}{2}     \right\} ,\quad &\xi \leq \frac{1}{\sqrt 6}  \\
\max\left\{ \frac{1}{4} - \frac{\xi^2}{2} ,  - \frac12 \right\} \quad &\xi \geq \frac{1}{\sqrt 6}
\end{dcases}
\end{equation}
and
\begin{equation} \label{eqn-lambda-upper}
\overline\lambda(\xi) :=
\begin{dcases}
\min\left\{    \frac{1}{4} - \frac{\xi^2}{2} ,  \sqrt{2}\xi    \right\} ,\quad &\xi \leq \frac{1}{\sqrt 6}  \\
\min\left\{  \left(\sqrt{\frac{5}{2}} - \frac{1}{\sqrt 6} \right) \xi  - \frac{\sqrt{15} -2}{6}    ,   1     \right\} ,\quad &\xi \geq \frac{1}{\sqrt 6} .
\end{dcases}
\end{equation}
\end{thm}
\begin{proof}
Recall from Lemma~\ref{lem-lambda-basic} that $\lambda(\xi) \in [-1/2, 1]$ for all $\xi\geq 0$.
Since $\lambda(0) = 0$, by setting $\widetilde\xi = 0$ in~\eqref{eqn-lambda-mono} and~\eqref{eqn-dg-mono} and solving for $\lambda(\xi)$, we get
\begin{equation} \label{eqn-around0}
- \frac{\xi^2}{2}  \leq  \lambda(\xi)  \leq \sqrt{2}\xi   ,\quad\forall \xi \geq 0.
\end{equation}
Since $\lambda(1/\sqrt 6) = 1/6$, by setting $\widetilde\xi = 1/6$ in~\eqref{eqn-lambda-mono} and~\eqref{eqn-dg-mono} and solving for $\lambda(\xi)$, we get
\begin{equation} \label{eqn-around-4}
 \frac{1}{4} - \frac{\xi^2}{2} \leq  \lambda(\xi) \leq  \left(\sqrt{\frac{5}{2}} - \frac{1}{\sqrt 6} \right) \xi  - \frac{\sqrt{15}-2}{6} ,\quad\forall\xi \geq 1/\sqrt 6 .
\end{equation}
By instead setting $\xi = 1/6$ and solving for $\lambda(\widetilde\xi)$, we get~\eqref{eqn-around-4} with the inequality signs flipped for $\xi \leq 1/\sqrt 6$.
Combining these inequalities gives~\eqref{eqn-lambda-bound}.
\end{proof}

See Figure~\ref{fig-lambda-bound}, left, for a plot of the bounds~\eqref{eqn-lambda-bound}.
At the time this paper was written, the bounds for $\lambda(\xi)$ from Theorem~\ref{thm-lambda-bound} were the best known except when $\xi$ is very small (non-explicit), in which case~\cite{ding-goswami-watabiki} gives $\lambda(\xi) \geq  c \xi^{4/3}/\log(1/\xi) $ for a non-explicit universal constant $c>0$. However, very recently improved lower bounds have been obtained in some cases; see Remark~\ref{remark-ang}. From Corollary~\ref{cor-lambda-mono} and Theorem~\ref{thm-lambda-bound}, we get the following.

\begin{cor} \label{cor-watabiki-contradict}
The extended Watabiki prediction~\eqref{eqn-watabiki-extend} is false on a dense subset of $\{ \xi \geq 1/\sqrt 3 : \lambda(\xi) < 1 \}$ and for all $\xi\in \left(\sqrt{\frac52}-\sqrt{\frac23} , \frac{4 +\sqrt{15}}{ \sqrt 2 ( 3 \sqrt{5} - \sqrt 3)} \right)\approx \left(0.7646 ,  1.1187  \right)$.
\end{cor}
\begin{proof}
If $\lambda(\xi) = \xi^2$ holds on a neighborhood of $\xi$ and $\xi > 1/\sqrt 3$, then $\lambda'(\xi) = 2\xi < \sqrt{2+2\xi^2 + \xi^2} -\xi$, contrary to~\eqref{eqn-lambda-deriv}. The second statement follows since the upper bound in~\eqref{eqn-lambda-upper} is strictly less than $\xi^2$ for $\xi > \sqrt{\frac52}-\sqrt{\frac23}$ and strictly less than 1 for $\xi < \frac{4 +\sqrt{15}}{ \sqrt 2 ( 3 \sqrt{5} - \sqrt 3)} $.
\end{proof}

\begin{comment}
Solve[2 xi == Sqrt[2 + 3 xi^2] - xi , xi]

lambdaBound[xi_] := 1/6 (2 - Sqrt[15] - Sqrt[6] xi + 3 Sqrt[10] xi)
Solve[lambdaBound[xi] == xi^2 , xi];
FullSimplify
N
Solve[lambdaBound[xi] == 1 , xi]
N

FullSimplify[(Sqrt[5/2] - 1/Sqrt[6]) xi - (Sqrt[15] - 2)/6 - ((1/6) (2 - Sqrt[15] - Sqrt[6] xi + 3 Sqrt[10] xi))]

\begin{lem}
Suppose $\lambda'(\xi) \leq \sqrt{2+2\lambda(\xi) + \xi^2} - \xi$.
Then $\lambda(\xi) \leq   \xi_0^2 -  \xi \xi_0 +   \sqrt{ 2 + 2 \lambda (\xi_0) + \xi_0^2 } (\xi-\xi_0) +     \lambda (\xi_0) $ for each $\xi \geq \xi_0$.
\end{lem}
\begin{proof}
Let $u(\xi) := 2 + 2 \lambda (\xi) + \xi^2$. Then $u'(\xi) = 2\lambda'(\xi) + 2\xi$, so $\lambda'(\xi) = (u'(\xi) - 2\xi)/2$.
Therefore, $u'(\xi) \leq 2\sqrt{u(\xi)}$. Hence for $\xi\geq \xi_0$, we have $2\sqrt{u(\xi)} \leq 2 (\xi - \xi_0) + 2 \sqrt{u(\xi_0 )}$ and hence $u(\xi)  \leq   ( \xi - \xi_0 + \sqrt{u(\xi_0)} )^2   $.
Substituting back in $\lambda$ gives
\begin{equation*}
2 + 2 \lambda (\xi) + \xi^2
\leq (\xi - \xi_0 + \sqrt{ 2 + 2 \lambda (\xi_0) + \xi_0^2 } )^2
= \xi^2 + 2 \xi_0^2 - 2\xi \xi_0 + 2 \sqrt{ 2 + 2 \lambda (\xi_0) + \xi_0^2 } (\xi-\xi_0) + 2 + 2 \lambda (\xi_0)
\end{equation*}
Hence
\begin{equation*}
\lambda(\xi) \leq      \xi_0^2 -  \xi \xi_0 +   \sqrt{ 2 + 2 \lambda (\xi_0) + \xi_0^2 } (\xi-\xi_0) +     \lambda (\xi_0)    .
\end{equation*}
\end{proof}

\end{comment}

As explained in Section~\ref{sec-cc}, the conditions in Corollary~\ref{cor-watabiki-contradict} correspond to central charge in $(17,25)$ and in $(21.741\dots,25)$, respectively.
The combination of Corollary~\ref{cor-watabiki-contradict} and~\cite{ding-goswami-watabiki} shows that if Watabiki's prediction is true for a non-trivial interval of $\xi$-values, then $\lambda(\xi)$ must fail to be analytic at at least two different values of $\xi$ with $\lambda(\xi )  <1$.
Since the existence of two non-analytic points would be more surprising than the existence of just one such point, this provides further evidence against the statement that Watabiki's prediction is true for a non-trivial interval of $\xi$-values.

In contrast, all known bounds for $\lambda(\xi)$ (including those of Corollary~\ref{cor-lambda-mono}, Theorem~\ref{thm-lambda-bound}, and~\cite{ding-goswami-watabiki}) \emph{are} consistent with the alternative guess for the dimension of $\gamma$-LQG from~\cite[Equation (1.16)]{dg-lqg-dim}, namely $d_\gamma^{\operatorname{DG}} = 2 + \frac{\gamma^2}{2} + \frac{\gamma}{\sqrt 6}$,
which is equivalent to $\lambda(\xi) = \xi/\sqrt 6$ for $\xi \in [0,2/d_2]$ and corresponds to the extended guess
\begin{equation} \label{eqn-dg-guess}
\lambda^{\operatorname{DG}}(\xi) = \min\{\xi/\sqrt 6 , 1\} ,\quad \forall \xi \geq 0.
\end{equation}
Note that if $\lambda(\xi) = \lambda^{\operatorname{DG}}(\xi)$, then the Ding-Goswami bound $\lambda(\xi) \geq  c \xi^{4/3}/\log(1/\xi)$ for small $\xi$ would be far from optimal. We emphasize, though, that there is currently no theoretical justification for the above alternative guess, even at a heuristic level.

\begin{figure}[t!]
 \begin{center}
\includegraphics[scale=.55]{figures/fig-lambda-bound.pdf} \hspace{15pt} \includegraphics[scale=.55]{figures/fig-geodesic-bound.pdf}
\vspace{-0.01\textheight}
\caption{ \textbf{Left.} Graph of the lower bound $\underline \lambda(\xi)$ (red) and the upper bound $\overline \lambda(\xi)$ (blue) from Theorem~\ref{thm-lambda-bound} together with the extended Watabiki prediction $\lambda(\xi) = \max\{\xi^2,1\}$ (green) and the alternative guess $\lambda(\xi) = \max\{\xi/\sqrt 6 , 1\}$ (orange) for $\xi \in [0,1]$.
\textbf{Right.} Graph of the upper bound for the geodesic dimension $g(\xi)$, obtained from plugging in the lower bound for $\lambda(\xi)$ from Theorem~\ref{thm-lambda-bound} into Corollary~\ref{cor-geodesic-dim-upper}, on the interval $[0,\sqrt{5/2}]$ (the range on which they are non-trivial). The upper bound is constant at $\frac16(4+\sqrt{15})\approx 1.31216$ for $\xi \in \left[ 0.241\dots , 1/\sqrt 6 \right]$ but we expect that $g(\xi)$ is strictly increasing, at least on $[0,2/d_2]$. The dashed red line is at our upper bound $2 - \sqrt{5/2} \approx 0.4189$ for $2/d_2$. The vertical coordinate where it crosses the graph is $2\sqrt{10} - 5\approx 1.3246$, which is (at least heuristically) an upper bound for the largest possible Euclidean dimension of a $\gamma$-LQG geodesic for $\gamma \in (0,2]$.
}\label{fig-lambda-bound}
\end{center}
\vspace{-1em}
\end{figure}

Using~\eqref{eqn-lambda-gamma}, we can translate Theorem~\ref{thm-lambda-bound} for $\xi\in (0,2/d_2]$ into bounds for the $\gamma$-LQG dimension.

\begin{cor}[Bounds for $d_\gamma$] \label{cor-d-bound}
For $\gamma \in (0,2)$, one has
\begin{equation}
\underline d_\gamma \leq d_\gamma \leq \overline d_\gamma
\end{equation}
for
\begin{equation} \label{eqn-d-lower}
\underline d_\gamma :=
\begin{dcases}
\max\left\{  \frac{ 12 - \sqrt 6 \gamma + 3 \sqrt{10} \gamma + 3 \gamma^2}{4 + \sqrt{15}}    ,  \frac{2\gamma^2}{4+\gamma^2-\sqrt{16 +\gamma^4}}         \right\} ,\quad &\gamma \leq \sqrt{8/3}  \\
\frac13 \left( 4 + \gamma^2 +\sqrt{16 + 2 \gamma^2 + \gamma^4} \right) ,\quad &\gamma \geq \sqrt{8/3}
\end{dcases}
\end{equation}
and
\begin{equation}  \label{eqn-d-upper}
\overline d_\gamma :=
\begin{dcases}
\min\left\{    \frac13 \left( 4 + \gamma^2 +\sqrt{16 + 2 \gamma^2 + \gamma^4} \right)  ,  2 + \frac{\gamma^2}{2} + \sqrt 2 \gamma \right\} ,\quad &\gamma \leq \sqrt{8/3}  \\
\frac{ 12 - \sqrt 6 \gamma + 3 \sqrt{10} \gamma + 3 \gamma^2}{4 + \sqrt{15}}  ,\quad &\gamma \geq \sqrt{8/3}
\end{dcases} .
\end{equation}
\end{cor}

See Figure~\ref{fig-d-bound} for a plot of the bounds for Corollary~\ref{cor-d-bound}, the previous best known bounds from~\cite{dg-lqg-dim}, and the Watabiki prediction~\eqref{eqn-watabiki}.
The new bounds are still consistent with Watabiki's prediction for $\gamma \in (0,2]$ (since $2/d_2  < 1/\sqrt 3$).
The upper (resp.\ lower) bound from Corollary~\ref{cor-d-bound} is strictly better than previously known bounds in the case when $\gamma \geq \sqrt{8/3}$ (resp.\ $\gamma \in (0.4981,\sqrt{8/3})$).
For $\gamma \geq \sqrt{8/3}$, the upper bound differs from Watabiki's prediction by at most $0.008$.

\begin{figure}[t!]
 \begin{center}
\includegraphics[scale=.55]{figures/fig-d-bound.pdf} \hspace{15pt} \includegraphics[scale=.55]{figures/fig-d-bound2.pdf}
\vspace{-0.01\textheight}
\caption{ \textbf{Left.} Graph of the lower bound $\underline d_\gamma$ (red) and the upper bound $\overline d_\gamma$ (blue) from Corollary~\ref{cor-d-bound} together with the Watabiki prediction $d_\gamma^{\operatorname{Wat}}$ from~\eqref{eqn-watabiki} (green) and the previous best known bounds (dashed blue and dashed red). Note that the bounds $\underline d_\gamma \leq d_\gamma \leq \overline d_\gamma$ are consistent with the Watabiki prediction (so the only contradiction to Watabiki for $\gamma \in (0,2)$ is still~\cite{ding-goswami-watabiki}).
\textbf{Right.} Graph of the same functions but restricted to the interval $[\sqrt 2 ,2]$.
}\label{fig-d-bound}
\end{center}
\vspace{-1em}
\end{figure}

\begin{remark} \label{remark-ang}
The recent paper~\cite{ang-discrete-lfpp} shows that $\lambda(\xi) \geq 0$ for a slightly different definition of $\lambda(\xi)$, using distance between the inner and outer boundaries of an annulus, rather than distance across a circle. All of our arguments still work with this alternative definition of $\lambda(\xi)$ (the definitions are known to be equivalent when $\xi \leq 2/d_2$, and we expect that they are equivalent for all $\xi \geq 0$).
The bound $\lambda(\xi) \geq 0$ improves on our lower bound for $\lambda(\xi)$ from Theorem~\ref{thm-lambda-bound} in the case when $\xi \in (0,0.2661\dots) \cup (1/\sqrt 2 , \infty)$.
By~\eqref{eqn-lambda-gamma}, for $\gamma \in (0,2)$, $\lambda(\xi ) \geq 0$ implies that $d_\gamma \geq 2+\gamma^2/2$, which improves on our lower bound for $d_\gamma$ from Corollary~\ref{cor-d-bound} in the case when $\gamma \in (0, 0.5765\dots)$.
\end{remark}

\begin{comment}

Solve[(Sqrt[5/2] - 1/Sqrt[6]) xi - (Sqrt[15] - 2)/6 == 0, xi]
N

Solve[1/4 - xi^2/2 == 0, xi]

Solve[(12 - Sqrt[6] gamma + 3 Sqrt[10] gamma + 3 gamma^2)/(4 +
     Sqrt[15]) == 2 + gamma^2/2, gamma]
N

\end{comment}

\subsection*{Geodesic dimension}

In addition to the Hausdorff dimension/distance exponent, another natural quantity associated with the LQG metric is the Euclidean dimension of its geodesics.
Theorem~\ref{thm-length-compare} also leads to a bound for this dimension.
Let $P_h^{\xi,\varepsilon}$ be the a.s.\ unique path in $\mathbbm S^\varepsilon$ connecting the left and right boundaries with minimal $D_h^{\xi,\varepsilon}$-length.\footnote{The uniqueness of $P^{\xi,\varepsilon}$ follows since there are only finitely many simple paths in $\mathbbm S^\varepsilon$ and a.s.\ no two such paths have the same $D_h^{\xi,\varepsilon}$-length, which in turn is a consequence of the fact that a.s.\ $h_\varepsilon(z)\not=h_\varepsilon(w)$ for each distinct $z,w\in \mathbbm S^\varepsilon$.} We define the LFPP geodesic dimension
\begin{equation} \label{eqn-geo-dim-def}
g(\xi) := \inf\left\{ \alpha  > 0 : \liminf_{\varepsilon\rightarrow 0} \mathbbm P\left[ \# P_h^{\xi,\varepsilon} \leq \varepsilon^{-\alpha}   \right] = 1\right\} ,
\end{equation}
where here and in what follows we write $\# P$ for the number of vertices in a path $P$.
Then $g(\xi)$ is a reasonable notion of the ``Euclidean dimension" of the path $P_h^{\xi,\varepsilon}$ since $\# P$ is the number of Euclidean squares of side length $\varepsilon$ needed to cover $P$. If (as expected) $D_h^{\xi,\varepsilon}$ converges to a limiting metric as $\varepsilon\rightarrow 0$, then $g(\xi)$ should be the Euclidean Minkowski dimension of a typical geodesic for this limiting metric.

In~\cite{ding-zhang-geodesic-dim} it is shown that $g(\xi) > 1$ whenever $\lambda(\xi) > 0$, which we know is the case for $\xi$ sufficiently small by~\cite{ding-goswami-watabiki} and for $\xi  > 0.266\dots$ by Theorem~\ref{thm-lambda-bound}.
We also note that~\cite[Proposition 4.8]{mq-geodesics} shows that the Hausdorff dimension of geodesics for the continuum $\gamma$-LQG metric is strictly less than 2.
Neither of these works prove a non-trivial explicit bound.
Here we prove the first non-trivial explicit bound for $g(\xi)$.

\begin{cor}[Geodesic dimension upper bound] \label{cor-geodesic-dim-upper}
For each $\xi > 0$ and each $\zeta >0$, it holds with probability tending to 1 as $\varepsilon\rightarrow 0$ that each simple path $P$ in $\mathbbm S^\varepsilon$ with $D_h^{\xi,\varepsilon}$-length $L_h^{\xi,\varepsilon}(P) \leq \varepsilon^{\lambda(\xi) - \zeta}$ satisfies
\begin{equation} \label{eqn-geodesic-dim-upper}
\# P \leq \varepsilon^{\lambda(\xi)   -  \xi \left( \sqrt{2 + 2 \lambda(\xi) + \xi^2}  -\xi  \right)  - 1  - \zeta } .
\end{equation}
In particular,
\begin{equation} \label{eqn-g-upper}
g(\xi) \leq 1 - \lambda(\xi)   +   \xi \left( \sqrt{2 + 2 \lambda(\xi) + \xi^2}  -\xi  \right) .
\end{equation}
\end{cor}
\begin{proof
Apply Theorem~\ref{thm-length-compare} with $\widetilde\xi =  0$ and note that $L_h^{0,\varepsilon}(P) = \varepsilon \# P$.
\end{proof}

The upper bound~\eqref{eqn-g-upper} is a decreasing function of $\lambda(\xi)$ whenever $\lambda(\xi) \geq -1$, and we know that $\lambda(\xi) \geq -1/2$ by Lemma~\ref{lem-lambda-basic}.
Plugging our lower bound for $\lambda(\xi)$ from Theorem~\ref{thm-lambda-bound} into~\eqref{eqn-g-upper} gives an upper bound for $g(\xi)$ in terms of $\xi$ which is non-trivial ($ < 2$) for $\xi  < \sqrt{5/2}$. We plot this bound in Figure~\ref{fig-lambda-bound}, right.
Recall from the discussion just after~\eqref{eqn-geo-dim-def} that $g(\xi)$ is expected to be the Euclidean dimension of geodesics with respect to the continuum limit of the metrics $D_h^{\xi,\varepsilon}$. By setting $\xi = \gamma/d_\gamma$ for $\gamma \in (0,2)$ in~\eqref{eqn-g-upper} and using~\eqref{eqn-lambda-gamma}, we get the following heuristic bound:
\begin{equation}  \label{eqn-geodesic-gamma}
\left( \text{Euclidean dimension of $\gamma$-LQG geodesics}\right)\leq \frac{\gamma}{d_\gamma}\left( \frac{2}{\gamma} + \frac{\gamma}{2} - \frac{\gamma}{d_\gamma}    + \sqrt{2 + 2 \frac{\gamma}{d_\gamma} + \frac{\gamma^2}{d_\gamma^2} }    \right)  .
\end{equation}
For $\gamma =\sqrt{8/3}$, in which case $d_\gamma = \sqrt{8/3}$, the right side of~\eqref{eqn-geodesic-gamma} is
$\frac{1}{6} (4 + \sqrt{15}) \approx 1.31216$.
In~\cite{gp-kpz}, we prove~\eqref{eqn-geodesic-gamma} for geodesics of the continuum $\gamma$-LQG metric from~\cite{gm-uniqueness}.

\section{Proof of Theorem~\ref{thm-length-compare}}
\label{sec-proof}

The only estimate needed for the proof of Theorem~\ref{thm-length-compare} is the following lemma.

\begin{lem} \label{lem-bad-circle-count}
For $\alpha >0$,
\begin{equation*}
\mathbbm E\left[ \#\left\{ z \in \mathbbm S^\varepsilon : h_\varepsilon(z) <  \alpha\log\varepsilon  \right\}  \right] = O_\varepsilon\left(  \varepsilon^{ -(2-\alpha^2/2)  } \right) .
\end{equation*}
\end{lem}
\begin{proof}
The calculations in~\cite[Section 3.1]{shef-kpz} show that for each vertex $z \in \mathbbm S^\varepsilon$, the circle average $ h_\varepsilon(z)$ is centered Gaussian with variance $\log \varepsilon^{-1} + O_\varepsilon(1)$, where the $O_\varepsilon(1)$ is uniform over all $z\in\mathbbm S^\varepsilon$.\footnote{The calculations in~\cite[Section 3.1]{shef-kpz} are carried out for the zero-boundary GFF on a proper subdomain of $\mathbbm C$, but similar calculations work in the whole-plane case. Alternatively, the whole-plane case can be extracted from the zero-boundary case using either the Markov property of the whole-plane GFF~\cite[Proposition 2.8]{ig4}; or the fact that $h|_{(-1,2)^2}$ can be expressed as the limit (in the total variation sense) of $(\mathring h^n - \mathring h^n_1(0))|_{(-1,2)^2}$ as $n\rightarrow\infty$, where for $n\in\mathbbm N$, $\mathring h^n$ is the zero-boundary GFF on the ball $B_n(0)$~\cite[Proposition 2.10]{ig4}.}
The lemma now follows by applying the Gaussian tail bound to each of these random variables, then summing over the $O_\varepsilon (\varepsilon^{-2})$ vertices of $\mathbbm S^\varepsilon$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm-length-compare}]
Fix $\alpha > 0$ to be chosen later, in a manner depending only on $\xi$ and $\widetilde\xi$.
Our strategy for bounding the $\widetilde{\xi}$-LFPP length of a path $P$ in terms of its $\xi$-LFPP length is to partition the set of points $z \in P$ according to whether $h_{\varepsilon}(z) < \alpha \log \varepsilon$ or $h_\varepsilon(z) \geq \alpha\log\varepsilon$.   The idea is that the contribution of points $z$ with $h_{\varepsilon}(z) \geq \alpha \log \varepsilon$ to the $\widetilde{\xi}$-LFPP length can be bounded in terms of their contribution to the $\xi$-LFPP length; on the other hand, we can crudely bound the set of $z \in P$ with $h_{\varepsilon}(z) < \alpha \log \varepsilon$ in terms of the total number of such points in $\mathbbm S^{\varepsilon}$.

Let us now proceed with the details.
By Lemma~\ref{lem-bad-circle-count}, it holds with probability tending to 1 as $\varepsilon\rightarrow 0$ that
\begin{equation} \label{eqn-use-bad-circle-count}
\#\left\{ z \in \mathbbm S^\varepsilon : h_\varepsilon(z) <  \alpha\log\varepsilon \right\} \leq  \varepsilon^{ -(2-\alpha^2/2)  - \zeta } .
\end{equation}
Henceforth assume that~\eqref{eqn-use-bad-circle-count} holds. We will show that~\eqref{eqn-length-compare} holds.

Let $P : \{0,1,\ldots,N\} \rightarrow \mathbbm S^\varepsilon$ be a simple path in $\mathbbm S^\varepsilon$ with $L_h^{\xi,\varepsilon}(P)$-length at most $\varepsilon^{\lambda(\xi) -\zeta} $.
We partition the points in the sum defining $L_h^{\widetilde\xi,\varepsilon}(P)$ according to whether $h_\varepsilon(P(j)) < \alpha\log\varepsilon$ or $h_\varepsilon(P(j)) \geq \alpha\log\varepsilon$:
\allb \label{eqn-path-split}
L_h^{\widetilde\xi,\varepsilon}(P)
= \sum_{j=0}^N \varepsilon e^{\widetilde\xi h_\varepsilon(P(j) )}
&= \sum_{j : h_\varepsilon( P(j) )  <   \alpha\log\varepsilon } \varepsilon e^{\widetilde\xi h_\varepsilon(P(j) )}
 + \sum_{j : h_\varepsilon( P(j) )  \geq \alpha\log\varepsilon } \varepsilon e^{\widetilde\xi h_\varepsilon(P(j) )} \notag \\
&\leq  \varepsilon^{1 + \alpha \widetilde\xi} \#\left\{ j : h_\varepsilon(P(j) )   <   \alpha\log\varepsilon   \right\}
 +   \sum_{j : h_\varepsilon(P(j) )  \geq \alpha\log\varepsilon } \varepsilon e^{\widetilde \xi h_\varepsilon(P(j) )}  .
\alle

Since $P$ is a simple path, the bound~\eqref{eqn-use-bad-circle-count} shows that the first term on the right in~\eqref{eqn-path-split} is at most $\varepsilon^{  \widetilde\xi \alpha   + \alpha^2/2 - 1   - \zeta }$.
As for the second term, since $\widetilde\xi\leq \xi$, if $h_\varepsilon(P(j) ) \geq \alpha\log\varepsilon$, then $e^{\widetilde \xi h_\varepsilon(P(j) )} \leq \varepsilon^{- (\xi-\widetilde\xi)\alpha } e^{ \xi h_\varepsilon(P(j) )}$.
Plugging these two estimates into~\eqref{eqn-path-split} shows that
\begin{equation} \label{eqn-split-exponents}
L_h^{\widetilde\xi,\varepsilon}(P)
\leq \varepsilon^{  \widetilde\xi  \alpha  + \alpha^2/2 - 1  + o_\varepsilon(1)} +  \varepsilon^{-(\xi-\widetilde\xi) \alpha } L_h^{\xi,\varepsilon}(P)
\leq \varepsilon^{ \widetilde\xi \alpha + \alpha^2/2 - 1  - \zeta} + \varepsilon^{\lambda(\xi) - (\xi-\widetilde\xi) \alpha  -\zeta} .
\end{equation}
We now choose $\alpha >0$ so that the two powers on $\varepsilon$ on the right in~\eqref{eqn-split-exponents} are equal, i.e.,
\begin{equation}
\alpha =   \sqrt{2 + 2 \lambda(\xi) + \xi^2}  -\xi .
\end{equation}
Plugging this into~\eqref{eqn-split-exponents} gives~\eqref{eqn-length-compare}.
\end{proof}

\section{Relating $\xi$ to the central charge}
\label{sec-cc}

The exponent $\lambda(\xi)$ of~\eqref{eqn-lambda-def} gives rise to a notion of ``central charge" for LFPP with exponent $\xi$, as we will now explain.
Following~\cite{shef-kpz}, one can define a \emph{$\gamma$-Liouville quantum gravity (LQG)} surface for $\gamma \in (0,2]$ to be an equivalence class of pairs $(U,h)$ where $U\subset\mathbbm C$ is a simply connected domain and $h$ is a random distribution on $U$ (always taken to be a realization of some variant of the GFF on $U$), with two pairs $(U,h )$ and $(\widetilde U  ,\widetilde h)$ considered to be equivalent if there is a conformal map $\phi : \widetilde U \rightarrow U$ such that
\begin{equation} \label{eqn-lqg-coord}
\widetilde h = h\circ \phi  + Q\log |\phi'| \quad \text{for} \quad Q = \frac{2}{\gamma} +\frac{\gamma}{2} \geq 2 .
\end{equation}
We think of equivalent pairs as different parametrizations of the same surface.
Objects associated with LQG are invariant under coordinate changes of the form~\eqref{eqn-lqg-coord}.
This is proven for the measure in~\cite[Proposition 2.1]{shef-kpz} and the metric in~\cite[Theorem 1.1]{gm-coord-change}.

The parameter $Q$ in~\eqref{eqn-lqg-coord} is called the \emph{background charge}.
It is related to the so-called \emph{central charge} by $\mathbf{c} = 25 - 6Q^2$.
We have $\mathbf{c} \in (-\infty,1]$ for $\gamma \in (0,2]$.
In the physics literature, the parameter $\mathbf{c}$, rather than the parameter $\gamma$, is often viewed as the more natural one.
The above definition of an LQG surface makes sense for any value of $Q  >0$ (not just $Q \geq 2$) and hence for any central charge $\mathbf{c} \in (-\infty, 25)$.
See \cite{ghpr-central-charge} for further discussion of LQG with $\mathbf{c}  \in (1,25)$.

It is not hard to see (see~\cite[Proposition 2.3]{dg-lqg-dim}) that if $D_h^{\xi,\varepsilon}$ has a scaling limit, then at least for complex affine maps $\phi$ the limiting metric must be invariant under coordinate changes of the form~\eqref{eqn-lqg-coord} for $Q = (1-\lambda(\xi))/\xi$.
This leads us to define the background charge and central charge, respectively, for LFPP with parameter $\xi$ by
\begin{equation} \label{eqn-Q-c}
Q(\xi) := (1-\lambda(\xi)) / \xi \quad \text{and} \quad \mathbf{c}(\xi) := 25-6Q(\xi)^2 .
\end{equation}
It is shown in~\cite[Theorem 1.5]{dg-lqg-dim} that for $\gamma\in (0,2)$, one has $Q(\gamma/d_\gamma) = 2/\gamma+\gamma/2$, as expected.
Since $\lambda(\xi) \leq 1$, one always has $Q(\xi) \geq 0$.

For $\xi = 1/\sqrt 3$, the extended Watabiki prediction~\eqref{eqn-watabiki-extend} gives $\lambda(\xi) = 1/3$ and hence $Q(\xi) = \sqrt{4/3}$ and $\mathbf{c}(\xi) = 17$.
Similarly, under~\eqref{eqn-watabiki-extend}, $\xi = \sqrt{5/2} - \sqrt{2/3}$ corresponds to $\mathbf{c}(\xi) = 21.741\dots$.
Combined with Corollary~\ref{cor-watabiki-contradict} and Lemma~\ref{lem-Q-mono} just below, this means that the extended Watabiki prediction~\eqref{eqn-watabiki-extend} is false for a dense subset of central charge values in $(17,25)$ (resp.\ for all $\mathbf{c}  \in ( 21.741\dots , 25)$).

\begin{lem} \label{lem-Q-mono}
The background charge $Q(\xi)$ is strictly decreasing on $\left(0 , 0.7  \right)$, non-increasing on $[0.7,\infty)$, and satisfies $\lim_{\xi\rightarrow\infty} Q(\xi) = 0$.
\end{lem}
\begin{proof
Since $\lambda(\xi) \in [-1/2,1]$ (Lemma~\ref{lem-lambda-basic}), it is obvious that $\liminf_{\xi\rightarrow\infty} Q(\xi) = 0$.

Since $Q(\xi)$ is a Lipschitz continuous function of $\xi$ (Lemma~\ref{lem-lambda-basic}), it is absolutely continuous and so is differentiable Lebesgue-a.e.
Hence to show that $Q(\xi)$ is strictly decreasing on $(0,0.7)$ it suffices to show that its derivative is strictly negative there.
By~\eqref{eqn-dg-mono}, $\lambda'(\xi) \geq -\xi$ and hence
\begin{equation*}
Q'(\xi)  = \frac{1}{\xi^2} \left( -\lambda'(\xi) \xi - 1 + \lambda(\xi) \right) \leq \frac{1}{\xi^2} \left( - \xi^2 - 1 + \lambda(\xi) \right) .
\end{equation*}
Plugging in our upper bound for $\lambda(\xi)$ from Theorem~\ref{thm-lambda-bound} shows that this is negative for $\xi < \sqrt{2 - \frac{1}{6} \sqrt{113 - 8 \sqrt{15}}} \approx 0.70044$.

Finally, we show $Q(\xi)$ is always non-increasing. Let $ 0 \leq \widetilde\xi \leq \xi$.
For $\varepsilon > 0$, let $\widetilde P =  P_h^{\widetilde\xi,\varepsilon}$ be the minimal-$D_h^{\widetilde\xi,\varepsilon}$-length path between the left and right boundaries of $\mathbbm S^\varepsilon$.
Since $x\mapsto x^{\widetilde\xi/\xi}$ is subadditive,
\begin{equation*}
\left[ L_h^{\xi , \varepsilon}(\widetilde P) \right]^{\widetilde\xi/\xi}  =   \varepsilon^{\widetilde\xi/\xi} \left( \sum_{j=0}^N   e^{ \xi h_\varepsilon(\widetilde P(j) )} \right)^{\widetilde\xi/\xi} \leq  \varepsilon^{\widetilde\xi/\xi - 1} L_h^{\widetilde\xi , \varepsilon}(\widetilde P)  .
\end{equation*}
By the definition~\eqref{eqn-lambda-def} of $\lambda$, it holds with probability tending to 1 as $\varepsilon\rightarrow 0$ that
\begin{equation*}
\left[ D_h^{\xi,\varepsilon}\left(\partial_{\operatorname{L}} \mathbbm S , \partial_{\operatorname{R}} \mathbbm S \right) \right]^{\widetilde\xi/\xi} \leq \left[ L_h^{\xi,\varepsilon}(\widetilde P) \right]^{\widetilde\xi/\xi} \leq   \varepsilon^{\widetilde\xi/\xi  +  \lambda(\widetilde\xi)   - 1  +o_\varepsilon(1)}  .
\end{equation*}
Consequently, $\lambda(\xi) \widetilde\xi/\xi \geq \widetilde\xi / \xi + \lambda(\widetilde\xi)   - 1$. Re-arranging gives
$(1 -  \lambda(\xi))/\xi \leq (1-\lambda(\widetilde\xi))/\widetilde \xi  $.
\end{proof}

\begin{remark} \label{remark-square-subdivision}
The paper \cite{ghpr-central-charge} introduces another natural discretization of LQG which works for all $\mathbf{c}  < 25$ (equivalently, $Q > 0$), based on a dyadic tiling $\mathcal S_h^{Q,\varepsilon}$ of the plane consisting of squares which all have ``LQG size $\varepsilon$" with respect to $h$. We expect that this model is related to LFPP as follows: if $ \xi(Q)  > 0$ is chosen so that the graph distance in $\mathcal S_h^{Q,\varepsilon}$ between $\partial_{\operatorname{L}} \mathbbm S$ and $\partial_{\operatorname{R}}\mathbbm S$ grows like $\varepsilon^{-\xi(Q)}$ as $\varepsilon\rightarrow 0$, then $\lambda(\xi(Q)) = 1 - \xi(Q) Q$; i.e., $Q(\xi(Q)) = Q$.
This relation for $Q > 2$ and $\xi \in (0,2/d_2)$ follows from~\cite[Theorem 1.5]{dg-lqg-dim}. We expect that the proof of that theorem could be adapted to treat the case when $Q   < 2$ and $\xi  >2/d_2$ as well.
\end{remark}

\begin{remark} \label{remark-Q-guess}
We expect that the function $\xi\mapsto Q(\xi)$ is injective, at least up until some ``critical point" $\xi_*$ at which it becomes constant.
By Remark~\ref{remark-square-subdivision}, each value of $Q\in (0,2)$ should correspond to some value of $\xi > 0$, so if such a critical point $\xi_*$ exists then $\xi_*$ should be the smallest $\xi   > 0$ for which $\lambda(\xi) = 1$, equivalently $Q(\xi) = 0$.
However, we do not know whether there exists $\xi > 0$ for which $Q(\xi) = 0$, so it could be the case that $\xi\mapsto Q(\xi)$ is strictly decreasing on all of $(0,\infty)$.
Note that both the extended Watabiki prediction~\eqref{eqn-watabiki} and the extended alternative guess~\eqref{eqn-dg-guess} would say that $Q(\xi) = 0$ for some finite $\xi$, but we know that the former is not correct (Corollary~\ref{cor-watabiki-contradict}) and, as noted above, we have no theoretical justification for the latter.
\end{remark}

redpajama_set_name: RedPajamaArXiv


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John Lydon talks wife's Alzheimer's battle and says she shouts 'he's trying to murder me'
Loose WomenSex Pistols star John Lydon has been married to wife Nora Forster since 1979 and he is now a full-time carer for her as her condition has been deteriorating

redpajama_set_name: RedPajamaCommonCrawl


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The revised exam blueprint shows the content that will be covered on each of the new LARE sections.
CURRENT EXAM (through August 2023):
The current LARE is a four (4) part, fully computerized examination designed to test applicants for the minimum knowledge, skills and abilities required to provide landscape architecture services in a manner protective of public health, safety and welfare. The LARE is prepared, administered, and scored by the Council of Landscape Architectural Registration Boards (CLARB) in accordance with current standards for fairness and quality of licensure exams. CLARB is a non‐profit association, and state and provincial registration boards that regulate the profession of landscape architecture are members of CLARB.
CLARB contracts with a testing vendor to have exams administered at independent testing facilities. There are multiple sites in Oregon and also facilities in neighboring states. Candidates select the facility during the exam registration process. CLARB also offers remote proctoring. Candidates facing potential barriers with getting to a testing facility are encouraged to contact CLARB for information about the remote proctoring option.
The LARE currently consists of the following exam sections:
Section 1: Project and Construction Management
covering pre-project management, project management, bidding, construction, and maintenance
Section 2: Inventory and Analysis
covering site inventory, physical analysis, and contextual analysis
Section 3: Design
covering stakeholder processes, master planning, and site design
Section 4: Grading, Drainage and Construction Documentation
covering site preparation plans, general plans and details, specialty plans, and specifications
Each section is independent of the others. Sections can be taken in any order. Sections may be taken one at a time or multiple sections may be taken in one administration. Sections 1 and 2 consist of multiple-choice and multiple-response questions. Sections 3 and 4 consist of advanced item types ("drag and place" and "hot spot" items), multiple-choice and multiple-response questions. Most exam applicants find it best to start with Sections 1 and 2 but may need to obtain practical work experience before finding success with Sections 3 and 4. For more information about the exam structure and question formats, the Board recommends reading the LARE blueprint (for April 2017 - August 2023) and the LARE Orientation Guide available from CLARB.
The LARE content is based on job analysis surveys conducted by CLARB every five to seven years. Survey respondents are asked to rate all job tasks on three separate scales: how frequently the tasks are performed, how important the tasks are to successful job performance, and whether successful performance of each task was required at initial licensure. The tasks performed most often are considered the most important and are required at the initial point of licensure and thus form the basis of the LARE. The current exam is based on a survey administered by CLARB in early 2016 which involved participation by more than 5500 practicing Landscape Architects from the United States and Canada. Survey results are analyzed by subject matter experts working with CLARB and comprised of licensed (registered) Landscape Architects representing the diversity (areas of practice, geographic, etc.) of the profession.
The Board will also accept the equivalent of the LARE from previous versions of the national exam, such as the CLARB "UNE" or earlier CLARB versions of the LARE. The Board works with CLARB to determine equivalency of past vs. current exam forms. The Board does NOT accept any other past or current state or provincial exams in lieu of the LARE or a previous version of the national exam.
The following resources are provided to help applicants in preparation for the LARE.
Taking the LARE
Optimize LARE Performance (Source: CLARB)
Candidates for examination are strongly encouraged to visit the CLARB website to access various resources provided by CLARB to assist with preparation for the LARE. These resources include YouTube videos, the LARE Orientation Guide, a recommended reading list, and related materials.
See also the Board's Exam and Registration FAQs for additional guidance.
707 13th Street SE, Suite 114
Public office hrs 9 AM-3 PM Tues-Thurs. Other days/times by appointment
Christine Valentine, Board Administrator
The Board regulates the practice of landscape architecture in Oregon to help assure the safety, health, and welfare of Oregonians.
Board contact
The Board considers complaints about landscape architecture practice and title use in Oregon.


The Golden Rule teaches us to treat others as we would like to be treated. This tenet of good faith holds true across a diversity of religious traditions.
As clergy and civil rights leaders, we stand united against efforts to codify discrimination into law, and firmly oppose any attempts to give license to discriminate under the guise of protecting religious freedom.
We believe freedom of religion is one of our most fundamental rights as Americans, which is why it is safeguarded by both the U.S. Constitution and Georgia's State Constitution. But religious freedom does not give any of us the right to harm or exclude others.
Our diverse faith traditions call us to love others as God loves us, to treat all living beings with respect and dignity, and to uplift those in need.
That is why we stand in unity with our LGBT brothers and sisters, in support of their equal rights, and in condemnation of their unfair treatment.


Sacrificed on behalf of his queen.
This entry was posted in photo prompt, Photography and tagged #writephoto, Poetry, writing prompt. Bookmark the permalink.
Great poem! I clicked through and read the whole poem.


I know from my Portland relationship counseling clients that few things can damage a marriage as much as an affair. Learning that your partner cheated on you is reason enough for anyone to get marriage help. But it's not just the fact that one of you sought out physical or romantic intimacy from another person, it's also that they engaged in lies and deception to cover it up and maybe even to keep it going.
In short, it's often less about the act and more about the betrayal. Someone put their complete trust in their spouse and allowed themselves to open up and relax, and then had that repaid by having things deliberately kept from them. When my clients ask me what is needed to repair what has been broken, my answer is simple: time, and complete honesty.
Because your problems stem from betrayal and dishonesty, the solution needs to be complete truth. Many people nod their heads when I say this, but a lot of them aren't prepared for what it really means. If your partner has questions about the affair, you have to answer them as honestly as possible. If they ask where you're going and what you're doing, you need to tell them without bending the truth or hesitating. And if they want proof, you should strive to provide it to them – without complaint.
Why? Because you're trying to rebuild trust. It's the foundation upon which everything else is built, and you can only get there with honesty. Without trust, you can't expect your spouse to feel the affection for you that they once did, and you can forget about physical and emotional intimacy. Those things grow out of the feeling that you are safe and at home with another person, and immediately after an affair that just isn't going to happen.
It's not going to happen in a week or even a month, but if you're honest with each other over time – even about the stuff you'd rather not say – this will help to show that you're not just trying to tell your significant other what he or she wants to hear and move you closer to where you used to be. In the long run, you might both find that you can now be more honest with each other because you've passed through the worst thing.
For more advice on how honesty can help you recover after an affair, contact Portland relationship counseling.


AAMCO is putting the care back into car care. Sure, you know AAMCO as your guys for transmissions and we remain the experts in transmission diagnostics, service and repair, but we're also your guy for all your car care needs. Brakes, exhaust, engine trouble, even factory scheduled maintenance and a whole lot more.
Here at AAMCO Lake Charles, we know your transmission is the last thing you have time to worry about. At AAMCO, our transmission repair specialists make sure you leave our centers with peace of mind knowing that your transmission is working perfectly all at the right price.
AAMCO Lake Charles is proud to be one of hundreds of AAMCO centers across North America that have serviced millions of vehicles for more than 50 years and earning AAMCO a reputation as experts in transmission repair as well as Total Car Care providers.
AAMCO has become an American icon, representing trust, quality and value in the eyes of motorists everywhere, we have also become one of the fastest growing names in complete car care. Our AAMCO certified technicians are specialists, who have dedicated years to repairing the most complicated system in your car — your transmission and now give you the same peace of mind servicing and repairing all of your car or truck's major systems. It's expert service at prices a lot less than the dealers charge. And we honor most warranties too!
From brakes, to air conditioning service, we use only the best parts — meeting or exceeding most manufacturers' recommendations. With our 36-point Multi-Point Check, we'll check all your car's systems before any work is done. That means, we'll diagnose what's wrong and fix it right. We know your car's maintenance schedule and can provide "dealer-level" service to your vehicle at less than dealer prices — without the wait for appointments. Whatever your car or truck needs to keep running in peak condition (or get back to running in peak condition), you've got a guy at AAMCO.
Plus many more services from the guys you trust - AAMCO.
* Each AAMCO center is independently owned and operated and not every dealer may offer all services listed.
Click here: for centers & additional information.


One of my favourite things about my own school days was the presence of a 'nature table' in several classes, and this is something that children in my class still enjoy today. It's bright, colourful, changeable and interactive. With so many colours appearing and nature changing daily, Spring is an especially good time for this. One nice idea can be to provide some spotter cards or scavenger hunts for children to follow if they want to, either at playtimes or with a TA perhaps. You could even use such an activity as a stimulus for art or writing perhaps. Here is an example, focusing on some of the commonly seen signs of Spring present in many school grounds. Feel free to use as you wish; I've laminated some and keep them on my nature table.
The 'Working Scientifically' strand underlies the entire new Science Curriculum.
The closest comparison to the familiar outgoing curriculum is the erstwhile SC1 knowledge and understanding' unit. With the new dawn comes a more explicit intention to develop pupil's methodical and investigative thinking skills. For KS1, this involves plenty of chances to observe and experiment, working up to designing investigations, collecting data and reporting back on findings. In a new curriculum that can seem top-heavy on subject knowledge at the expense of wider skills, this is welcome relief. Over the next few weeks, we'll deliver a series of posts on how to integrate these skills, and how to make them seem more engaging for children. Today's post deals with the proposed use of 'science talk' to this end.
The national curriculum for science reflects the importance of spoken language in pupils' development across the whole curriculum – cognitively, socially and linguistically. The quality and variety of language that pupils hear and speak are key factors in developing their scientific vocabulary and articulating scientific concepts clearly and precisely. They must be assisted in making their thinking clear, both to themselves and others, and teachers should ensure that pupils build secure foundations by using discussion to probe and remedy their misconceptions.
Such use of language is clearly of especial revelance to this idea of 'working scientifically', allowing time to share, discuss and compare ideas for scientific investigation. Even though the new curriculum is so full of facts, facts and more facts, it is important to allow a breathing space for children to discover these facts themselves through the use of talk and exploration.
The use of talk partners and 'talk time' has been widely recommended in schools for some time now, and science lessons can provide an excellent forum for this. In my Y3 class, I've taken this one step further with the introduction of specific 'Science Partners'. This is hardly a revolutionary idea; lab partners are widespread further up the key stages, and mixed ability pairs can work very well in any lesson. What having specific science partners at lower levels does provide is an excitement in class, as well as all the established advantages of mixed ability pairs. It also gets the children into a mindset that, with their science partner, they will be required to investigate, talking and working scientifically. I make sure I use these phrases in class, and the children have started to use them too.
Of course, to get the most out of Science talk partners, it is necessary to establish spaces in planning and in lessons for it. Many schools explicitly include 'talk time' and specific questions in their planning, and this is a good idea for Science too. (I have begun specifically marking 'Science Talk' in colour on my Science planning. Future planning available here will show this).
To make talk as productive and focused as possible, talk prompt cards can be helpful. These could include sentence starters ("I predict that…", "I've noticed a difference…", "I'm wondering if…") or could simply be key words for types of talk ("prediction", "observation", etc). Organisations such as Sheffield's ESCAL have produced such resources aimed at guided reading, and adapting and tweaking for Science could be very useful.
Making space for so much talk can lead to lessons becoming a bit elongated when times is of the essence. Cross-curricular lessons can be helpful to kill more birds with one stone. Especially helpful is incorporating Science Talk into Speaking and listening aspects of English/Literacy. let's not forget that NC quote above… science talk "reflects the importance of spoken language in pupils' development across the whole curriculum". The transition from spoken to written language could be a focus for an English lesson, especially if an explanation text of a scientific experiment or process was being written. If patterns or data are being looked at, the science talk could provide a useful link into discussing mathematical relationships.
Why not give science partners a try? I'll be posting some examples of talk prompts and planning incorporating talk time in the next few days.
Ideas of your own? Please drop us a line in the comments box below. Happy Talk times!


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Home News Jammu and Kashmir Jammu Centre constitutes three-member SEIAA for J&K
Centre constitutes three-member SEIAA for J&K
JAMMU: The Central Government on Thursday constituted the State Level Environment Impact Assessment Authority (SEIAA) for Jammu and Kashmir.
"In exercise of the powers conferred by sub-section (3) of Section 3 of the Environment (Protection) Act, 1986 (29 of 1986) and in pursuance of the notification of the Government of India, in the erstwhile Ministry of Environment and Forests, No S.O. 1533(E), dated September 14, 2006, the Central Government hereby constitutes the State Level Environment Impact Assessment Authority, Jammu and Kashmir with three members," reads the notification issued by Ministry of Housing and Urban Affairs, Government of India.
The State Level Environment Impact Assessment Authority will be headed by Lal Chand as its Chairman while Nazir Ahmed will be its Member and Director, Ecology, Environment and Remote Sensing, Jammu and Kashmir will be its Member-Secretary.
"The Chairman and members of the Authority shall hold office for a term of three years from the date of publication of this order in the Official Gazette," the notification stated and added "The Authority shall exercise such powers and follow such procedures as specified in the aforesaid notification."
"The Authority shall base its decision on the recommendations of the State Level Expert Appraisal Committee constituted under paragraph 5 of this order.
"For the purposes of assisting the Authority, Jammu and Kashmir, the Central Government, in consultation with the Government of Jammu and Kashmir, hereby constitutes the State Level Expert Appraisal Committee, Jammu and Kashmir comprising 12 members," the notification further stated.
Subhash Chandra Sharma will be the chairman of State Level Expert Appraisal Committee while M.A Tak, Braj Bhushan Sharma, Abdul Rashid Makroo, Prof Arvind Jasrotia, Dr. Gulam Muhammad Dar, Irfan Yasin, Prof M. A. Khan, Prof Falendra Kumar Sudan, Prof Anil K. Raina and Prof Shakil Ahmad Romshoo will be its Members and Scientist, Incharge Remote Sensing and Geographic Information System Lab will be its secretary.
"The Chairman and members of the State Level Expert Appraisal Committee, Jammu and Kashmir shall hold office for a term of three years and it shall function on the principle of collective responsibility and the Chairman shall endeavour to reach a consensus in each case, and if consensus cannot be reached, the views of the majority shall prevail," reads the notification.
"The Government of Jammu and Kashmir shall notify an agency to act as Secretariat for the Authority and the State Level Expert Appraisal Committee, Jammu and Kashmir and shall provide all financial and logistic support including accommodation, transportation and other such facilities in respect of all their statutory functions," notification stated and added, "The sitting fee, travelling allowance and dearness allowance to the Chairman and members of the Authority and the Chairman and members of the State Level Expert Appraisal Committee, Jammu and Kashmir shall be paid in accordance with the relevant rules of the Government of Jammu and Kashmir."
Three SPs transferred to ACB


Federal prosecutors in Portland, Oregon, obtained a 19-count grand jury indictment in mid-March against Winston Shrout, a Hillsboro, Oregon, resident and one of the most prominent sovereign citizen gurus in the United States, a man whose videos and seminars have attracted thousands of people to the anti-government extremist movement.
Shrout was charged with 13 counts of using fictitious financial instruments in connection with an alleged debt elimination scheme. Fictitious financial instruments are bogus checks, money orders, or similar documents that purport to be payments of money but are not in fact genuine. Since the early 1980s, sovereign citizens have been fascinated with fictitious financial instruments, using them for everything from escaping their own debts to perpetrating major frauds and scams, especially against indebted property owners. Passing them became a federal crime thanks to a law passed after the 1996 Montana Freemen standoff; the Freemen having been energetic promoters of such bogus instruments.
Debt elimination schemes are also extremely common within the sovereign citizen movement; sovereigns use their pseudo-legal language and concepts to convince victims that, for a fee, their mortgages or other debts can simply be made to vanish. Often, fictitious financial instruments and debt elimination schemes go hand in hand.
The federal indictment accuses Shrout of creating and spreading more than 300 bogus "International Bills of Exchange" and "Non-Negotiable Bills of Exchange," instruments with a combined face value of over $100 trillion (but worthless in fact). The indictment claims that Shrout used such instruments himself and also marketed them as a way for others to pay off their debts. Shrout is also charged with 6 counts of willful failure to file income tax returns.
The indictment is a superseding indictment, adding the fictitious instrument charges to the tax charges, which were originally filed against Shrout in December 2015. Since that original indictment, Shrout has declined to use an attorney—a common tactic for sovereigns—and has defended himself using sovereign citizen filings that, among other things, declare his refusal to consent to the jurisdiction of the federal court or to be taxed by the IRS.
Shrout, 67, has been one of the most influential leaders of the sovereign citizen movement in the 21st century. Sovereign citizens believe that, long ago, an evil conspiracy infiltrated and replaced the original "de jure" government with an illegitimate, tyrannical "de facto" government. They claim that the "de facto" government has no authority or jurisdiction over them, which allows them to rationalize ignoring or breaking virtually any tax, law, regulation, or court order. The movement is dominated by a coterie of gurus, the people who come up with the movement's pseudo-legal theories—as well as its often-illegal tactics—and teach them to their followers.
Following this early "retirement," Shrout encountered the sovereign citizen movement at a time when it was enjoying a burst of popularity. This was due to a new compilation of sovereign citizen theories and tactics, often referred to as "redemption" or "straw man theory," which swept through the movement in 1999 like a wildfire and still remain quite popular to this day. Longtime sovereign citizen guru Roger Elvick came up with redemption theory—acting on it would eventually land him in prison—and shared it with a group of disciples who became gurus travelling the country, holding seminars and selling manuals and videos explaining redemption theory and its various associated tactics. Many of those disciples are now themselves in federal or state prison.
Shrout has at times seemed to imply that he learned redemption theory from Roger Elvick, but Shrout's earliest sovereign citizen filings appear to date from 2000, by which time redemption theory was already quite popular in the movement. In August 2000, he filed a notarized document explaining—in sovereign citizen pseudo-legalese—how he had refused to sign or accept a traffic citation from a Washington County, Utah, sheriff's deputy.
Two months later, Shrout filed his first bogus lien—a common harassing tactic that sovereigns use against perceived opponents or enemies. Shrout filed a $1,340,000 lien dubbed an "Affidavit of Obligation" against Unified Industries, Inc., which is a corporation that holds resources and business enterprises associated with the Apostolic United Brethren (AUB), one of the major fundamentalist Mormon polygamist sects in Utah. The lien stemmed from some sort of dispute Shrout had with El Rancho Motoqua, a subsidiary company of Unified Industries established to create a polygamist community in southern Utah. In such communities, properties are often not owned by individual business owners or residents but rather by a holding company or trust run by the sect. Shrout seems to have been permitted a residence in Motoqua and came into conflict with the polygamists running the community. In the bogus lien, Shrout complained that he had been threatened with "removal" from his house and that he was "excluded from participation in the religious ceremonies and usages" held inside a community building.
Was Shrout himself an adherent of the polygamist AUB? It is not entirely clear from the lien document, though at one point Shrout refers to himself "and several thousand other fundamentalist Mormons." However, non-adherents have sometimes resided in AUB communities. Some adherents of polygamists sects have gotten involved with the sovereign citizen movement. Shrout's stepdaughter, April Rampton, was a resident of Motoqua as late as 2012, before she herself was convicted in federal court on nine counts of filing false tax claims while engaged in a common sovereign citizen and tax protest scheme.
Before the end of 2000, Shrout filed a series of redemption-related documents and he soon became a redemption guru, teaching the theory to sovereign audiences. In 2004, Shrout and Patricia Bekken formed an entity called Solutions in Commerce to market sovereign seminars and workshops. Shrout proved to be a popular speaker, with a folksy demeanor that he deliberately played up, referring to it once as "hillbilly shtick." The debt elimination schemes that Shrout promoted were also popular; one admiring extremist in 2004 referred to Shrout as a "top dawg" among such promoters.
Shrout's reputation within the sovereign citizen movement grew in the mid-2000s, but it was the social media revolution that really helped propel him to the top ranks of sovereign gurus, as YouTube videos of some of his appearances and seminars began to circulate widely by 2008-2009, bringing him far greater attention and popularity. This same time period corresponded with the beginnings of the great recession and the mortgage crisis, events that helped spawn a major resurgence of the sovereign citizen movement by creating a large pool of angry and desperate people who were potential recruits.
Shrout held seminars across the country but hardly limited his activities to the United States. In the 1990s, the sovereign citizen movement had taken root in Canada, so Shrout spoke to attentive audiences in that nation—until Canadian authorities finally prohibited him from entering the country. But Shrout found other international audiences, holding seminars in Australia, New Zealand, and Great Britain, helping to bring the sovereign citizen movement to all of those countries—to the dismay of authorities.
More recently, Shrout expanded beyond strictly sovereign notions to express himself on UFOs and other "New Age" issues, claiming that in the early 2000s he and friends of his have been visited by alien "Pleiadians." He's made references to everything from incarnated fairies to "Hollow Earth" theory. Several years ago, Shrout set up a new website, dubbed Exo-Commerce, to provide "insight from a universal perspective," which appears to be an attempt to blend sovereign citizen theories and tactics with "New Age" beliefs. It is difficult to determine whether these are sincerely held beliefs or merely a cynical attempt on the part of Shrout to expand to another gullible audience.
The new federal charges against Shrout could put an end to such opportunism, cynical or sincere. If convicted on all charges, Shrout may face what could be an effective life sentence.


text:
Tara je ženské křestní jméno nejasného významu.

Tara je jednak anglická forma irského místa jménem Teamhair, které možná znamená hora, vrch ve skotské keltštině. Jménu se dostala popularita díky knize Severu proti Jihu.

Tara také znamená hvězda v sanskrtu. Je to jméno hinduistické bohyně a rovněž jméno jednoho z bóddhisattvů.

Známé nositelky jména
 Tara Reid – americká herečka
 Carmen Electra – americká modelka a herečka
 Tara Strong – americká dabérka
 Tara Fitzgerald – britská herečka
 Tara Lipinski – americká krasobruslařka
 Tara Rapoš – dcera české herečky Evy Vejmělkové a slovenského režiséra Dušana Rapoše
 Tara Maclay – postava ze seriálu Buffy, přemožitelka upírů

Literatura
 Miloslava Knappová, Jak se bude vaše dítě jmenovat?. Praha: Academia, 2006.

Externí odkazy

Tara na Behind The Name

Ženská jména
Ženská jména indického původu
Ženská jména irského původu

redpajama_set_name: RedPajamaWikipedia


Groveton is a community located just south of Alexandria along Richmond Highway. It's compromised largely of single-family homes centering around the Beacon Hill Shopping Center. Residents at Groveton enjoy a suburban lifestyle, close to town in an organically grown eclectic neighborhood.
Groveton is a census-designated place in Fairfax County, Virginia. Groveton is at the border of the Fairfax County Districts of Mount Vernon and Lee. The heart of Groveton is found along the northbound Route 1 Corridor commercial area south of the city of Alexandria, including the Beacon Hill Shopping Center and surrounding retail stores all of which run right down to Huntington.
Homes in Groveton have an Alexandria address although all of Groveton is in Fairfax County. Groveton includes many neighborhoods such as Bucknell Manor, Jefferson Manor, Virginia Hills and Stoneybrooke. Huntley Meadows Park, Fairfax County's largest park, is located nearby.
Groveton High School, which served the community since 1959, was renamed in the mid-1980s as West Potomac High School, which also serves Fort Hunt. Groveton Elementary School also serves Groveton.
We can help you achieve your real estate goals in Groveton.
Are you thinking of selling your home in Groveton?
The U.S. Department of Housing and Urban Development (HUD) says that since so many details, especially financial ones, are involved in buying and selling a home, using a real estate agent is a good idea because that professional can be a guide throughout the process, making the entire process and experience so much easier.
The key to a successful and positive experience with a real estate transaction is finding the right professional to assist you. While a real estate agent can meet your needs, the services of a Realtor® will suite your needs even better.
Of the more than two million people who have earned real estate licenses, only a small percentage of them are Realtors. Realtors, who are members in good standing of the National Association of Realtors, their state association of Realtors, and a local real estate board, are held to high standards of ethical behavior and undergo continuing education annually to remain accredited.
perform necessary and customary acts, in good faith and with reasonable care, to assist in the purchase or sale of real estate.
Realtors can assist buyers in many ways. One such way is keeping buyers informed on all the latest listings, to which Realtors gain access as soon as they are listed on the market. Realtors will also be familiar with all the information you may want regarding the area of your potential home, which may include school quality, community safety, traffic volume and commute time, accessibility to shops and public transportation, and proximity of local parks and recreation. When a buyer is ready to purchase a home, Realtors can help you pre-qualify for a mortgage, weigh the pros and cons of the varying types of mortgages, and walk you through the complicated paperwork involved in real estate transactions.
Realtors can also assist sellers with listing and preparing to sell a home, screening potential buyers, and showing investors properties that suit their investing profiles.
Through efficient organization and coordination of meetings with potential buyers and sellers, Nesbitt Realty can quickly locate the perfect home for you or sell your home in the shortest time possible.


Possibly in the collection of the painter Georges Chenard-Huché (1864–1937).
Collection of Alphonse Kann (1870–1948).
Collection of Dikran Khan Kélékian (1868–1951) by 1920.
Purchased by Isabella Stewart Gardner at the American Art Association auction of Dikran Khan Kélékian's collection, New York on 30-31 January 1922 for $3400, lot 104, through American artist Louis Kronberg (1872–1965).
Arséne Alexandre. Collection Kélékian: Tableaux de L'Ecole Française Moderne (Paris, 1920), no. 45.
American Art Association. The Notable Private Collection of Modern Pictures belonging to the widely known antiquarian Dikran Khan Kélékian of Paris and New York (New York, 30-31 January 1922), lot 104, ill.
Catalogue. Fenway Court. (Boston, 1924 or 1925), p. 1.
Philip Hendy. Catalogue of Exhibited Paintings and Drawings (Boston, 1931), pp. 225-26.
Gilbert Wendel Longstreet and Morris Carter. General Catalogue (Boston, 1935), p. 25.
Philip Hendy. European and American Paintings in the Isabella Stewart Gardner Museum (Boston, 1974), pp. 152-53.
Denis Rouart and Daniel Wildenstein. Edouard Manet. Catalogue raisonné (Lausanne-Paris, 1975), pp. 254-55, no. 328.


Additional Description We will not actually be meeting. But all board members should still send in a report. This report should include a summary of your goals and which ones you met. Also, please include any advice or recommendations for next year's board.


text:
Coplay: Christmas tree lighting set for Dec. 1
Published November 30. 2022 07:35AM
The Coplay Borough Recreation and Welfare Association invites all to the tree lighting and holiday concert, set for 6:30 p.m. Dec. 1 at borough hall, 98 S. Fourth St.
Santa will be there, with Lorri Woodward and Rob Ballonoff providing entertainment.
Light refreshments will be served, and kids receive a small gift.
Coplay Sports is holding a Toys for Tots donation drive. Donations will be accepted 6-8 p.m. Dec. 9 and 10 a.m.-2 p.m. Dec. 10 at the Wildcat Den, 666 Potter Terrace.
See more details on the Coplay Sports Facebook page.
The Michael Ann and Saquon Barkley Hope Foundation designer purse bingo is planned for Dec. 9 at American Club of Coplay, 300 Cherry St.
Doors open 5 p.m., and bingo starts 6 p.m.
There will be food and beverages for sale. All proceeds benefit the foundation.
For more information, call Alberta at 610-295-3600.
The Coplay Sports Santa parade is scheduled for Dec. 11, starting 10:30 a.m. on Front Street.
Gift drop-off is 6-8 p.m. Dec. 9 and 10 a.m.-2 p.m. Dec. 10 at the Wildcat Den, 666 Potter Terrace.
Be sure to clearly mark your gift with a name, address and phone number.
Coplay Food Bank is seeking donations for the holiday season. Much-needed items are canned sweet potatoes, canned fruit, Jell-O mixes, cereal, peanut butter, jelly and canned meats like SpaghettiOs, ravioli, etc.
Drop off items at St. John's Lutheran Church, 18 S. Third St.
The food bank, located in the lower level of the municipal building, is open 8:30-10 a.m. the second Tuesday of every month.
If unable to attend the scheduled Tuesday morning time frame, call Jodi at 610-262-0928 to make other arrangements.
Coplay Saengerbund, 205 S. Fifth St., is planning its New Year's Eve celebration called the Sylvester dance.
More information is to come.
St. John's Lutheran Church is planning a chicken potpie pickup dinner Jan. 21, 2023.
The church is assembling a yearlong celebration for 2023 as the church celebrates its 150th-year anniversary.
Coplay Public Library is seeking a library director to work 32 hours a week, as well as an assistant library director.
Send a resume and three references to Janet Eisenhauer, 98 S. Fourth St.
American Club of Coplay has restarted its monthly hoagie sales. Hoagies (regular or tuna) and sandwiches (ham and cheese or tuna) are available.
Call 610-261-2884 for the cost and to order, or order online via email to americanclub@rcn.com.
Looking ahead, the club is planning a steak and cash raffle event Feb. 5, 2023.
Leaves will continue to be picked up through Dec. 30. Yard waste and grass can be dropped off 7:30 a.m.-3 p.m. at 1 Bridge St.
If you have a street department question, call Paul at 610-262-8378.
The police department's office number is 610-262-2288.
The nonemergency number is 610-437-5252. This number connects you with Lehigh County Communication Center.
Recycling electronics can be dropped off at DCR Technologies, 205 Main St., Northampton, by appointment only. Visit drc-tech.net for a list of items accepted and any related costs.
Call 610-502-0854 to arrange a drop-off.
Coplay Athletic Club is seeking 2023 new and renewal memberships - $90 per year and $80 for renewals. The gym is located on the second floor of the municipal building.
Email coplayac@gmail.com for information.
Coplay American Legion Post 426 has a special mailbox in front of the Post, 134 S. Second St. Worn-out American flags can be deposited there for the Post to properly retire the flags.
To join the Legion, call 610-262-7556.
To have an announcement or to share something in this column, email billleiner@aol.com or text 610-751-2745.


One of the most powerful men in late antiquity, Attila's peerless Hunnic empire stretched from the Ural mountains to the Rhine river. In a series of epic campaigns dating from the AD 430s until his death in AD 453, he ravaged first the Eastern and later the Western Roman Empire, invading Italy in AD 452 and threatening Rome itself. Lavishly illustrated, this new analysis of his military achievements examines how Attila was able to sweep across Europe, the tactics and innovations he employed and the major battles he faced, including one of his few major setbacks, the defeat at the battle of the Catalaunian Fields in AD 451.
Dr Nic Fields started his career as a biochemist before joining the Royal Marines. Having left the military, he went back to University and completed a BA and PhD in Ancient History at the University of Newcastle. He was Assistant Director at the British School at Athens, Greece, and then a lecturer in Ancient History at the University of Edinburgh.Steve Noon was born in Kent, UK, and attended art college in Cornwall. He's had a life-long passion for illustration, and since 1985 has worked as a professional artist. He has provided award-winning illustrations for the publishers Dorling Kindersley, where his interest in historical illustration began. Steve has illustrated over 30 books for Osprey.


There are many great courses in the Costa del Sol region but there is no doubt that La Brisas is one of the best and also one of the oldest. Designed by Trent Jones Snr in 1968 the course includes numerous water obstacles, tricky raised greens and strategically placed bunkers. Perfectly maintained, Las Brisas oozes quality from beginning to end.


Electronic discovery plays a central role in litigation where parties claim violations of trade secrets and breaches of noncompete agreements. Electronic discovery and forensic investigations often reveals extremely damaging evidence against the former employee, including acts such as downloading or e-mailing valuable company information.
The sheer number of emails that must be collected and reviewed can be an overwhelming task. VeriSign (the exclusive registry for .com and .net domains) estimates that there are approximately 2.25 billion email queries per day, and that is only a fraction of the number of emails sent each day.
When litigation commences, a question may arise as to whose responsibility it is to ensure all emails and other forms of electronic documents have been preserved and collected. Delaware's Court of Chancery has, not surprisingly, had several occasions to set forth how the Court intends to have litigants handle electronic discovery.


This lovely collar necklace is hand fabricated Zuni work. It is set with 16 hand cut natural, turquoise cabochons that are carefully set in fine hand crafted bezels. This necklace dated to the 1970's, and has a wonderful design. Note the tiny, tiny stamped silver drops between each stone. Beautifully crafted and lovely on the neck. Hallmarked VC.
The total necklace length measures 19" long, which includes a hand made link chain that allows for your desired length.


I was playing solitaire ("regular" single-deck solitaire) on my phone the other day and got dealt an unusual hand. All of a sudden I noticed a very weird combination of cards. Then another. Then another. Do you see it?
Click the picture to see a larger one.
I just had my favorite conversation ever with my little girl (2yo).
Makes me wish there was a DVR for real life.


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'The only superstar is Jesus': Kanye West speaks at Joel Osteen's Lakewood Church
KTRK
HOUSTON, Texas -- In one of his most spiritual collaborations yet, rap superstar Kanye West took the stage with Joel Osteen at the popular preacher's megachurch in Houston Sunday.
The morning service was one of two opportunities to see West at Lakewood Church. Though entry is free for the evening event, some scalpers who scooped up tickets were selling them for $500 each.
A church spokesperson said the 42-year-old Grammy award winner attended the 11 a.m. CST service for a conversation with Osteen on West's faith journey and overcoming adversity.
"God has been calling me for a long time," West said. "The devil has been distracting me for a long time."
West spoke about how he feels many producers, musicians, artists, designers, and business people aren't working towards their faith.
"The greatest artist that God ever created is now working for Him," said West, referring to himself. "I'm here in service to God. No weapon formed against me shall prosper."
On a more humble note, West said, "The only superstar is Jesus."
Kanye West has taken the stage at Lakewood with Pastor Joel Osteen. "The greatest artist God has ever created is working for Him." https://t.co/lFTcM9TjnZ pic.twitter.com/QcPxjDM5Px
— Mycah Hatfield (@MycahABC13) November 17, 2019
Later in the evening, West is scheduled to hold his own "Sunday Service" at Lakewood. The free tickets, which became available on Saturday, were gone in less than 15 minutes. For those who were able to score tickets, doors open at 5:30 p.m., and the show is set to begin at 7 p.m.
"Sunday Service" started as a small gathering among West, wife Kim Kardashian, their family, and his choir, but has now grown into a cultural phenomenon. Sunday will be West's largest, where he's expected to reach over seven million viewers.
WATCH: Fans line up ahead of Kanye West Sunday Service at Lakewood Church
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Fans lining up to see Kanye West at Lakewood Church
Those who couldn't attend in person can watch the service on the church's Facebook and YouTube pages, website, app and on SiriusXM's Joel Osteen Radio channel 128 as well as the SiriusXM app.
The visit stems from a personal invite from Osteen on Oct. 25. A spokesperson said Osteen and West "do speak from time to time," and according to an article published by TMZ, Osteen is a fan of West, and believes his life is rooted in faith and love.
West began his Houston visit at a local jail, performing for more than 200 inmates, who were clearly moved by the mini concert.
In October, West released his ninth studio album "Jesus Is King," and the gospel-inspired album debuted at number 1 at the top of the Billboard 200.
I talked to a man outside of Lakewood this morning who has all of Kanye West's album artwork tattooed on his arm. He said Kanye is an inspiration to him because he's constantly breaking down barriers and proving that he can do what people says he can't. https://t.co/HtgDlr110O pic.twitter.com/9FbQnbgURg
RELATED: Fans resell Kanye West' Sunday Service tickets for up to $500
societyhoustonkanye westreligionchurchmusic newslakewood church
Copyright © 2020 KTRK-TV. All Rights Reserved.


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Fixed retirement age should be abolished, says report
By Marie Singer Published Feb 10, 2014 at 13:51 PM GMT
The fixed retirement age or default retirement age (DRA) should be abolished globally, say researchers from the Universities of Hull, East Anglia and Southampton in England.
They add that current pension systems worldwide are unsustainable.
The study, published in the peer reviewed journal Personnel Review, is titled "Retirement in a global labour market: A call for abolishing the fixed retirement age."
Professor Yehuda Baruch from the Southampton Management School, said:
"We have a global problem with funding pensions, which assume people will retire around their mid-60s. Young people are tending to start work later and in-turn start paying tax and pension contributions later. Older people are living longer, often in to their 90s, creating, in some cases, up to 30 years of retirement to provide for with their cover. This creates a funding gap."
"Our study advocates the abolition of the default retirement age worldwide by governments and private companies – allowing people to consider a range of options on how and when to stop work, as well as greater flexibility over their pension plans. This would help ease funding problems."
UK has no fixed retirement age
The fixed retirement age was abolished in the UK in 2011, although a number of organizations can set their own DRA if they can justify why it is necessary.
Several nations around the world have a fixed retirement age. In order to examine the benefits and disadvantages of having a more flexible approach to retirement, the researchers used the UK as a case study.
Professor Baruch, together with Dr Susan Sayce from the University of East Anglia, and Professor Andros Gregoriou from the University of Hull, carried out a financial analysis based on Monte Carlo simulation methodology to interpret pensions and other financial data.
A financial analysis assesses an entity's viability, solvency, profitability, and stability.
Current pensions systems are unstustainable
They found that in countries where a fixed retirement age is in place, the pensions system over the long-term is unsustainable. They recommend ending its use, in line with what the UK has done.
Baruch said:
"We would like to see a situation where people globally can be much more in control of when they retire. For example, they may want to work into their 70s or even 80s, but perhaps want to change the type or volume of work they take on."
"Our study suggests that old age can be seen as holding the prospect of long-term stable contributions to society, rather than a decline or preparation for giving up work altogether – which can lead to pressure on the public purse."
In order to abolish the default retirement age effectively worldwide, the researchers say there needs to be a change in cultural attitudes towards older employees and the perception of their productivity and value.
The labor market needs to help older workers find employment, as well as helping them out of jobs.
Although the policy of abolition is the best route, the investigators add that the effects of getting rid of a fixed retirement age will vary from-nation-to-nation.


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This is a listing of past and present correctional facilities run by the provincial government in Ontario, Canada. Provincial correctional facilities for adults are operated by the province's Ministry of the Solicitor General. Youth facilities have at various times been under the same jurisdiction, but currently fall under the Ministry of Children, Community and Social Services.

Types of facilities
Adult correctional facilities in Ontario are divided into four categories: correctional centres, jails, detention centres, and treatment centres. Some facilities are more than one type. Correctional centres house sentenced offenders who are serving a period of incarceration of up to two years, less a day. Provincial jails (historically spelled gaols) and detention centres house persons awaiting trial, offenders serving short sentences, or offenders awaiting transfer to other facilities. Jails are smaller and older facilities originally established by local governments while detention centres are larger, regional facilities.  Treatment centres are specialized facilities treating offenders for sexual misconduct, substance abuse, anger management, and other issues.

Youth correctional facilities in Ontario are also called "secure custody facilities" and hold young people who were between 12 to 17 years of age at the time of offence. Youths are held in secure custody facilities if they are sentenced to secure custody after being found guilty of a crime or if a youth is ordered to be held in custody before or during a trial. , the Ministry of Children, Community and Social Services directly operates six secure custody facilities.

Operational correctional facilities

Adult

Youth

Past jails and correctional facilities

See also
 :Category:Correctional Service of Canada institutions

Notes

References

 Correctional Services - Ontario Ministry of the Solicitor General
 Ontario Archives

Canada, Ontario
Correctional facilities


text:
Q: Count empty or not empty TD with jquery Usually input and textarea can be checkt and counted by its selector and length.
e.g. Counting empty inputs and textareas:
$('input[value=""]').length;

$('textarea[value=""]').length;

How this can be done with a Table ?

        aaa

        aaa

The count should be 2. I know I could use a $.each(...) but I would prefere a simple $(selector).length solution.
EDIT: corrected the topic a bit to get this small confusion away, sorry! :D

A: You can make use of :empty pseudo selector to check whether a DOM element contains data or not.
$("td:empty").length

This will give you empty td count
If you want to count the non empty td elements then:
$("td:not(:empty)").length

A: try this one
 $('td:not(empty)')

A: You can use the :empty selector:

console.log($('table td:empty').length);

    aaa

    aaa

As @Pointy mentions, you can invert the behaviour using :not() if you want to select td elements that do have some content:
$('table td:not(:empty)').length

redpajama_set_name: RedPajamaStackExchange