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\mysection{Introduction}
In this supplemental material  we derive the scaling of a typical matrix element {$V = \langle \alpha | O | \beta \rangle $} of the operator  $O = {\bf s}_1\cdot {\bf s}_L$ between  two tree-like states $| \alpha \rangle, |\beta\rangle$  presented in the main text. Figure~\ref{Fig:States} depicts the scenario considered, and we elaborate on the notation used below.

\mysubsection{Notations}

 We consider two states of a length $L=2^{{\cal {\cal K}}-1}$ spin chain given by two trees of depth ${\cal {\cal K}}$, and will henceforth use the term `state' and `tree' interchangeably.
We denote by $S_i^{\mathcal L}$ and $S_i^{\mathcal R}$, $i=1, \ldots {\cal {\cal K}}-1, $ the spins in the left-most and right-most branches of one tree, respectively.  We count the level $i$ of a tree from the top and in our notations $S_{{\cal K}-1}^{\mathcal L }\equiv s_1=\frac12$. We also introduce the notations $\tilde{S}_i^{\mathcal L}$ and $\tilde{S}_i^{\mathcal R}$ for the corresponding spins in the left-most and right-most branches of the second tree.  By the selection rules  (see Sec. \ref{SubSec:Symmetry}) $\tilde{S}^\eta_i=S_i^\eta\,, S_i^\eta\pm 1$ and the differences $\Delta S^\eta_i\equiv S^\eta_i-\tilde{S}^\eta_i$ can only take values $-1$, $0$ and $1$.

 We denote the spin at the top of both trees by $S_0$ (this has to be the same in order for there to be a non-trivial matrix element). In most of this supplemental material we set $S_0=0$, implying that
$S_1^{\mathcal R}=S_1^{\mathcal L}$ and  $\tilde{S}_1^{\mathcal R}=\tilde{S}_1^{\mathcal L}$. This assumption is relaxed in Sec.~\ref{Sec:S0}.
Finally, we denote by $J^{\mathcal L}_i$ and  $J^{\mathcal R}_i$, $i=2\ldots {\cal K}-1$, the spins in the branches next to that of left-most and right-most branches
. They are shared by both states under consideration in order for the matrix element to be potentially non-zero.  Note that the notations of the present Supplemental material are slightly different from that of the main text. Specifically, in the notations of the manuscript, $S^{\mathcal R }_i$ corresponds to $S^{({\cal K}-i)}_1$ while  $S^{\mathcal L }_i$ to  $S^{({\mathcal K}-i)}_{2^i}$.

\mysubsection{Structure and summary of supplemental material}

This supplemental material is organzied as follows: in Sec.~\ref{Sec:TwoStates} we derive recursion relations that allow us to compute the matrix element for any given pair of states; in Sec.~\ref{Sec:StatisticsOverBaseState}  we convert those into recursion relations for the distribution function of the matrix elements connecting a given base state to other states allowed by the selection rules; in Sec. \ref{Sec:Statistics} we analyze the statistics of the random quantities $\overline{\zeta}$ and $\sigma$ in the large-$L$ limit within the sector of the theory with $S_0=0$ and show that the typical matrix element possesses the scaling
\begin{equation}
V_{\rm typ.}\propto L^{-\beta}\,, \qquad \beta=\frac{17+4\ln 2}{9\ln 2}.
\end{equation}
Finally, in Sec. \ref{Sec:S0} we demonstrate that our results for the asymptotic scaling of the matrix elements hold for arbitrary values of $S_0$.

\begin{figure}
\includegraphics[width=500pt]{matrixElements_sup.pdf}
\caption{Two states of the system connected by an operator ${\bf s}_1\cdot{\bf s}_L $ }
\label{Fig:States}
\end{figure}

\mysection{Matrix element for a given pair of states}
 \label{Sec:TwoStates}
\mysubsection{Preliminary symmetry considerations}
\label{SubSec:Symmetry}
Let us first consider a system of spins $(1,2,\cdots,L-1,L)$ interacting via rotationally invariant interactions, that is, we consider a system with $SU(2)$ symmetry, and derive certain constraints on matrix elements of eigenstates due solely to symmetry considerations.  This scenario encompasses the tree-like states that we have considered, but is more general than that.

Due to the $SU(2)$ symmetry, the eigenstates of the system form multiplets
\begin{equation}
 |S, M\rangle,\qquad M=-S,\ldots S
   \end{equation}
labeled by the spin $S$.  Next we consider two such multiplets and construct a matrix
\begin{equation}
g^\alpha_{\tilde{M}, M}(\tilde{S}, S)=\langle \tilde{S},\tilde{M}| s_1^\alpha |S, M\rangle,
\end{equation}
where $s_1^\alpha$ is the $\alpha$-spin operator of the first site.

Importantly,  the form of $g^\alpha_{\tilde{M}, M}$ is fully fixed by symmetry.
Let us show first that  $g^\alpha_{\tilde{M}, M}(\tilde{S}, S)$  vanishes whenever $|S-\tilde{S}|>1$ or $S=\tilde{S}=0$.
Indeed, the matrix element $g^\alpha_{\tilde{M}, M}$  can be considered as a linear function on the triple $(~|\tilde{S}, \tilde{M}\rangle\,, |S, M\rangle\,, s_1^\alpha ) $ which transforms under the $\tilde{S} \otimes S \otimes(s=1)$ representation of the $SU(2)$ group. However since the matrix element $g^\alpha_{\tilde{M}, M}$  is invariant under simultaneous rotations of all the three components of the triplet, i.e.~it transforms trivially under the $SU(2)$ group, the only way for this matrix element to be potentially non-trivial is such that the tensor product of the different representations $\tilde{S} \otimes S \otimes (s=1)$ contains the trivial representation $0$ as a direct summand. Exploiting the rules for angular momentum addition, we see that this can happen only for the cases $|S-\tilde{S}|\leq1$ and $\min(S, \tilde{S})>0$.

We now proceed to determine the form of the matrix $g^\alpha_{\tilde{M}, M}$.
 For the case $S=\tilde{S}$ obvious symmetry arguments imply that
\begin{equation}
g^\alpha_{\tilde{M}, M}(S, S)\propto S^\alpha_{\tilde{M}, M}
\end{equation}
where $S^\alpha_{\tilde{M}, M}$ are the spin operators in the spin-S representation.
Introducing $g^\pm=g^x\pm i g^y$ we write explicitly
\begin{equation}
g^+_{\tilde{M}, M}=\frac{\xi}{\sqrt{\gamma(S, 0)}} \delta_{\tilde{M}, M+1}S^+_{M}\,, \qquad g^+_{\tilde{M}, M}=\frac{\xi}{\sqrt{\gamma(S, 0)}} \delta_{\tilde{M}, M-1}S^-_{M}, \qquad g^z_{\tilde{M}, M}=\frac{\xi}{\sqrt{\gamma(S, 0)}} \delta_{\tilde{M}, M}M.
\label{Eq:gss}
\end{equation}
Here
\begin{equation}
 S^-_{M}=\sqrt{(S+M)(S-M+1)}\,, \quad   S^+_{M}=\sqrt{(S-M)(S+M+1)},
\end{equation}
and we have also introduced a convenient  normalization factor
\begin{equation}
\gamma(S, S-\tilde{S}=0)=2 S (S+1).
\end{equation}
$\xi$ is a numerical coefficient that depends on the precise internal structure of the states in the multiplets and is where the physics lies (i.e.~details not able to be determined by only symmetry considerations). Different structures give different numerical values; our aim in the supplemental material, which is to understand the scaling of the typical matrix element $V$, thus amounts to understanding $\xi$ for the tree-like states we are considering.

The case of  $|S-\tilde{S}|=1$ is slightly less obvious. However one  can employ   the  Wigner-Eckart theorem \cite{Wigner-Eckart} to find (we assume $S=\tilde{S}+1$; the case   $S=\tilde{S}-1$ is related to the present one by the Hermitian conjugation and the interchange of $S$ and $\tilde{S}$)
\begin{equation}
g^+_{\tilde{M}, M}=\frac{\xi}{\sqrt{\gamma(S, 1)}}\delta_{\tilde{M}, M+1}u^+_{M}\,, \qquad g^-_{\tilde{M}, M}=\frac{\xi}{\sqrt{\gamma(S, 1)}}\delta_{\tilde{M}, M-1}u^-_{M}, \qquad g^z_{\tilde{M}, M}=\frac{\xi}{\sqrt{\gamma(S, 1)}}\delta_{\tilde{M}, M}u^z_{M}.
\label{Eq:gsps}
\end{equation}
Here
\begin{equation}
u^z_{M}=\sqrt{S^2-M^2}, \qquad u^+_{M}=\sqrt{(S-M)(S-M-1)}\,, \qquad u^-_{M}=
-\sqrt{(S+M)(S+M-1)},
\end{equation}
and
\begin{equation}
\gamma(S, S-\tilde{S}=1)=2 S \sqrt{4 S^2-1}.
\end{equation}

\mysubsection{Recursion relations}
\mysubsubsection{Top level}
The symmetry considerations of the previous section allow one to establish recursion relations for the determination of the matrix element.
Let  us consider first the top level of our two trees. We have for the matrix element
\begin{multline}
V=\langle S_0, S_0| s^\alpha_1 s^\alpha_L| S_0, S_0 \rangle\\=
\sum_{\substack{m+M=S_0\\\tilde{m}+\tilde{M}=S_0}} C^*(\tilde{S}_1^{\mathcal  L}, \tilde{m}, \tilde{S}_1^{\mathcal  R}, \tilde{M}, S_0, S_0) C(S_1^{\mathcal L}, m, S_1^{\mathcal R}, M, S_0, S_0)\langle \tilde{S}_1^{\mathcal L}, \tilde{m} |s_1^\alpha
|S^{\mathcal L}_1, m\rangle  \langle \tilde{S}_1^{\mathcal R}, \tilde{M} |s_L^\alpha
|S^{\mathcal R}_1, M\rangle.
\label{Eq:Vbasic}
\end{multline}
Here $C(s_1, m_1, s_2, m_2, S, M)$ are the Clebsch-Gordan coefficients.  We now use the decomposition of the matrix elements given in the previous section and write
\begin{eqnarray}
\langle \tilde{S}_1^{\mathcal L}, \tilde{m} |s_1^\alpha
|S^{\mathcal L}_1, m\rangle=\xi_1^{\mathcal L}\overline{g}^\alpha_{\tilde{m}, m}(\tilde{S}_1^{\mathcal L}, S_1^{\mathcal L} ),\\
\langle \tilde{S}_1^{\mathcal R}, \tilde{m} |s_1^\alpha
|S^{\mathcal R}_1, m\rangle=\xi_1^{\mathcal R}\overline{g}^\alpha_{\tilde{m}, m}(\tilde{S}_1^{\mathcal R}, S_1^{\mathcal R} ),
\end{eqnarray}
to find
\begin{equation}
V=\xi_1^{\mathcal L}\xi_1^{\mathcal R}
\sum_{\substack{m+M=S_0\\\tilde{m}+\tilde{M}=S_0}} C^*(\tilde{S}_1^{\mathcal  L}, \tilde{m}, \tilde{S}_1^{\mathcal  R}, \tilde{M}, S_0, S_0) C(S_1^{\mathcal L}, m, S_1^{\mathcal R}, M, S_0, S_0)\overline{g}^\alpha_{\tilde{m}, m}(\tilde{S}_1^{\mathcal L}, S_1^{\mathcal L} )\overline{g}^\alpha_{\tilde{m}, m}(\tilde{S}_1^{\mathcal R}, S_1^{\mathcal R} ),
\label{Eq:VBasic1}
\end{equation}
where we have defined $\overline{g}$ to be a `normalized' g, i.e.~$\overline{g}$ is given by Eqs. (\ref{Eq:gss}) and (\ref{Eq:gsps}) with $\xi=1$.

Using the explicit expressions  for $\overline{g}$
together with the explicit expressions for the  Clebsch-Gordan coefficients, we can perform the summation in Eq.~(\ref{Eq:VBasic1}) and reduce the evaluation of the matrix element to the evaluation of the two factors $ \xi_1^{\eta}$, $\eta = \mathcal{L}, \mathcal{R}$. After long, tedious algebra, we find
\begin{equation}
V=\frac{\xi_1^{\mathcal L}\xi_1^{\mathcal R}}{\sqrt{\gamma(S^{\mathcal L}, \Delta S^{\mathcal L}) \gamma(S^{\mathcal R}, \Delta S^{\mathcal R})}} R_{\Delta S^{\mathcal L}_1, \Delta S^{\mathcal R}_1}(S_1^{\mathcal L}, S_1^{\mathcal R}, S_0).
\label{Eq:VGen}
\end{equation}
We explain the notation of the previous line. For a tree with some $(S_1^{\mathcal L}, S_1^{\mathcal R}, S_0)$, there are $3 \times 3$ possible trees it can connect to, which are given by $\Delta S^\eta_1=S^\eta_1-\tilde{S}^{\eta}_1 = -1, 0, +1$, for $\eta = \mathcal{L}, \mathcal{R}$. Thus, we can compactly store this information in a $3 \times 3$ matrix $R_{\Delta S^{\mathcal L}_1, \Delta S^{\mathcal R}_1}$:
\begin{multline}
 R_{\Delta S^{\mathcal L}_1, \Delta S^{\mathcal R}_1}(S_1^{\mathcal L}, S_1^{\mathcal R}, S_0)  =\\
\frac12\left(
\begin{array}{ccc}
-\sqrt{(j_3+1)(j_3+2)(\Sigma+2)(\Sigma+3)}& -\sqrt{(j_1+1)j_2(j_3+1)(\Sigma+2)} &
\sqrt{(j_1+1)(j_1+2)(j_2-1)j_2}\\
\sqrt{j_1(j_2+1)(j_3+1)(\Sigma+2)} & \left[j_1 j_2 -(2+j_1+j_2)j_3 -j_3^2\right]/2 & \sqrt{(j_1+1)j_2 j_3(\Sigma+1)}\\
\sqrt{(j_1-1)j_1(j_2+1)(j_2+2)} &-\sqrt{j_1(j_2+1)j_3(\Sigma+1)}&-\sqrt{(j_3-1)j_3\Sigma(\Sigma+1)}
\end{array}
\right)
\end{multline}
with
\begin{equation}
j_1=S_0+S^{\mathcal L}_1-S^{\mathcal R}_1, \qquad j_2=S_0-S^{\mathcal L}_1+S^{\mathcal R}_1\,, \qquad  j_3=-S_0+S^{\mathcal L}_1+S^{\mathcal R}_1\,, \qquad \Sigma=j_1+j_2+j_3.
\end{equation}

Equation (\ref{Eq:VGen}) takes an especially simple form for the case $S_0 = 0 $ where $S^{\mathcal L}= S^{\mathcal R}$ and   $\Delta S^{\mathcal L}= \Delta S^{\mathcal R}$. In this limit (which we are going to focus on in the following) we find
\begin{equation}
V=-\frac{1}{2}\xi_1^{\mathcal L}\xi_1^{\mathcal R}.
\label{Eq:VS0}
\end{equation}

\mysubsubsection{The factor $\xi$}
In the previous section, we reduced the problem to the evaluation of the two factors $\xi_1^{\mathcal L}$ and $\xi_1^{\mathcal R}$ coming from the left-most and right-most branches of a tree. They can be computed independently and in this section we focus on $\xi_1^{\mathcal L}$ omitting the superscript ${\mathcal L}$.

To establish the recursion relations let us consider the first level of the tree (just below the top node). We have
\begin{multline}
\xi_1 \overline{g}^{\alpha}_{\tilde{M}, M}(\tilde{S}_1,S_1)\equiv\langle \tilde{S}_1,\tilde{M}| s_1^\alpha | S, M\rangle=
\sum_{\substack{
m+n=M \\
\tilde{m}+\tilde{n}=\tilde{M}}} C^*(\tilde{S}_2, \tilde{m}, J_2, \tilde{n}, \tilde{S}_1, \tilde{M})C(S_2, m, J_1, n, S_1, M)
\langle \tilde{S}_2, \tilde{m}|s_1^\alpha|S_2, m\rangle \delta_{n, \tilde{n}}\\=
\xi_2
\sum_{\substack{
m+n=M \\
\tilde{m}+\tilde{n}=\tilde{M}}} C^*(\tilde{S}_2, \tilde{m}, J_2, \tilde{n}, \tilde{S}_1, \tilde{M})C(S_2, m, J_1, n, S_1, M)
\overline{g}^\alpha_{\tilde{m}, m}(\tilde{S}_2, S_2)\delta_{n, \tilde{n}}.
\end{multline}

Now, knowledge of $\overline{g}$ allows us to perform the summation explicitly and find the recursion relation for the factor $\xi$ (while we are currently working on the first level of the tree; it is obvious that the same procedure can be pushed further down the tree)
\begin{equation}
\xi_{k-1}=\frac{\xi_k}{\sqrt{\gamma(S_{k-1}, \Delta_{S_{k-1}}) \gamma(S_{k}, \Delta {S_{k}}})}f_{\Delta S_{k-1}, \Delta S_k}(S_{k-1}, S_k, J_k).
\label{Eq:xiExact}
\end{equation}
Here
\begin{equation}
\gamma(S, \Delta S)=\left\{ \begin{array}{cc}
2(S+1)\sqrt{(2S+1)(2S+3)}\,,  \quad& \Delta S=-1\\
2S(S+1)\,, \qquad& \Delta S=0 \\
2S\sqrt{4S^2-1}\,, \quad& \Delta S=1\\
\end{array}\right.
\label{Eq:gammaFinal}
\end{equation}
and we have stored the information in a $3 \times 3$ matrix $f_{\Delta S_1, \Delta S_2}$:
\begin{multline}
f_{\Delta S_1, \Delta S_2}(S_1, S_2, J_2) =\\
\left(
\begin{array}{ccc}
\sqrt{(j_1+2)(j_1+1) (\Sigma+2)(\Sigma+3)}& \sqrt{(j_1+1) (j_2+1) j_3(\Sigma+2)} &
-\sqrt{(j_2+1)(j_2+2)(j_3-1)j_3}\\
-\sqrt{(j_1+1)j_2(j_3+1)(\Sigma+2)} &  \left[j_1(\Sigma+2)-j_2  j_3\right]/2 &  -\sqrt{j_1(j_2+1)j_3(\Sigma+1)}\\
-\sqrt{j_2(j_2-1)(j_3+1)(j_3+2)} & \sqrt{j_1 j_2 (1+j_3)(1+\Sigma )}& \sqrt{j_1(j_1-1)\Sigma (\Sigma+1)}
\end{array}
\right)
\label{Eq:fExact}
\end{multline}
with
\begin{equation}
j_1=S_1+S_2-J_2\,, \qquad j_2=S_1-S_2+J_2\,, \qquad j_3=-S_1+S_2+J_2\,, \qquad
\Sigma=j_1+j_2+j_3.
\label{Eq:j}
\end{equation}

Equations (\ref{Eq:xiExact}) and (\ref{Eq:VS0}) together with Eqs. (\ref{Eq:gammaFinal}), (\ref{Eq:fExact}) and (\ref{Eq:j}) constitute the main result of this section.
They can be used for efficient numerical evaluation of the matrix elements.

In the limit of  large system size and for the situation when the level considered is not too close to the bottom of the tree, all spins involved are large and differ significantly one from another, one can significantly simplify the recursion relations. One finds
\begin{equation}
\xi_{k-1}=\tilde{f}_{\Delta S_{k-1}, \Delta S_k}(S_{k-1}, S_k, J_k)\xi_{k},
\label{Eq:xiLarge}
\end{equation}
where
\begin{equation}
\tilde{f}_{\Delta S_1, \Delta S_2}(S_1, S_2, J_2)=
\frac{1}{4S_{1} S_{2}}\left(
\begin{array}{ccc}
U& \sqrt{2UV} &
-V\\
-\sqrt{2UV} &  U-V &  -\sqrt{2UV}\\
-V & \sqrt{2UV}& U
\end{array}
\right)\,, \qquad U=j_1\Sigma,\qquad V=j_2 j_3.
\label{Eq:fLarge}
\end{equation}

We will employ Eqs. (\ref{Eq:xiLarge}) and (\ref{Eq:fLarge}) in our asymptotic analysis of the matrix elements for large systems.

Before closing this section let us draw the attention of the reader to the fact that according to Eq. (\ref{Eq:fExact}), matrix elements can still vanish despite being allowed by the selection rules . It is easy to see that the factor $\xi$ vanishes for the transitions with $\Delta S_{k-1}=\Delta S_k=0$ provided that
the three spins $S_{{\cal K}-1}$, $S_k$ and $J_k$ form a right-angled triangle
\begin{equation}
 S_{k-1}(S_{k-1}+1)+ S_{k}(S_{k}+1)-J_{k}(J_{k}+1)=0.
 \label{Eq:triangle}
 \end{equation}
However, these vanishing matrix elements do not influence the scaling of the connectivity with the system size presented in the main text, because the density of the configurations satisfying
Eq. (\ref{Eq:triangle}) vanishes in the limit of large system size.

\mysection{Statistics over a fixed base state}.
\label{Sec:StatisticsOverBaseState}
Let us now suppose that we choose randomly a a tree-like state (with top spin zero) and consider the distribution of non-zero matrix elements connecting this states to its neighbors.
More precisely we are interested in the distribution function $\zeta=\log_2 |V|$
In this section we show how this distribution can be computed.

\mysubsection{Top level}
We describe the states connected to the fixed base state by the set of $\Delta S^{\eta}_k=S^{\eta}_k-\tilde{S}_k^\eta$.
Let us introduce the generating function for the random variable $\zeta$
\begin{equation}
F_\zeta(\lambda)=\langle e^{i\lambda \zeta}\rangle=\frac{1}{K}\sum_{\rm all\; connected\; states}e^{i\lambda \zeta}=\frac{e^{-i\lambda}}{\cal K}
\sum_{\rm all\; connected\; states} e^{i\lambda (\omega^{\cal R}_1 + \omega^{\cal L}_1)}.
\label{Eq:Fdef}
\end{equation}
Here  $K$ is the number of states connected to the fixed base state, $\omega^\eta_1=\log_2 |\xi^\eta_1|$ and we have used Eq. (\ref{Eq:VS0}).
The summation goes over all the states for which the base state has non-zero matrix element with. This condition is local in $\Delta S$ in the sense that it can be expressed by an indicator function
\begin{equation}
\delta_{S_1^{\mathcal R} S_1^{\mathcal L}}\prod_{\eta={\mathcal R}, {\mathcal L}}\prod_{k={1}}^{\mathcal K-2}\psi_{J^\eta_{k+1}, S^\eta_{k+1}, S^\eta_k}(\Delta S^\eta_{k+1}, \Delta S^\eta_k)
\end{equation}
where the  Kronecker symbol is due to the fact that  we are considering trees with zero spin on top and the function
 $\psi_{J_2, S_2, S_1}(\Delta S_1, \Delta S_2)$ is only non-zero if the following conditions are fulfilled simultaneously
 \begin{enumerate}
 \item $S_2-\Delta S_2\geq 0$;
 \item $|J_2-S_2+\Delta S_2|\leq S_1-\Delta S_1\leq J_2+S_2-\Delta S_2 $;
 \item $ S_{1}(S_{1}+1)+ S_{2}(S_{2}+1)-J_{2}(J_{2}+1)\neq 0 $ if $\Delta S_1 =\Delta S_2=0$.
 \end{enumerate}

Note further that $\omega^{\eta}_1$ is a function of the corresponding $\Delta S^{\eta}$ only. Thus we can rewrite Eq.
(\ref{Eq:Fdef})  as
\begin{equation}
F_\zeta(\lambda)=\frac{e^{i\lambda}}{K}\sum_{\Delta S}F_{\omega_1}^{\mathcal R}(\lambda, \Delta S) F_{\omega_1}^{\mathcal L}(\lambda, \Delta S)
\label{Eq:FzetaFund}
\end{equation}
 with
 \begin{equation}
 F_{\omega_1}^{\eta}(\lambda, \Delta S)=\sum_{\Delta S^\eta_{k=1, \ldots {\cal K}-2}}
 \delta_{\Delta S^\eta_1, \Delta S}
 \exp\left[i\omega^\eta_1\right] \prod_{k={1}}^{\mathcal K-2}\psi_{J^\eta_{k+1}, S^\eta_{k+1}, S^\eta_k}(\Delta S^\eta_{k+1}, \Delta S^\eta_k).
 \end{equation}

\mysubsection{Recursion relations}
 The functions $F_\omega^{{\mathcal R}({\mathcal L})}$ introduced in the previous section  satisfy recursion
 relations that we are going to derive now. We omit the irrelevant superscript $\eta$ and rephrase the recursion relation (\ref{Eq:xiExact}) in terms of $\omega=\log_2|\xi|$
 \begin{equation}
 \omega_{1}=\omega_2+ r_{S_1, S_2, J_2}(\Delta S_1, \Delta S_2)\,, \qquad r_{S_1, S_2, J_2}(\Delta S_1, \Delta S_2)=\log_2\frac{f_{\Delta S_1, \Delta S_2}(S_1, S_2, J_2)}{\sqrt{\gamma(S_1, \Delta S_1) \gamma(S_2, \Delta S_2)}}.
 \label{Eq:omegaExact}
 \end{equation}
 We immediately get
 \begin{equation}
 F_{\omega_1}(\Delta S_1, \lambda)=\sum_{\Delta S_2} F_{\omega_2}(\Delta S_2, \lambda)\psi_{J_2, S_2, S_1}(\Delta S_1,\Delta S_2)\exp\left[i\lambda r_{S_1, S_2, J_2}(\Delta S_1, \Delta S_2)\right].
 \label{Eq:Fomega}
 \end{equation}

 \mysubsection{Mean and dispersion}
 The random process (\ref{Eq:omegaExact}) can be regarded as a kind of random walk with short-range correlations. Therefore, we can expect the probability distribution for the variable $\omega$ to approach a Gaussian form in the large-$L$ limit.
 This expectation is also supported by numerical simulations described in the main text. Thus, instead of analyzing the full generating functions  $F^{\eta}_\omega$ and $F_\zeta$ [see Eq. (\ref{Eq:FzetaFund})] we can limit ourself to the consideration of the mean value $\overline{\zeta}$ and dispersion $\sigma$ of the variable $\zeta$.

To that end, we introduce the power series expansion of the functions $F_{\omega_1}^{\eta}(\Delta S, \lambda)$
 \begin{equation}
 F_{\omega_1}^\eta(\Delta S, \lambda)=F_{\omega_1}^{\eta, (0)}(\Delta S)+i\lambda F_{\omega_1}^{\eta, (1)}(\Delta S)-\frac{\lambda^2}{2}F_{\omega_1}^{\eta, (2)}(\Delta S)+\ldots.
 \end{equation}
In terms of $F_{\omega_1}^{\eta, (i)}(\Delta S)$ we get
\begin{eqnarray}
K&=&\sum_{\Delta S}F_{\omega_1}^{{\mathcal L}, (0)}(\Delta S) F_{\omega_1}^{{\mathcal R}, (0)}(\Delta S),
\label{Eq:KExact}
\\
\overline{\zeta}&=&-1 +\frac1K\sum_{\Delta S}\left[ F_{\omega_1}^{{\mathcal R}, (0)}(\Delta S) F_{\omega_1}^{{\mathcal L}, (1)}(\Delta S)
+ F_{\omega_1}^{{\mathcal R}, (1)}(\Delta S) F_{\omega_1}^{{\mathcal L}, (0)}(\Delta S)\right],
\label{Eq:xibarExact}
\\
\sigma^2&=&\frac1K \sum_{\Delta S}\left[
F_{\omega_1}^{{\mathcal R}, (0)}(\Delta S) F_{\omega_1}^{{\mathcal L}, (2)}(\Delta S)+
F_{\omega_1}^{{\mathcal R}, (2)}(\Delta S) F_{\omega_1}^{{\mathcal L}, (0)}(\Delta S)+
2F_{\omega_1}^{{\mathcal R}, (1)}(\Delta S) F_{\omega_1}^{{\mathcal L}, (1)}(\Delta S)
\right]-(\overline{\zeta} +1)^2.\quad
\label{Eq:sigmaExact}
\end{eqnarray}

The recursion relations for $F_{\omega_1}^{\eta, (i)}(\Delta S)$  can be read off from Eq. (\ref{Eq:Fomega})
\begin{eqnarray}
F_{\omega_1}^{(0)}( \Delta S_1)&=&
\sum_{\Delta S_2}
\psi_{J_{2}, S_{2}, S_{1}}(\Delta S_{1}, \Delta S_2)F^{(0)}_{\omega_2}( \Delta S_2),
\label{Eq:F0Exact}
\\
F_{\omega_1}^{(1)}( \Delta S_1)&=&
\sum_{\Delta S_2}
\overline{\psi}_{J_{2}, S_{2}, S_{1}}(\Delta S_{1}, \Delta S_2)\left[F^{(1)}_{\omega_2}( \Delta S_2)+F^{(0)}_{\omega_2}( \Delta S_2)r(\Delta S_1, \Delta S_2)\right],
\label{Eq:F1Exact}
\\
F_{\omega_1}^{(2)}( \Delta S_1)&=&
\sum_{\Delta S_2}
\overline{\psi}_{J_{2}, S_{2}, S_{1}}(\Delta S_{1}, \Delta S_2)\left[F^{(2)}_{\omega_2}( \Delta S_2)+2F^{(1)}_{\omega_2}( \Delta S_2)r(\Delta S_1, \Delta S_2)+F^{(0)}_{\omega_2}( \Delta S_2)r^2(\Delta S_1, \Delta S_2)\right].\qquad\quad
\label{Eq:F2Exact}
\end{eqnarray}
Here we suppress the superscript $\eta$ and use the shorthand notation $r(\Delta S_1, \Delta S_2)\equiv r_{S_1, S_2, J_2}(\Delta S_1, \Delta S_2)$.

Equations (\ref{Eq:KExact}), (\ref{Eq:xibarExact}) and (\ref{Eq:sigmaExact}) together with  (\ref{Eq:F0Exact}), (\ref{Eq:F1Exact}) and (\ref{Eq:F2Exact}) constitute the main result of this section. They are exact and we have tested them against direct numerical simulations. In the next section we will specialize to the case of large system sizes.

\mysubsection{Large-$L$ limit}
In the large-$L$ limit constraints imposed by the indicator functions $\psi(\Delta S_1, \Delta S_2)$ are irrelevant.  Equation (\ref{Eq:F0Exact}) implies then that
$F^{(0)}_{\omega_1}$ is independent of $\Delta S$ and satisfies
\begin{equation}
F_{\omega_1}^{(0)}=3 F_{\omega_2}^{(0)}
\end{equation}
which with together with Eq. (\ref{Eq:KExact}) leads to the scaling of the connectivity
\begin{equation}
K\propto 3^{2 {\cal K}}=L^{\alpha}, \qquad \alpha=2\log_2 3.
\end{equation}

It is convenient to introduce $\overline{F}^{(i)}_{\omega_1}(\Delta S)= F^{(i)}_{\omega_1}(\Delta S)/ F^{(0)}_{\omega_1}$, $i=1, 2$. The recursion relations (\ref{Eq:F1Exact}) and (\ref{Eq:F2Exact}) now acquire the form
\begin{eqnarray}
\overline{F}_{\omega_1}^{(1)}( \Delta S_1)&=&
\frac13\sum_{\Delta S_2}
\left[\overline{F}^{(1)}_{\omega_2}( \Delta S_2)+r(\Delta S_1, \Delta S_2)\right],
\label{Eq:Fbar1}
\\
\overline{F}_{\omega_1}^{(2)}( \Delta S_1)&=&
\frac13\sum_{\Delta S_2}
\left[\overline{F}^{(2)}_{\omega_2}( \Delta S_2)+2\overline{F}^{(1)}_{\omega_2}( \Delta S_2)r(\Delta S_1, \Delta S_2)+ r^2(\Delta S_1, \Delta S_2)\right].
\label{Eq:Fbar2}
\end{eqnarray}

It proves useful to introduce
\begin{equation}
E_{\omega_k}=\sum_{\Delta S}\overline{F}_{\omega_k}^{(1)}(\Delta S)\,, \qquad
G_{\omega_k}=\sum_{\Delta S}\overline{F}_{\omega_k}^{(2)}(\Delta S)-\frac13E_{\omega_k}^2
\end{equation}
satisfying recursion relations
\begin{eqnarray}
E_{\omega_1}&=&E_{\omega_2}+\frac{1}{3}u,
\label{Eq:ERecFinal}
\quad\\
G_{\omega_1}&=&G_{\omega_2}+\sum_{\Delta S_1, \Delta S_2}\left[\overline{F}^{(1)}_{\omega_2}(\Delta S_2)-\frac13 E_{\omega_2}\right]r(\Delta S_1, \Delta S_2)-
\frac{1}{27}u^2+\frac13\sum_{\Delta S_1\Delta S_2}r^2(\Delta S_1, \Delta S_2).\quad
\label{Eq:GRecFinal}
\end{eqnarray}
Here
\begin{equation}
u=u(S_1, S_2, J_2)=\sum_{\Delta S_1, \Delta S_2}r_{S_1, S_2, J_2}(\Delta S_1, \Delta S_2).
\label{Eq:uDef}
\end{equation}

Expressions (\ref{Eq:xibarExact}) and  (\ref{Eq:sigmaExact}) for the mean and dispersion can now also be simplified. We get after straightforward but lengthy algebra
\begin{eqnarray}
\overline{\zeta}&=&-1 +\frac13\left[  E^{\mathcal L}_{\omega_1}+ E^{\mathcal R}_{\omega_1}\right]\,,
\label{Eq:xibarFinal}
\\
\sigma^2 &=&
\frac13 G^{\mathcal L}_{\omega_1}+ \frac13 G^{\mathcal R}_{\omega_1}+\frac{2}{27}
E^{\mathcal R}_{\omega_1}u^{\mathcal L}+ \frac{2}{27}
E^{\mathcal L}_{\omega_1}u^{\mathcal L}+\frac{2}{27}\sum_{\Delta S_1}v^{\mathcal L}(\Delta S_1) v^{\mathcal R}(\Delta S_1).
\label{Eq:sigmaFinal}
\end{eqnarray}
Here
\begin{equation}
v(\Delta S_1)=\sum_{\Delta S_2} r(\Delta S_1, \Delta S_2).
\end{equation}

\mysection{Statistics of $\overline{\zeta}$ and $\sigma$}
\label{Sec:Statistics}
In this section we turn to the main subject of this Supplemental Material.  In Sec. \ref{Sec:StatisticsOverBaseState} we have characterized the distribution of  the matrix elements connecting fixed tree-like state to other states by the $\overline{\zeta}$ and $\sigma$, the mean and the standard deviation of the logarith of the matrix element. Given different base states the quantities $\overline{\zeta}$ and $\sigma$  are themselves  random variables. We would like to characterize their statistics by computing the averages $\langle \xi\rangle$ and $\langle \sigma\rangle$ with respect to the chosen base state as well as the ``mesoscopic fluctuations'' of $\overline{\zeta}$, $\langle (\delta \overline{\zeta})^2\rangle$.

\mysubsection{Averaging of $\overline{\zeta}$}
\label{SubSec:xiAv}
\mysubsubsection{Recursion relations in large-$L$ limit.}
We start with the computation of $\langle \overline{\zeta}\rangle$. We limit ourself to the analysis in the large-$L$ limit where, according to Eq. (\ref{Eq:xibarFinal}),
\begin{equation}
\langle \overline{\zeta}\rangle\approx \frac{2}{3}\langle E^{\mathcal L}_{\omega_1}\rangle.
\label{Eq:xiaE}
\end{equation}
The quantity $E^{\mathcal L}_{\omega_1}$ satisfies the recursion relation (as usual we omit the superscript and restore the dependence on the spins in the base state, suppressed previously)
\begin{equation}
E_{\omega_1}=E_{\omega_2}+\frac{1}{3}u(S_1, S_2, J_2)
\label{Eq:ERecFinalExp}
\end{equation}
with $u(S_1, S_2, J_2)$ given by [see Eqs. (\ref{Eq:uDef}), (\ref{Eq:omegaExact}) and (\ref{Eq:fLarge})]
\begin{equation}
u(S_1, S_2, J_2)=\log_2 \frac{U^4 V^4 |U-V|}{2^{16} S_1^9 S_2^9}\,, \qquad U=(S_1+S_2)^2-J_2^2\,,\qquad V=J_2^2-(S_1-S_2)^2.
\label{Eq:uLarge}
\end{equation}

Let us introduce the number of trees of depth $k$ with spin $S$ on top, $N_k(S)$. Simple combinatorics gives
\begin{equation}
N_k(S)=\frac{L_k ! (2S+1)}{\left(\frac{L_k}{2}-S\right)! \left(\frac{L_k}{2}+S+1\right)!}.
\end{equation}
Here $L_k=2^{k-1}$ is the number of spins underlying a tree of depth $k$.

For the average we are interested in we can now write
\begin{equation}
\langle E_{\omega_1}\rangle=\frac{1}{N_{\cal K}(0)}\sum_{\{S, J\}} E_{\omega_1}(\{S, J\}) N_{{\cal K}-1}(S_1)\prod_{k=2}^{{\cal K}-2} N_{{\cal K}-k}(J_k).
\label{Eq:EaBasic}
\end{equation}
Here the summation goes over all consistent sets of $S_{k=1,\ldots  {\cal K}-1}$ and $J_{k=2, \ldots {\cal K}-1}$.

 We introduce now
\begin{equation}
R_{\omega_k}(S)=\frac{1}{N_{\cal K}(0)}\sum_{\{S, J\}_k} E_{\omega_k}(\{S, J\}) \delta_{S_k, S}\prod_{k=2}^{{\cal K}-k-1} N_{{\cal K}-k}(J_k)
\label{Eq:Rdef}
\end{equation}
and rewrite (\ref{Eq:EaBasic}) as
\begin{equation}
\langle E_{\omega_1}\rangle=\frac{1}{N_{\cal K}(0)}\sum_{S_1=0}^{L_{\cal K}/2} N_{{\cal K}-1}(S_1)R_{\omega_1}(S_1).
\label{Eq:EaBasic1}
\end{equation}
In Eq. (\ref{Eq:Rdef}) the summation runs over all consistent sets of $S_{i=k, \ldots {\cal K}-1}$ and $J_{i=k+1, \ldots {\cal K}-1}$.

The function  $R_{\omega_1}(S)$ satisfies the recursion relations that can be read off Eq. (\ref{Eq:ERecFinalExp})
\begin{equation}
R_{\omega_{k-1}}(S_{k-1})=
\sum_{S_k}D_{k}(S_{k-1}, S_k)R_{\omega_k}(S_k)+\frac{1}{3}\sum_{\substack{J_k, S_k\\ |S_k-J_k|\leq S_{k-1}\leq S_k+J_k}}u(S_{k-1}, S_k, J_k)N_{{\cal K}-k}(J_k) N_{{\cal K}-k}(S_k).
\label{Eq:Rrec}
\end{equation}
 with
 \begin{equation}
 D_{k}(S_{k-1}, S_k)=\sum_{\substack{J_k=|S_k-S_{k-1}|\\J_k\leq L_{{\cal K}-k}/2}}^{S_k+S_{k-1}} N_{{\cal K}-k}(J_k)
 \end{equation}

We are now in position to formulate the final set of equations describing $\langle E_{\omega_1}\rangle$ in the large-$L$ limit.
To achieve this goal we switch to a continuum description of the  spins.  In the continuum limit the function $N_k(S_k)$  is given by
\begin{equation}
N_k(S)=\sqrt{\frac{2}{\pi}}\frac{2^{L_k+2} S}{L_k^{3/2}}e^{-2S^2/L_k}=\sqrt{\frac{2}{\pi}}\frac{2^{L_{k}}}{L_k}n(S/\sqrt{L_k})\,, \qquad n(s)= 4 s
e^{-2s^2}\label{Eq:ns}
\end{equation}
We thus introduce the continuum variables $s_k=S_k/\sqrt{L_{{\cal K}-k}}$ and $j_k=J_k/\sqrt{L_{{\cal K}-k}}$  together with functions
\begin{equation}
\tilde{R}_{\omega_k}(s_k)=\frac{\sqrt{2}L_{{\cal K}-k}}{2^{L_{{\cal K}-k}}}R_{\omega_k}(s\sqrt{L_{{\cal K}-k}})
\end{equation}
and switch from summation to integration in Eq. (\ref{Eq:Rrec}). The result reads
\begin{equation}
\tilde{R}_{\omega_{k-1}}(s_{k-1})=
\int_0^\infty ds_k \tilde{D}(s_{k-1}, s_k) \tilde{R}_{\omega_k}(s_k)\\+
\frac{4\sqrt{2}}{3\pi} \int_{ |j_k-s_k|\leq \sqrt{2} s_{k-1}\leq j_k+s_k}^\infty dj_k d s_k
\tilde{u}(s_{k-1}, s_k, j_k )n(j_k) n(s_k).
\label{Eq:RecFinal}
\end{equation}
Here (see  Eq.~({\ref{Eq:uLarge}}))
\begin{eqnarray}
\tilde{D}(s_{k-1}, s_k)&=&\frac{2\sqrt{2}}{\sqrt{\pi}}\left[e^{-2(s_k+\sqrt{2}s_{k-1})^2}- e^{-2(s_k-\sqrt{2}s_{k-1})^2}\right],\\
\tilde{u}(s_{k-1}, s_k, j_k )&=&u(s_{k-1}\sqrt{L_{{\cal K}-k+1}}, s_k\sqrt{L_{{\cal K}-k}}, j_k \sqrt{L_{{\cal K}-k}})\nonumber\\&=& \log_2 \frac{\left[(\sqrt{2}s_{k-1}+s_k)^2-j_{k}^2\right]^4 \left[j_{k}^2-(\sqrt{2}s_{k-1}-s_k)^2\right]^4 \left|2s_{k-1}^2+s_k^2-j_k^2\right|}{2^{19}\sqrt{2} s_{k-1}^9 s_k^9}.
\end{eqnarray}
Finally, in the continuum limit  Eq. (\ref{Eq:EaBasic1}) takes the form
\begin{equation}
\langle E_{\omega_1}\rangle= \int_0^\infty ds n(s)\tilde{R}_{\omega_1}(s).
\label{Eq:EaFinal}
\end{equation}

Equations (\ref{Eq:EaFinal}) and  (\ref{Eq:RecFinal}) constitute  the main result of this section. We use them below to derive the asymptotic behavior of the average $\langle\overline{\zeta}\rangle$.

\mysubsubsection{Asymptotic behavior of  $\langle\overline{\zeta}\rangle$}
To find the asymptotic behavior  of $\tilde{R}_{\omega_1}(S)$ we need here several properties of the kernel $\tilde{D}$ in Eq. (\ref{Eq:RecFinal}). It is easy to check that
the function $n(S)$, Eq.~(\ref{Eq:ns}), is an eigenfunction of $\tilde{D}$ with eigenvalue $1$. Moreover, one can show that all other eigenvalues are smaller than 1 and the corresponding eigenfunctions $\psi_{2n+1}$ can be expressed in terms of the Hermite polynomials
\begin{equation}
\psi_{2n+1}(s)\sim H_{2n+1}(\sqrt{2} s)e^{-2s^2},
\end{equation}
which are orthorgonal on $(0, \infty)$ with weight $e^{2 s^2}$.

Expanding now $\tilde{R}_{\omega_k}$ in the eigenfunctions of $\tilde{D}$ as
\begin{equation}
R_{\omega_k}(s)=c_k n(s)+\ldots
\end{equation}
we find from Eq. (\ref{Eq:RecFinal})
\begin{equation}
c_{k-1}=c_k+\frac{4\sqrt{2}}{3\pi |n(s)|^2} \int_{ |j_k-s_k|\leq \sqrt{2} s_{k-1}\leq j_k+s_k}^\infty ds_{k-1} dj_k d s_k
\tilde{u}(s_{k-1}, s_k, j_k )n(j_k) n(s_k)n(s_{k-1})e^{2 s_{k-1}^2}
\end{equation}
with
\begin{equation}
 |n(s)|^2=\int_0^\infty ds e^{2 s^2}n^2(s)=\sqrt{2\pi}.
\end{equation}

Computation of the integral leads to
\begin{equation}
c_{k-1}=c_k -\frac{17+4\ln 2}{3\sqrt{\pi}\ln 2}
\end{equation}
implying
\begin{equation}
\tilde{R}_{\omega_1}(s)\sim -\frac{(17+4\ln 2){\cal K}}{3\sqrt{\pi}\ln 2}n(s).
\label{Eq:Rass}
\end{equation}

Equation (\ref{Eq:Rass}) together with Eqs. (\ref{Eq:EaFinal}) and (\ref{Eq:xiaE}) gives us now the central  result of this section.
\begin{equation}
\langle\overline{\zeta}\rangle\sim -\beta \log_2 L\,, \qquad \beta=\frac{17+4\ln 2}{9\ln2}.
\label{Eq:mainAns}
\end{equation}
This is the scaling stated in the main text.

\mysubsection{Averaging of $\sigma^2$}
\label{SubSec:sigmaAv}
Let us now discuss the averaging of the dispersion $\sigma^2$. Such an averaging can be performed along the lines of Sec. \ref{SubSec:xiAv} on the basis of Eqs. (\ref{Eq:sigmaFinal}), (\ref{Eq:ERecFinal}) and (\ref{Eq:GRecFinal}) that we collect here for the convenience of the reader
\begin{eqnarray}
\sigma^2 &=&
\frac13 G^{\mathcal L}_{\omega_1}+ \frac13 G^{\mathcal R}_{\omega_1}+\frac{2}{27}
E^{\mathcal R}_{\omega_1}u^{\mathcal L}+ \frac{2}{27}
E^{\mathcal L}_{\omega_1}u^{\mathcal L}+\frac{2}{27}\sum_{\Delta S_1}v^{\mathcal L}(\Delta S_1) v^{\mathcal R}(\Delta S_1),
\label{Eq:sigmaFinalCopy}\\
E_{\omega_1}&=&E_{\omega_2}+\frac{1}{3}u,
\label{Eq:ERecFinalCopy}
\quad\\
G_{\omega_1}&=&G_{\omega_2}+\sum_{\Delta S_1, \Delta S_2}\left[\overline{F}_{\omega_2}(\Delta S_2)-\frac13 E_{\omega_2}\right]r(\Delta S_1, \Delta S_2)-
\frac{1}{27}u^2+\frac13\sum_{\Delta S_1\Delta S_2}r^2(\Delta S_1, \Delta S_2).\quad
\label{Eq:GRecFinalCopy}
\end{eqnarray}

The analysis is quite involved and we do not present it here. Straightforward inspection of Eq. (\ref{Eq:GRecFinalCopy}) shows that $G_{\omega_1}$, very much like $E_{\omega_1}$ discussed in the previous section, grows linearly with the ${\cal K}=\log_2 L$. It is easy to see then from Eq. (\ref{Eq:sigmaFinalCopy}) that
\begin{equation}
\langle\sigma^2\rangle\propto \log_2 L
\end{equation}
which is the result stated in the main text.

\mysubsection{Mesoscopic fluctuations of $\overline{\zeta}$}
Let us now study the mesoscopic fluctuations of $\overline{\zeta}$. According to Eq. (\ref{Eq:xibarFinal}) we have
\begin{equation}
\langle(\delta\overline{\zeta})^2\rangle=\frac{2}{9}\left[\left\langle \left(E_{\omega_1}^{\mathcal L}\right)^2\right\rangle +\left\langle E_{\omega_1}^{\mathcal L}E_{\omega_1}^{\mathcal R}\right\rangle -
2\langle E_{\omega_1}^{\mathcal L}\rangle^2\right].
\label{Eq:deltaxiBase}
\end{equation}
We would like to show here that $\langle(\delta\overline{\zeta})^2\rangle$ scales linearly with ${\cal K}=\log_2 L$.
Since (see Sec. \ref{SubSec:xiAv} )
\begin{equation}
\langle E_{\omega_1}\rangle\sim \frac{3}{2}\beta {\cal K},\qquad \beta=\frac{17+4\ln 2}{9\ln2},
\label{Eq:EaFinalFinal}
\end{equation}
we need to analyze the average
\begin{equation}
\left\langle \left(E_{\omega_1}^{\mathcal L}\right)^2\right\rangle +\left\langle E_{\omega_1}^{\mathcal L}E_{\omega_1}^{\mathcal R}\right\rangle
\end{equation}
to the order ${\cal K}^2$ and show that the quadratic scaling cancels in Eq. (\ref{Eq:deltaxiBase}).

\mysubsubsection{Averaging  $ \left(E_{\omega_1}^{\mathcal L}\right)^2$}
Along the lines of Sec. \ref{SubSec:xiAv} we can write the average in question as
\begin{equation}
\langle E_{\omega_1}^2\rangle=\frac{1}{N_{\cal K}(0)}\sum_{S_1=0}^{L_{\cal K}/2} N_{{\cal K}-1}(S_1)U_{\omega_1}(S_1)
\label{Eq:Ea2Basic1}
\end{equation}
 with $U_{\omega_k}(S_1)$ given by
 \begin{equation}
U_{\omega_k}(S)=\frac{1}{N_{\cal K}(0)}\sum_{\{S, J\}_k} E^2_{\omega_k}(\{S, J\}) \delta_{S_k, S}\prod_{k=2}^{{\cal K}-k-1} N_{{\cal K}-k}(J_k)
\label{Eq:Udef}
\end{equation}
 and satisfying recursion relations
 \begin{multline}
U_{\omega_{k-1}}(S_{k-1})=
\sum_{S_k}D_{k}(S_{k-1}, S_k)U_{\omega_k}(S_k)+
\frac{2}{3}\sum_{\substack{J_k, S_k\\ |S_k-J_k|\leq S_{k-1}\leq S_k+J_k}}u(S_{k-1}, S_k, J_k)N_{{\cal K}-k}(J_k) R_{\omega_k}(S_k)
\\+\frac{1}{9}\sum_{\substack{J_k, S_k\\ |S_k-J_k|\leq S_{k-1}\leq S_k+J_k}}u^2(S_{k-1}, S_k, J_k)N_{{\cal K}-k}(J_k) N_{{\cal K}-k}(S_k).
\label{Eq:UrecBasic}
\end{multline}

As we are interested only in the leading ${\cal K}^2$ term in the asymptotic we can omit term in Eq. (\ref{Eq:UrecBasic}). Going to the continuum description of spins we get
\begin{eqnarray}
\langle E^2_{\omega_1}\rangle&= &\int_0^\infty ds n(s)\tilde{U}_{\omega_1}(s)
\label{Eq:EaSFinal}\\
\tilde{U}_{\omega_{k-1}}(s_{k-1})&=&
\int_0^\infty d s_k \tilde{D}(s_{k-1}, s_k)\tilde{U}_{\omega_k}(s_k)+
\frac{4}{3}\sqrt{\frac{2}{\pi}}\int_{ |j_k-s_k|\leq \sqrt{2} s_{k-1}\leq j_k+s_k}^\infty dj_k d s_k n(j_k) \tilde{R}_{\omega_k}(s_k)\tilde{u}(s_{k-1}, s_k, j_k).\qquad\qquad
\label{Eq:URecSimple}
\end{eqnarray}

The scaling of $\tilde{R}_{\omega_k}(s_k)$ is give by (cf.~Eq.~(\ref{Eq:Rass}))
\begin{equation}
\tilde{R}_{\omega_k}(s)\sim -\frac{(17+4\ln 2)({\cal K}-k)}{3\sqrt{\pi}\ln 2}n(s).
\end{equation}
Projecting now Eq. (\ref{Eq:URecSimple}) onto $n(s)$ (cf.~Sec.~\ref{SubSec:xiAv}) we get
\begin{equation}
c_{k-1}=c_k+\frac{1}{\sqrt{\pi}} \left(\frac{17+4\ln 2}{3\ln 2}\right)^2({\cal K}-k)
\end{equation}
implying
\begin{equation}
U_{\omega_1}(s)\sim \frac{1}{2\sqrt{\pi}} \left(\frac{17+4\ln 2}{3\ln 2}\right)^2{\cal K}^2 n(s).
\end{equation}
With the aid of Eq. (\ref{Eq:EaSFinal}) we  finally get
\begin{equation}
\left\langle\left(E_{\omega_1}^{\mathcal L}\right)^2\right\rangle \sim \frac{1}{4}\left(\frac{17+4\ln 2}{3\ln 2}\right)^2 {\cal K}^2=\frac{9}{4}\beta^2 {\cal K}^2.
\label{Eq:EaSFinalFinal}
\end{equation}

\mysubsubsection{Averaging  $ E_{\omega_1}^{\mathcal L} E_{\omega_1}^{\mathcal R}$}
The average $\langle  E_{\omega_1}^{\mathcal L} E_{\omega_1}^{\mathcal R}\rangle$ can be presented as
\begin{equation}
\langle  E_{\omega_1}^{\mathcal L} E_{\omega_1}^{\mathcal R}\rangle=\frac{\sqrt{\pi}}{2}\int_0^\infty \tilde{R}^2_{\omega_1}(s).
\end{equation}
Using Eq. (\ref{Eq:Rass}) we find
\begin{equation}
\langle  E_{\omega_1}^{\mathcal L} E_{\omega_1}^{\mathcal R}\rangle\sim \frac{\sqrt{\pi}}{2} \left(\frac{(17+4\ln 2){\cal K}}{3\sqrt{\pi}\ln 2}\right)^2\int_0^\infty ds n^2(s)= \frac{1}{4}\left(\frac{17+4\ln 2}{3\ln 2}\right)^2 {\cal K}^2=\frac{9}{4}\beta^2 {\cal K}^2.
\label{Eq:EREL}
\end{equation}

Equations (\ref{Eq:deltaxiBase}), (\ref{Eq:EaFinalFinal}), (\ref{Eq:EaSFinalFinal}) and (\ref{Eq:EREL}) immediately imply that the mesoscopic fluctuations scale linearly with $\log_2 L$, i.e.
\begin{equation}
\langle(\delta\overline{\zeta})^2\rangle\propto \log_2 L.
\end{equation}

\mysection{Non-zero top spin}
\label{Sec:S0}
To conclude this supplemental material, let us now show that our results on the asymptotic scaling of matrix elements, most notably Eq. (\ref{Eq:mainAns}), hold also
outside the sector of the theory with $S_0=0$.

Indeed, led us consider the sector with $S_0$ of the order of typical its typical value, $S_0\sim \sqrt{L}$. A straightforward generalization of the derivation presented in Sec. \ref{Sec:StatisticsOverBaseState} shows that in the asymptotic large-$L$ limit we still have (cf.~Eq.~(\ref{Eq:xibarFinal}))

\begin{equation}
\overline{\zeta}\approx \frac13\left( E^{\mathcal L}_{\omega_1}+ E^{\mathcal R}_{\omega_1}\right).
\end{equation}

Averaging with respect to the base state we then get (cf.~Eq.~(\ref{Eq:EaBasic1}))
\begin{eqnarray}
\langle \overline{\zeta}\rangle\approx \frac{2}{3}\langle E_{\omega_1}\rangle,\\
\langle E_{\omega_1}\rangle=\frac{1}{N_{\cal K}(S_0)}
\sum_{
\substack{S_1^{\mathcal L}, S_1^{\mathcal R}\\
|S_1^{\mathcal L}- S_1^{\mathcal R}| \leq S_0\leq S_1^{\mathcal L}+S_1^{\mathcal R}
}}^{L_{\cal K}/2} N_{{\cal K}-1}(S_1^{\mathcal R})R_{\omega_1}(S_1^{\mathcal L}).
\end{eqnarray}

In the continuum limit we thus get (with $s_0=S_0/\sqrt{L}$)
\begin{equation}
\langle E_{\omega_1}\rangle=\frac{e^{2s_0^2}}{2 \sqrt{2}s_0}\int_{|s_1-s_1^\prime|\leq \sqrt{2} s_0\leq s_1+s_1^\prime} d s_1 ds_1^\prime n(s_1^\prime)\tilde{R}_{\omega_1}(s_1).
\end{equation}
Also, using Eq. (\ref{Eq:Rass}) we find
\begin{equation}
\langle E_{\omega_1}\rangle=
-\frac{(17+4\ln 2){\cal K}}{3\sqrt{\pi}\ln 2}
\frac{e^{2s_0^2}}{2 \sqrt{2} s_0}\int_{|s_1-s_1^\prime|\leq \sqrt{2} s_0\leq s_1+s_1^\prime} d s_1 ds_1^\prime n(s_1^\prime)n(s_1)=-\frac{(17+4\ln 2)
}{6\ln 2} {\cal K}
\end{equation}
implying
\begin{equation}
\langle\overline{\zeta}\rangle\sim -\beta \log_2 L\,, \qquad \beta=\frac{17+4\ln 2}{9\ln2}.
\end{equation}

In a similar manner one can show that the scaling laws for the the average  dispersion $\langle \sigma^2\rangle $ and mesoscopic fluctuations $\langle(\delta\overline{\zeta})^2\rangle$ remain unchanged in the case of arbitrary spin $S_0$

\begin{equation}
\langle \sigma^2\rangle\sim \langle\overline{\zeta}\rangle\sim  \log_2 L.
\end{equation}

\thebibliography{100}
\bibitem{Wigner-Eckart}
B. Hall,  \textit{ Lie groups, Lie algebras, and representations: An elementary introduction} (Springer, 2015).
\end{document}

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text:
Q: Why is convergence of measures tested against functions? This question is to help my intuition. Why do we test the convergence of measures against different classes of functions and not use definitions like:
If $(B,\mathcal{B})$ is a measurable space then $\mu_n\to\mu$ if $\mu_n(A)\to\mu(A)$ for every $A\subseteq B$?
Are there examples where say, $\int_Bf \, d\mu_n\to\int_Bf \, d\mu$ but $\mu_n(A)\to\mu(A)$ fails?

redpajama_set_name: RedPajamaStackExchange


This is a well-established general practice with a stellar reputation that has served this suburban community for the past 35 years. This restorative-driven practice has consistently generated gross receipts in excess of $950,000 for the past 5 years on 30-32 hours per week of chair time with the majority of specialty work being referred out to specialists in the area. Practice has 5 treatment rooms with room for a 6th. This is a proven, productive and profitable practice where the potential exists for even greater growth in patient flow and in the bottom line. This is an exceptional practice that's prepared for a smooth and seamless change of ownership. The opportunistic purchaser of this practice will be busy from the start and will be in position to realize a timely return on his/her investment.


In light of the recent fire incident at Gadani ship-breaking yard that left seven workers injured, labour rights activists and unions held a rally on Tuesday demanding that Section 144 be lifted from the shipyard to provide immediate support to labourers.
At a press conference held at the Karachi Press Club, Nasir Mansoor of the National Trade Union Federation said that the ship-breaking industry was on its way to a slow demise as the number of labourers is decreasing owing to dismal work conditions.
He added that since November 2016, more than 100 workers have been injured in fire-related incidents and around 30 of them have lost their lives. Speaking about the October 14 incident, Mansoor said that a fire had erupted in an oil tanker anchored at plot number 9-10, which resulted in injuries to seven workers; three of them are in critical condition.
The fire erupted due to a lack of safety measures and a similar incident was witnessed in plot number 6-7 as well, he said.
Representing Ship-breaking Workers' Union Gadani, Bashir Ahmed said that the implementation of Section 144 and halting work in the entire shipyard, instead of just the affected plots, meant doom for all workers. "Hundreds of labourers are now trying to find work because the yard has been shut," he said.
Ahmed added that following the last grave incident two years ago, around 3,000 to 4,000 workers didn't have work because the ship-breaking yard was shut.
He requested the government to look into this and give the required rights under the International Labour Organization to labourers, especially death grants to the families of those who have lost lives and health provisions to those who are injured.
Zehra Khan of Home-Based Workers Association said that the contract system has spread in almost all sectors which allows further exploitation of labour. The system takes away the right to EOBI, minimum wage and other provisions from the workers, and the state should look into all industries using it to get away from labour laws, she said.
Pakistan Labour Bureau's Habibullah Junaidi said that it seems that the labourers in some sectors are still living in times some 300 years back because of their preposterous and exploitative labour laws.
Karamat Ali of PILER stressed that many times the anchored ship isn't checked by the authorities for any possibility of explosives and workers are pushed to work without any protective gear, which results in untoward incidents.
The unionists said that a petition has been filed to release the payment for all those who have suffered because the right to life is the constitutional right of every citizen in the country. Workers from Gadani shipyard also arrived at KPC and protested against the inhumane working conditions in the ship-breaking industry.


Making a difference for nearly forty years!
Miami University's gerontology academic programs date back to the 1970s. The Master of Gerontological Studies program was founded in 1977, the BA in Gerontology in 2000, the PhD in Social Gerontology in 2005 and the Master of Population and Social Gerontology in 2010. We now have nearly 500 gerontology alumni from those four programs, and an estimated 700+ alumni with the gerontology minor. Our alumni are out there making a difference around the world!
This page is devoted to our gerontology alumni. Here we will highlight your accomplishments, post alumni photos, invite you to connect with us online or at upcoming events, and make it easy to update your contact information and support current students.
If you are an alumni and want more information about how to connect, please contact Pamela Mayberry, MGS.
Share your latest contact information so we can stay in touch.
Heather Menne, (MGS '01) shares the story of her academic and career path and offers advice to students.
Quanhong "Cho" Qiu (MGS '01) discussed her career in gerontology with us and shared advice to current students who are studying gerontology.
"Where Miami really stands out is the ability to work directly with some of the top leaders in the field."
Marisa Scala-Foley (MGS '95) explains why Miami was her top choice for her masters in gerontology. Marisa is the Director for the Office of Policy Analysis & Development at the Administration for Community Living.
Our alumni are part of Scripps Gerontology Center's legacy.
On August 1, 2016, Scripps Gerontology Center convened the inaugural meeting of the Gerontology Alumni Advisory Group.
Read more about this meeting.
We are pleased to announce that Heather Menne is the recipient of the 2017 Cottrell Distinguished Alumni Award.
Read more about initiatives to make connections.


He came into prominence following enjoying Daario Naharis at Game of Thrones and Frank Martin at The Transporter: Refueled. His forthcoming most significant job is Deadpool, that will be set to be published in 2016. As a rapper, he's well famous for his or her 2007's record The Eat. He's of Austrian English and Jewish descent. In regards to his private life, he's always remained low profile. Until he hasn't familiar some details about his private life, including his family history, youth status and information regarding his current connection status. According to the source, he's reportedly a married person and has a boy also.
He is living in London with his loved ones. We can remain attached to him through Twitter @edskrein.


Lace and cotton kimono with wide sleeves, contrast sleeve cuffs and lapel.
This kimono is made with a black cotton fabric and overplayed with soft beautiful white lace.
Perfect layering piece for all weather.


All researchers affiliated with the Danish universities and the Royal Danish Library can get access and use the ABACUS 2.0 supercomputer under the same conditions.
To request access, please fill out the Account Request Form.
Proposers for any kind of allocation schemes must hold at least the position of postdoc within their institution.
PhD students request access in consultation with, and under the auspices of, their supervisor.
Users are billed according to how many node hours they could be allocated on ABACUS 2.0.
The maximum amount is 10000 node*hours on slim nodes and 2000 node*hours on fat and GPU nodes.
The maximum allowed period to use the compute time is 8 months.
Requests can be done at any time, and they will be processed as soon as possible.
This kind of allocation provide access for a period of one year.
Requesters are required to have a track-record of research based on the use of HPC platforms.
The allocation of the available resources for regular projects follows a schedule of four months' periods, called "allocation periods". The three yearly allocation periods are: 1st of March to 30th of June, 1st of July to 31st of October, 1st of November to last day of February.
Up to a 20% of the original allocated resources for a period will automatically be carried over to the next period, if the project has not used all available resources. Unused resources in excess of 20% of the original allocated amount will be lost.


Liang Feng received his Ph.D. in Electrical Engineering from UCSD in 2010, and was subsequently a postdoc researcher in the Department of Electrical Engineering at California Institute of Technology and the NSF Nanoscale Science and Technology Center at UC Berkeley. Prior to joining Penn in 2017, he was an assistant professor of SUNY Buffalo from 2014 to 2017. Currently his research interests include optical and photonic materials, metamaterials, nanophotonics, and optoelectronics.


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You've got the odd blemish, a bit of dry skin, a razor cut and chapped lips? You need a miracle. Well hold on there look what we have here, a product that fixes all of them in one! The Proto-col Miracle Balm.
Enough of the cheesy intro to this post and onto the important stuff. Proto-col is a brand I was introduced to a few months back with their Collagen Facial Serum. This product literally blew me away so when I was asked to try out their new release I was over the moon. The Miracle Balm is a jack of all trades, that can be used to treat blemishes, heal cuts and sooth dry skin including chapped lips. It's definitely one of those products to keep in your backpack or hand bag for all times. Using a collagen enriched formula it takes no time to get right to work at the source of the problem.
I've used this on a few different areas of my body for different problem, including spots on my face, spots on my back, dry skin on my arm and dry lips… and I can say now it has helped every one of them. I can't believe how effective this formula actually is and wonder, why aren't there other products out there like this!
Now lets get to the in depth part, what does it actually contain. You'll be surprised to know a lot of the ingredients are natural and quite familiar.
Plant Derived Collagen: A nourishing high-grade ingredient that plumps, firms and hydrates the skin leaving it feeling soft, supple and firm.
Hyaluronic Acid: Capable of holding up to 6000 times its own weight in water, hyaluronic acid delivers intense hydration to the skin leaving it feeling smooth and plump.
Beeswax: Known for its anti-bacterial and anti-inflammatory properties, Beeswax can help to balance the skin and reduce areas of redness and irritability.
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Vitamin E: A powerful antioxidant that protects against radical damage and environmental aggressors.
Orange Oil: A natural anti-inflammatory and antiseptic that helps to balance and detox decongested skin whilst soothing dry areas.
Bergamot Oil: A powerful antiseptic with healing properties to treat an array of skin problems such as acne, excess oil, psoriasis and can also be used as a natural insect repellent.
Coral Seaweed:Rich in regenerating Amino Acids, Calcium and Magnesium, Coral Seaweed will help to retexture, brighten and soften skin.
It has a light zesty fragrance to it which isn't at all overpowering, and feels soothing why applying to any area of the body. The best thing about this product, is that it suits all skin types, so unless you have a really bad skin condition or allergies, you'll be fine with this one!
At £14.95, it's such a great price for literally a miracle in a jar.
Have you tried any photo-col products? What were your thoughts?
Next Post Speed Up Your Skincare.


What is the difference? Actually, a lot! OK, maybe not a whole lot—oil that is. The object of sautéing foods is to produce a flavorful exterior with the best possible texture and color. This technique cooks food rapidly in a small amount of fat over relatively high heat. The juices released forms the base for a sauce.
Because sautéing is a rapid technique and does not have the tenderizing effect of other methods, any food to be sautéed must be naturally tender.
The object of pan-frying is to produce a flavorful exterior with a crisp, brown crust, which acts as a barrier to retain juices and flavor. Pan-frying requires more oil. Although this technique shares similarities with sautéing, it has some important differences. A sautéed item is often lightly dusted with flour and quickly cooked over high heat in a small amount of oil, pan-fried food is usually coated with batter or breaded and cooked in a larger amount of oil over less intense heat. The product is cooked more by the oil's heat than by direct contact with the pan. In pan-frying, the hot oil seals the food's coated surface and thereby locks the natural juices inside instead of releasing them. Because no juices are released and a larger amount of oil is involved, any accompanying sauce is made separately whereas, in sautéing the bits that remain in the pan are used to flavor the sauce for the sautéed item.
Alton Brown's Pan-Fried Chicken (worth the 5 minutes).
The following gives you a good walkthrough of a typical sauté dish.


On December 11, 2018, the Fairfax City Council approved an application submitted by Capstone Collegiate Communities, LLC (Capstone) to allow the development of a 275 unit multifamily building on a six-acre parcel located in Old Town Fairfax. A market leader in the student housing industry, Capstone specializes in the development, construction and management of high quality student housing communities in proximity to major colleges and universities throughout the country.
Land Use Attorneys Lynne Strobel and Robert Brant worked diligently with the development team, City officials and the community to obtain approval of a Comprehensive Plan Amendment, Rezoning, Special Exception and Certificate of Appropriateness. The approvals will allow the development of a purpose-built student housing community, the first of its kind in Northern Virginia, which will be marketed to graduate and undergraduate university students. The fully amenitized building will provide housing for up to 825 students in a secure, professionally managed setting. Due to the continued growth of nearby George Mason University (GMU), which has publicly acknowledged a need for additional off-campus housing options for its students, there is a growing demand for student housing in the region. The proposed development will help to meet that need.
The project provides a number of benefits to the City by adding residential density to support the business community in the City, strengthening the relationship between the City and GMU, and promoting further investment in Old Town Fairfax. This approval represents another step towards the revitalization of downtown Fairfax.


text:
Q: Is it possible to return nested records from pl/pgsql? I have the following tables in pgsql 8.4:
user
permission
role
user_permission (n:m)
role_permission (n:m)
user_role (n:1)

I have a REST service, by GET /user/1 I have to get something like this from the database:
{
    id:1,
    name: "John Smith",
    roles: [
        {id: 1, name: "Administrator"},
        {id: 2, name: "Editor"}
    ],
    permissions: [
        {id: 1, method: "GET", resource: "^/$"}
    ]
}

Where roles are from tables role & user_role and permissions are from tables permission & user_permission. By PUT /user/1 I want to send similar data to the database.
Is it possible to do that with pl/pgsql functions, or I have to write simple queries, and execute them one-by-one from php?
I don't necessarily want to send json to the pgsql, if this is possible with php arrays without concatenated strings, etc..., it would be more than enough.
I can't upgrade the server to a higher version.
Relational mapping is not good for me, I use stored procedures for everything because I don't want to write sql templates in php. I need just raw data, the only thing I do with that is json_encode or json_decode, so the classic ORM way is useless here I think because I have the business logic in pl/pgsql functions in the database, and php just validates and converts messages.

A: Be aware user and role are database reserved words.
Just ask the database what you want with one query:
SELECT
    my_user.id
    my_user.name,
    array_agg(my_role) AS "roles",
    array_agg(permission) AS "permissions"
FROM
  "user" my_user
    LEFT JOIN user_permission up ON my_user.id = up.user_id
    LEFT JOIN permission ON up.permission_id = permission.id
    LEFT JOIN user_role ur ON user.id = ur.user_id
    LEFT JOIN "role" my_role ON my_role.id = ur.role_id

And you will get an ugly string with dobule escaped objects into array. Using Pomm layer will allow you to use the converter to get arrays of PHP objects directly from this query.
<?php

public function getUserPermissionsAndRoles($user_id)
{
    $sql = <<<SQL
SELECT
    :user_fields_as_my_user,
    array_agg(my_role) AS "roles",
    array_agg(permission) AS "permissions"
FROM
  ":user_table" my_user
    LEFT JOIN :user_permission_table up ON my_user.id = up.user_id
    LEFT JOIN :permission_table ON up.permission_id = permission.id
    LEFT JOIN :user_role_table ur ON user.id = ur.user_id
    LEFT JOIN ":role_table" my_role ON my_role.id = ur.role_id
WHERE
  my_user.id = ?
SQL;

    $sql = strtr($sql, array(
        ":user_fields_as_my_user" => $this->formatFieldsWithAlias('getSelectFields', 'my_user'),
        ":user_table" => $this->getTableName(),
        ":user_permission_table" => $this->getConnection()->getMapFor("\Db\App\UserPermission")->getTableName(),
        ":permission_table" => $this->getConnection()->getMapFor("\Db\App\Permission")->getTableName(),
        ":user_role_table" => $this->getConnection()->getMapFor("\Db\App\UserRole")->getTableName(),
        ":role_table" => $this->getConnection()->getMapFor("\Db\App\Role")->getTableName(),
        ));

    return $this->query($sql, array($user_id));
}

In order to tell Pomm it can be converter of user_role, user_permission, permission and role database objects, when instanciating your database, add the following :

$database = new \Pomm\Database(array("dsn" => "pgsql://user:pass@host:port/db_name", "name" => "app"));

$cnct = $database->getConnection();

$database
    ->registerConverter('UserPermission', new \Pomm\Converter\PgEntity($cnct->getMapFor('\Db\App\UserPermission')), array('user_permission'))
    ->registerConverter('Permission', new \Pomm\Converter\PgEntity($cnct->getMapFor('\Db\App\Permission')), array('permission'))
    ->registerConverter('UserRole', new \Pomm\Converter\PgEntity($cnct->getMapFor('\Db\App\UserRole')), array('user_role'))
    ->registerConverter('Role', new \Pomm\Converter\PgEntity($cnct->getMapFor('\Db\App\Role')), array('role'))
    ;

Now the database knows theses converters can be used but the UserMap has to know when it parses a result field named "roles" and "permissions", it must handle them as an array of Role and Permission entities. This is what the initialize() function of the UserMap class stands for:
<?php

class UserMap extends \Db\App\Base\UserMap
{
    public function initialize()
    {
        parent::initialize();

        $this->addVirtualField('roles', 'Role[]');
        $this->addVirtualField('permissions', 'Permission[]');
    }

Now, in your controller you just have to call the model's method and cast it to a JSON response:
    <?php

    $database = new \Pomm\Database(array("dsn" => "pgsql://user:pass@host:port/db_name", "name" => "app"));

    $cnct = $database->getConnection();

    $database
        ->registerConverter('UserPermission', new \Pomm\Converter\PgEntity($cnct->getMapFor('\Db\App\UserPermission')), array('user_permission'))
        ->registerConverter('Permission', new \Pomm\Converter\PgEntity($cnct->getMapFor('\Db\App\Permission')), array('permission'))
        ->registerConverter('UserRole', new \Pomm\Converter\PgEntity($cnct->getMapFor('\Db\App\UserRole')), array('user_role'))
        ->registerConverter('Role', new \Pomm\Converter\PgEntity($cnct->getMapFor('\Db\App\Role')), array('role'))
        ;

    $collection = $cnct->getMapFor('\Db\App\User')
        ->getUserPermissionsAndRoles($_GET['user_id']) // <- get an iterator over fetched users with extra info.
        ;

    echo json_encode($collection->extract()); // <- dump an array from the iterator and cast it to json.

Of course the pure PHP part would be a way easier with a micro-framework like Silex where your controller would be:
<?php //...

$app->get('/user/{user_id}', function($user_id) use ($app) {
    $collection = $app['pomm.connection']
        ->getMapFor('\Db\App\User')
        ->getUserPermissionsAndRoles($user_id);

    if ($collection->count() === 0)
    {
        $this->abort(404, sprintf("No such user '%s'.", $user_id));
    }

    return $app->json($collection->extract());
});

Cheers !

A: It is possible: "Returning a nested composite type from a PL/pgSQL function", but since there is no built-in associative array in pgsql, it is way too hard to do the data conversion to php associative arrays and json after that. So I think the best solution here to read the data with multiple queries and build the associative array with php.
In pgsql 9.2 it would be easier to do this with json data type and a custom setof record -> json aggregate function. In 8.4 I don't think it is worth the effort...


Madrid is renowned for its bustling restaurants, but behind of every mouth-watering bite the city offer us, there is a hard-working food-producer.
Madrid is renowned for its chefs, restaurants and bustling markets, but as people feast upon delicious meals and carefully cultivated produce, they should spare a thought for the hard-working food-producers in the countryside. These skilled farmers have made a life out of producing quality food and are the true backbone of the city's thriving metropolitan food scene.
In a river basin that weaves through the mountains of Torremocha de Jarama, beekeeper Luis Escudero from honey brand La Abeja Meli is busy ensuring his colony of pint-sized workers are both happy and well-fed. Stemming from a family of honey producers with experience that stretches back three decades, Escudero opened his own colony of beehives in 2001 in one of Madrid's most unspoilt patches of countryside filled of rosemary, thyme and lavender. The pristine conditions provided by this high altitude oasis mean that the organic honey produced here is as pure as the surrounding mountain, bringing delight to breakfast tables in barrios around the city.
Miraflores Cheesery—that takes its nome from the near village located in La Sierra, is as committed to making quality cheese as it is to nurturing an allegiance to the local region. José Luis Santé leads a team of skilled cheese-makers who carefully forge wheels of queso from sheep's and goat's milk sourced from a farm in the Sierra de Guadarrama. Working closely with local farmers, distributors, and businesses, the Miraflores Cheesery strives to cultivate a more sustainable local approach to food production. Their range of specialty cheeses have a strong connection to the land in both nature and name.
A little closer to the city, on a small plain at the foot of the Madrid's stony Sierra mountain range, a family-run finca cultivates some of the region's finest cuts of meat. The verdant fields of the Sierra de Guadarrama provide the perfect feeding ground for Raúl de Lema Turégano's prized cattle. His family has been toiling this land for four generations—Raúl continues in the footsteps of his great-grandfather Hilario, and created the official brand in 1999. With his homestead perched just a few paces from the grazing animals, the proud farmer and butcher is perfectly poised to nurture his herd of cattle around the clock producing some of the most prized pick of customers perusing gastro-markets in the city centre.
So, as you sit down for a meal in one of the Madrid's inner-city eateries; marvel at a mouth-watering morsel in a colourful market, or enjoy some cured cheese with your wine as you soak up the festive air of a taberna—pause for a moment and pay your respects to the tireless army living beyond the city limits, quietly toiling the land to make every mouth-watering bite and delicious sip possible.
Sometimes all we need to balance the gruelling monotony of the daily grind is one grand unbridled party and celebrate everything worth celebrating.
A city is an infinite patchwork of people and places. The perfect city fertilises this garden so that businesses can grow and its citizens can flourish.
I open the window and a salty breeze caresses my face. I leave the house, accompanied by the rustling sound of the sand under my feet and the swaying leaves of the palm tree.
Commissioned by his sovereign, Francisco de Goya restored country life to its rightful position, which had been—and should continue to be, the king's residence.
Achieving a harmonious integration of the urban, rural and natural spheres into a continuous whole, avoiding landscape fragmentation … That is the real point.
The consumption of local produce is a sound environmental practice that providing us with quality food and ensuring sustainable production models.
Let's take a look at the faces and hands of those who farm the land and produce our food—we'll immediately understand a life filled with sacrifice.
The countryside is a scenario from which we may defend the return to the origins of total artists, their constant exploration of the land as a place from which to create.


The Aramith Crown Standard Ball Set features the quality you'd expect from Aramith balls at a lower price. This set features 15 regulation pool balls and an Aramith cue ball. Aramith phenolic balls last up to five times longer than other balls made from polymers or polyester. You're sure to get your money's worth out of this ball set.
This graphic first created by GameWorld Planet for customer use only. No reproduction or copy for commercial use permitted.


Saint Blaise's feast day is celebrated on February 3rd. In the morning I set up our St. Blaise peg doll and we read about St. Blaise during our Couch Catechism time from Saints for Young Readers for Every Day, Vol. 1. The St. Blaise prayer card was given to my 3 year old during her Good Shepherd class that she takes at our parish.
St. Blaise lived in the 4th century and it's a tradition to have your throat blessed on his feast day, but why?
When Blaise was sent to prison to be beheaded because he was a persecuted Christian, a poor mother rushed up to him and begged him to save her child who was chocking to death on a fishbone. The saint said a prayer quietly and blessed the child and the child's life was spared through this miracle. So on St. Blaise's feast day throats are blessed by the priest as he crosses two candles because he is the patron of those with throat diseases.
"Through the intercession of Blessed Blaise, Bishop and Martyr, may you be delivered from all illnesses of the throat and any other ailments. In the Name of the Father, and of the Son, and of the Holy Spirit. Amen."
The Church is decorated with red linens and the priest wears red vestments because St. Blaise was a martyr for the faith. His death was gruesome to say the least. From Franciscan Media: Agricolaus, governor of Cappadocia, tried to persuade Blaise to sacrifice to pagan idols. The first time Blaise refused, he was beaten. The next time he was suspended from a tree and his flesh torn with iron combs or rakes that are used to comb wool (and why he is the patron of not only those with throat ailments, but also English wool combers.) Finally, he was beheaded.


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NEWSLETTER: Advertising, Marketing & Promotions, Summer 2010
Dear Clients, Colleagues and Friends,
We hope that you are keeping cool with the incredible temperatures outside as well as the new regulatory restrictions circulating around the industry. Indeed, there have been a number of important developments in the advertising and promotional marketing legal fronts since our last newsletter, including potential new substantiation requirements for health products, enhanced privacy restrictions, new restrictions on negative marketing and telemarketing, and mobile marketing. These and other timely topics are discussed in this newsletter.
As always, if you would like to discuss any of these developments, have concerns about their impact on your business or marketing campaign, or have any questions about the legal aspects of advertising and promotional marketing, please feel free to give us a call.
FTC Debt Settlement Telemarketing Restrictions to Take Effect in September and October 2010
FTC Expands Claim Substantiation Requirements in Two Settlements
Am I Liable for a Blogger's Review of My Products?
FTC Issues FAQs on Endorsement and Testimonial Guides
FTC Considers "Do Not Track" Registry for Behavioral Activity
FTC in Ongoing Review of COPPA
Senator Rockefeller Introduces Legislation Banning Certain Internet Sales Practices
FTC Expansion Omitted in Final Wall Street Reform Bill
New Legislation Targets "Cramming"
Telemarketer Scores Legal Victory in New York Lawsuit
FTC Stops Robocalling Operation
Rhode Island to Regulate Discount Buying Clubs
FTC Extends Deferral of Red Flags Enforcement - December 31, 2010
T-Mobile Florida Attorney General Mobile Content Settlement Announced
Why is the Mobile Industry So Susceptible to Shakedowns?
The Mobile Marketer Article Americans Want to Move Toward an Opt-in Paradigm
Overcoming Legal Hurdles of Mobile Marketing
Class Action Lawsuit Over New England Patriots' Cheating Is Dismissed
OLSHAN BUZZ
Chambers and Partners USA Names Andrew B. Lustigman for 3rd Consecutive Year as Notable Practitioner in the Field of Advertising Law: Transaction & Regulatory
Andrew Lustigman Joined a Panel of Experts Discussing the Latest Legal Developments in Mobile Marketing Law
Meet us at DMA2010 in San Francisco
Attorneys from Olshan will be attending DMA2010 conference from October 10-12, 2010 in San Francisco. The show is always a great opportunity to network and hear the latest developments. If you are planning on attending, please let us know so that we can catch up in person!
New and Improved SweepstakesLaw.com Site Launched
We are proud to announce the re-launching of our popular website sweepstakeslaw.com. Check it out to learn the latest developments in Internet sweepstakes and contests promotions, social media promotions and sweepstakes basics.
Social Media Law Roundtable: Los Angeles, September 21, 2010
The ABA Special Committee of Intellectual Property Law Special Committee on Promotion and Marketing Law, chaired by Andrew Lustigman, invites you to attend a complimentary Social Media Law Roundtable to be held on September 21, 2010 at 6pm at the UCLA Anderson School of Management. Panelists include: Tammy Brandt, Toyota Motor Sales, USA, Bakari Brock, Twitter, Laura Berger, Federal Trade Commission, and Professor Sanjay Sood, UCLA. The conference is co-chaired by Tanya Forsheit of Infolawgroup LLP and Alan Friel of Wildman Harold.
Sweepstakes and Contests: All you Need to Know for Legal Compliance
In the Global Business & Legal Seminars continuing legal education (CLE) seminar "Sweepstakes and Contests: All You Need To Know for Legal Compliance" Andrew B. Lustigman and Adam Z. Solomon of Olshan, discuss how to structure a legally compliant sweepstakes promotion. Some of the issues covered are how a compliant promotion differs from a lottery, eligibility considerations, drafting the rules, awarding prizes and selecting winners. They also discuss skill contests, essay contest rules, user-generated content, copyright issues (including the Digital Millenium Copyright Act) and the Communications Decency Act. This audio CLE program is on line at here.
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The Federal Trade Commission has announced sweeping amendments to the Telemarketing Sales Rule that significantly restrict the marketing of what it broadly characterizes as "debt settlement" programs. The restrictions include an advance fee ban and mandatory disclosures. As part of its announcement, the FTC has published a comprehensive explanation - Debt Relief Services & The Telemarketing Sales Rule: A Guide For Business.
In two recent settlements announced the same day, the Federal Trade Commission expanded the type of substantiation required for a marketer making certain health claims. In those instances, the FTC requires that the claims be backed by at least two adequate and well-controlled human clinical studies. Moreover, under those settlements, certain disease or illness claims must now first be pre-approved by the Food and Drug Administration.
Why should companies care about claims being made about their products in social media? As Andrew Lustigman recounts in this ABA feature, companies should care because there is potential liability surrounding those third-party claims.
The FTC recently issued an FAQ for business: The FTC's Revised Endorsement Guides: What People are Asking. The FAQs are intended to provide further clarity and explanation as to compliance with the revised Endorsement Guides that went into effect on December 1, 2009. The FAQs devote provide a significant emphasis on social media, blogs, Twitter, as well as the use of hashtags and other button-type disclaimers. Importantly, FTC states that merely providing a button that states "DISCLOSURE, LEGAL, or something like that" would not be sufficient disclosure. Instead, the FTC has opined that marketers should provide the disclosure as part of the message.
At a recent Congressional hearing, Federal Trade Commission Chairman Jon D. Leibowitz said that the agency is considering assessing the viability of a do-not-track list for online advertising modeled on the national "do not call" list utilized in telemarketing. Details on the proposal are scare. However, the FTC plans to release a privacy report this fall which may include the agency's conclusions on such a program.
As we had first mentioned in April 2010 after our ABA Roundtable on Social Media Law, the FTC is conducting a review of its rules under the Children's Online Privacy Protection Act of 1998 ("COPPA"), to determine whether changes to technology since its adoption warrant changes to the rule. Under COPPA and the FTC's rules, Web sites may not knowingly collect personally identifiable information from children under the age of 13 without verifiable parental consent.
Since the adoption of the FTC's rules, the expansion of communication technologies (to "mobile communications, interactive television, interactive gaming, and similar activities", as the request for comments states) and increased adoption of such technologies by children (many of whom now regularly carry and use smartphones) have raised concerns about whether the COPPA rules give sufficient guidance to both parents and companies seeking to comply with them. The issues include both practical (how to request and receive verifiable parental consent) and technical (is a text message considered a "Web site"), and the overall concerns about how best to preserve children's privacy while enabling companies to communicate with (and market to) them.
On May 19, 2010, Senator Rockefeller (D-W. Va.), head of the Senate Commerce Committee, introduced draft legislation that would ban certain internet sales tactics and restrict others. The legislative effort comes after a hearing investigation where Senator Rockefeller and other members of the committee determined that certain internet industry practices were unfair and deceptive. The legislation is cited as the "Restore Online Shoppers' Confidence Act."
Foremost among those activities deemed unfair and deceptive was Data Pass. The draft legislation defines "Data Pass" as the practice of an online retailer or website sharing their customers' billing information, including credit card and debit numbers, with third party sellers in exchange for "bounties" and other payments. The legislation would prohibit companies from enrolling consumers in paid-subscription programs unless the consumers entered their credit card numbers and agreed to the program. The measure would also restrict online companies' ability to use "negative options" which often involves offering people free trials and then charging them unless they cancel their accounts within a short time frame.
The major pending financial regulatory reform bill (the Dodd-Frank Bill (HR 4173/S3217), was finalized without providing the FTC with additional enforcement and rulemaking powers. As we had previously written, the House version of the bill included provisions granting the FTC greater discretion in initiating rulemaking, permitting the FTC the right to impose civil monetary penalties without the involvement of the Department of Justice, and increasing the exposure for suppliers to marketers.
Following on the heels of increased class action activity, state legislatures are increasing the focus on attacking allegations of cramming. Virginia recently enacted a law effective July 1, 2010 requiring all authorizations for third-party charges to be verified by an independent third party. Maryland recently passed a law effective October 1, 2010 that requires "express authorization" for third-party services. New York is considering legislation that would require "valid proof" of authorization of third-party services. These laws combined with private enforcement, are increasing a service provider's obligation to ensure that authorization is valid and properly verified.
A recent federal court ruling in the Southern District of New York was a blow to plaintiffs who seek to sue telemarketers under the Telephone Consumer Protection Act ("TCPA"). In the case of Leyse v. Bank of America, a June 14th ruling indicated only the "called party" can assert claims under the TCPA. The facts of the case were as follows: in 2005, a telemarketing call containing a pre-recorded message was made to a Ms. Dutriaux, who at the time shared an apartment with a Mr. Leyse. The roommate, Leyse, happened to answer the call and claimed his privacy was disturbed by the call. Both Dutriaux and Leyse filed separate class-action lawsuits under the TCPA. Dutriaux filed in New York, where her case was dismissed in 2008 because New York has a rule that does not allow class actions in TCPA cases. Leyse filed his case in North Carolina, and it was transferred to New York after some legal wrangling. The New York court threw out Leyse's case because he was not the intended recipient of the call or the "called party". The court stated, "the uncontroverted evidence shows that DialAmerica, the entity that placed the call on behalf of Bank of America, placed the call to Dutriaux, Leyse's roommate and the telephone subscriber. DialAmerica's records demonstrate that it associated the phone number with Dutriaux, not with Leyse. To the extent that Leyse picked up the phone, he was an unintended and incidental recipient of the call." In light of this ruling, any telemarketer threatened with a TCPA lawsuit should immediately check to see who was the intended recipient of the call in question.
The Federal Trade Commission obtained a temporary restraining order against a telemarketing operation that allegedly made millions of illegal phone calls pitching extended auto warranties and credit card interest rate-reduction programs.
Effective June 12, 2010, Rhode Island has enacted the Discount Buying Clubs Act (2009 RI HB 7597). The law requires Discount Buying Clubs to have a signed written contract with the consumer. The contract will be required to disclose certain terms and provide for a 3-day right of cancellation.
The Federal Trade Commission announced that it is further extending its deferral of enforcement of the Identity Theft Red Flags Rule through December 31, 2010, while Congress continues to consider legislation that would impact the scope of entities covered by the Rule.
Florida Attorney General Bill McCollum has reached an agreement with T-Mobile concerning unauthorized billing for third-party charges on consumers' mobile phone bills. The settlement is the third in a continuing series of enforcement actions against mobile carriers relating to the charges assessed by third-party mobile content providers on customer's phone bills. The settlement requires T-Mobile to enforce what the state has determined to be various best practices regarding the advertising and marketing of content provider services. Read the Mobile Marketer article here.
The Mobile Marketer article "Why is the mobile industry so susceptible to shakedowns? discusses some of the reasons why the mobile industry is facing so many different kinds of lawsuits, ranging from patent infringement to consumer protection issues.
This article summarizes "Mobile Media and Law: Legal Dos and Don'ts" seminar at the Mobile Marketing and Media 2010 conference on June 30, 2010 at the Direct Marketing Association in New York, in which Andrew Lustigman participated. The panel discussed the fact that mobile marketing messages require express opt-in, as well as the proposed federal Boucher-Stearns privacy legislation.
Mobile marketing is different from any other form of marketing because it requires express consent, or opt-in, from consumers to be able to contact them. Other forms of marketing such as email or telemarketing only require that consumers have the option to opt out. This MobiMatter article "Overcoming legal hurdles of mobile marketing", discusses the greatest legal compliance challenges in this new and developing marketing channel.
A class-action lawsuit filed by a football fan angered over one team's documented cheating has been dismissed on the basis that the fans did not suffer any loss by watching an NFL game that was not played according to the rules. The lawsuit was filed by a New York Jets season ticket-holder over the "Spygate" videotaping scandal of 2007, which arose when the New England Patriots were caught secretly videotaping the sideline signals of the Jets and relaying those signals down to the sidelines. The fan, Carl Mayer, sued the Patriots, the team's head coach, Bill Belichick, and the NFL for over $100 million because, on September 9, 2007, an agent of the Patriots videotaped the Jets coaches and players on the field in order to illegally capture and steal the Jets' signals and visual coaching instructions to their players. The Patriots were subsequently found by the NFL to have engaged in improper conduct.
Sheldon Lustigman to Speak at ACI's 4th Annual Focus on Sweepstakes Contests and Promotions
On Friday, September 24, 2010 Sheldon Lustigman will present along with Leonard Gordon, Northeast Regional Director of the Federal Trade Commission at ACI's 4th Annual Focus on Sweepstakes, Contests and Promotions. Sheldon will be speaking at the session entitled: "Adapting to Changing Government Enforcement Priorities: Complying with New Sweepstakes Legislation and FTC Regulatory Requirements."
The session will cover the following topics:
Sweepstakes and contest enforcement activities
Applying the FTC's Endorsement Guidelines and CAN-SPAM rules to sweepstakes, contest, and promotions
Balancing the company's information storage proactives with state data breach notification laws
Organizing legally-compliant sweepstakes when dealing with 50 different state laws
Aligning in-house practices with changing (and at times conflicting) laws
Bonding and registration requirements
We strive to stay on top of all relevant legal issues to provide our clients with the most effective and efficient legal advice. If you find any legal marketing news of interest, send it to us!
Please pass this newsletter on to your friends and colleagues and invite them to join our mailing list. As always, please feel free to contact us with any comments, questions or recommendations. Also, please follow us on http://twitter.com/advlaw.
Sheldon S. Lustigman
Andrew B. Lustigman
Scott A. Shaffer
Adam Z. Solomon
Jonathan I. Ezor
This newsletter is not, nor is it intended to be, legal advice. You should consult an attorney for advice regarding your individual situatio
n. We invite you to contact us and welcome your calls, letters and electronic mail. Contacting us does not create an attorney-client relationship. Please do not send any confidential information to us until such time as an attorney-client relationship has been established.
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There was a time not so long ago when childhood offered the space and freedom for children to explore their world. To be curious. To be creative. To be bold. To be bored. To just . . . be. Today, our overscheduled and overstimulated children are using their imaginations less than ever. It's a crisis in the making. And it's why we're committed to making products that inspire open-ended thinking and encourage kids to see new possibilities. It's part of our bigger vision to Take Back Childhood. Because by giving children a true childhood, we give them a path to realizing their full potential!
It has become increasingly clear over the years that it's not enough to simply make engaging toys. We feel it is our obligation to change the world's way of thinking about childhood and play. Free time, those precious moments when kids have the freedom to create something from nothing and to follow their curiosity, is in short supply in today's world. Real learning and growth happen when kids are given the unstructured time and space to explore the outer reaches of their imaginations. To turn the tide, we're starting a movement to fight for free time and all the countless ways to enjoy it! It's time to Take Back Childhood.
Plan for Play. Put free time on your calendar just like you'd pencil in soccer practice.
Let Kids Lead. Your role is to guide and facilitate, not direct or dictate.
Keep It Free. The best play, discovery, and creation truly begins with nothing but imagination.
Check our blog every month to read our co-founder Melissa's take on play, imagination, and childhood. Share your thoughts and pictures on how your family is embracing the movement, at #TakeBackChildhood.
As part of our mission to Take Back Childhood, we partnered with Gallup for a study on parents' perspectives on childhood play. Experts agree that child-led play is essential to development, but do parents recognize and value the benefits of this kind of play? Go here to read more about what we learned and to find links to ways you can make time for play in your child's life!
It's when we're bored and have to dig deep to find ways to fill our time that imagination is born.
Without play, children will develop without imagination and creativity.
Free play is an essential part of childhood. All children should be afforded ample, unscheduled, independent, nonscreen time to be creative, to reflect, and to decompress.
Kids who are just passively stimulated by a glowing screen don't have to do the neural heavy lifting to create those images... thus stunting their own creative abilities.
It's through play that kids discover interests, passions, and talents. We want our kids to be happy and, ideally, building a career when they grow up that's true to their core selves. Figuring out who they truly are and what they love to do begins with play.
Given time and space and removed from distractions, kids gravitate to creativity and creative thinking. When they are able to let their minds and bodies roam, they can come up with imaginative solutions to problems.
Kids at play have the freedom to try out their ideas, test them, and try again. It's a safe space for trial and error.
Playful interactions - with friends and grownups alike - help kids navigate social terrain, negotiate terms, define norms, develop empathy, and so much more.
Play is important for healthy brain development as it is how children learn to master the world around them. It's a critical tool for language acquisition and also a way kids develop new competencies that build confidence.


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zohaib
design writing about
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Empower Your Datacenter
Growing up, I played video games all the time. The first video games I remember playing were Super Mario Bros and Teenage Mutant Ninja Turtles on the original NES. When I was a bit older, my parents finally let me play fighting games (after I begged incessantly). In middle school and high school, I entered the world of RPGs, and I pretty much never left.
The point is: I love video games, so designing the shirt for NEC's attendance at SpiceWorld was a blast!
The TurboGrafx-16 was NEC's home gaming system that released back in 1987. SpiceWorld attendees are generally fans of all things tech, so we thought it would be fun to invoke a bit of nostalgia by building a theme around the game console. Our presentations and talks would include references and imagery from various games such as Fighting Street (the home version of the iconic Street Fighter) and Dragon Spirit. What's more is we were able to setup a fully functioning TurboGrafx-16 at our booth for visitors to play.
Designing for Myself
I've found that the best approach to designing a t-shirt is to make something you would actually want to wear. I wanted to design a shirt that I would get excited about. We were used to requestors providing us a laundry list of requirements, often leading to a design-by-committee situation. This time, however, things were different. We were given only three requirements:
Messaging: Empower Your Datacenter
Inspired by Fighting Street and/or Dragon Spirit
One-color screen printed
The result was a shirt attendees were rushing to NEC's booth to get their hands on.
Check out more of my work!
© Copyright 2020 Zohaib Waliany & Respective Owners