text:
\section{Introduction}

The aim of this paper is to study possible connections between two
perspectives on the glass transition \cite{Reviews}.  One is the
mosaic, or entropic droplet, view that follows from the random
first-order transition theory of Kirkpatrick, Thirumalai and Wolynes
(KTW) \cite{KTW,Xia-Wolynes}: a deeply supercooled liquid is viewed as
a patchwork of correlated mesoscopic regions which relax by
entropically driven nucleation events.  These ``mosaics'' are
characterised by a length scale that diverges on approach to a
temperature $T_K$, where a Kauzmann \cite{Kauzmann,Rome} entropy
crisis occurs.  The growth of static correlations, under certain
assumptions for droplet nucleation, leads to a divergence of
relaxation times of the Vogel-Fulcher kind \cite{Reviews} in three
dimensions.

The second, in principle very different, approach is based on the idea
of dynamic facilitation \cite{Fredrickson-Andersen,Palmer-et-al}:
glassiness is not due to any precipitous thermodynamics, but is a
consequence of effective constraints in the dynamics.  Here, the
central feature is dynamic heterogeneity \cite{DH,Other-theories}, and the
corresponding dynamic scaling, i.e., growing times are accompanied by
growing dynamic, but not necessarily static, correlation lengths
\cite{Garrahan-Chandler}.  In this view there are no finite
temperature singularities, and scaling properties are controlled by
zero temperature critical points where dynamic lengths and times
diverge \cite{Whitelam-et-al}.  The simplest models that realize this
perspective are kinetically constrained lattice models, such as the
facilitated spin models of Fredrickson and Andersen (FA) and Jackle's
East model \cite{Fredrickson-Andersen,East,Ritort-Sollich}.

Here we study lattice spin models with plaquette interactions
\cite{Lipowski,Newman-Moore,Garrahan-Newman,Garrahan} from the mosaic
or entropic droplet perspective.  The two models we consider are the
square plaquette model (SPM) \cite{Lipowski,Garrahan} and the
triangular plaquette model (TPM) \cite{Newman-Moore,Garrahan-Newman}.
These models have exact dual descriptions: one in terms of interacting
spin variables with standard single-spin Glauber dynamics; another in
terms of free excitations with dynamics subject to kinetic
constraints.  The dynamics of the SPM is like that of the
facilitated FA model \cite{Garrahan}, the dynamics of the TPM, like
that of the East model \cite{Garrahan-Newman}.

While plaquette models are realizations of the dynamic facilitation
scenario, they also allow for a detailed and explicit
analysis using the mosaic approach.  They therefore allow a direct
analysis of the similarities and differences between the facilitation and
the mosaic perspectives on glasses.  In this work, we apply to these
models the procedure recently suggested by Bouchaud and Biroli (BB)
\cite{Bouchaud-Biroli}, by which the mosaic lengthscale is estimated
from the partition function of finite droplets within a larger system.
We show that low temperature states do resemble mosaic states, and
that the associated lengths can be extracted using the BB procedure.
We also show that these thermodynamic lengthscales have the same
scaling as those extracted from bulk many-point static correlations,
and from multi-point dynamical correlation functions which measure
dynamic heterogeneity.  In the case of the SPM, the static analysis
gives two typical caging lengths, a feature not anticipated in BB.
Moreover, the relation between caging lengths and relaxation times in
these models is not that expected from the mosaic approach, as the
nucleation assumptions miss the relevant transition states and
overestimate the free-energy barriers to droplet melting.

The paper is organized as follows.  In section II we give a summary of
the arguments  of KTW and BB,  and make general  observations about the
applicability of this approach to the models studied in this work.  In
section III we apply the mosaic procedure to the SPM, and compare the
lengthscales obtained  with other relevant static  and dynamic lengths
of the  system.  In section  IV we repeat  the analysis for  the TPM.
Finally, in section V we give our conclusions.

\section{Background}
\label{sec:BB}

\subsection{Mosaics and their lengthscales}

The basic arguments underlying the BB procedure to compute a mosaic
lengthscale are the following \cite{KTW,Bouchaud-Biroli}.

Imagine a glassy state in which an atom is prevented from moving by
the fact that its neighbours are fixed in position. This picture
is consistent only insofar as the neighbours of the original
atom are fixed by their neighbours, and so on.
Consider now the original atom and its neighbours as a small droplet.
One can compute
the probability that the droplet can relax, assuming that its
boundaries are fixed.  If this probability is small then
rearrangements require correlated motion over distances larger than
this droplet.  One must then increase the droplet size and recalculate
the probability of rearrangement.
In a system with finite-ranged interactions, as the droplet is
made larger, we eventually cross over to a regime where the
rearrangement probability is large.  All larger droplets will be then
able to rearrange by sequential moves of this type.

This crossover defines the mosaic lengthscale
\cite{KTW,Bouchaud-Biroli}.  The result is a picture of a glassy state
in which droplets of small sizes are jammed, or frozen, by their
boundaries, whereas those with large sizes are unjammed, or melted. As
the temperature decreases, the typical lengthscale separating jammed
and unjammed droplets increases.  (The idea of using finite size
scaling to extract a lengthscale is not at all new, of course.  See
Refs.~\cite{Scheidler-et-al,Berthier} for applications to glass
formers.)

The above remarks can be made quantitative by way of three assumptions
\cite{KTW,Bouchaud-Biroli}:

(i) A droplet of size $\xi$ contains very many metastable states whose
 number scales as $e^{s_c(T) \xi^d}$, where $s_c$ is the
 configurational entropy density which vanishes at the Kauzmann
 temperature, $T_K$.

(ii) The probability of finding a droplet in a given state $i$, given
that its boundary conditions are in the state $\alpha$ is
\begin{equation}
p_{i\alpha} = \mathcal{Z}^{-1} e^{-\beta \left( \xi^d f_i +
\xi^{\theta}\Upsilon_{i\alpha} \right) } ,
\end{equation}
where $\mathcal{Z}$ is a partition sum, $f_i$ is the free energy
density of state $i$ and $\xi^{\theta}\Upsilon_{i\alpha}$ is the free
energy cost of matching the bulk state $i$ to the boundary state
$\alpha$.  The exponent $\theta$ should satisfy $0\leq\theta<d$ in a
short ranged system.

(iii) Moves involving co-operative motion over a lengthscale $\xi$
occur on a timescale
\begin{equation}
\tau \sim e^{\beta \Delta \xi^\psi} ,
\label{equ:tau_psi}
\end{equation}
where $\Delta$ is a microscopic energy scale, and $\psi$ is some
exponent.  If we assume that these moves involve nucleation
events of one state in a background of an uncorrelated state then
the exponent $\psi$ is expected to obey $\psi\geq\theta$.

The lengthscale of the mosaic state can be extracted by comparing the
probability of a droplet state satisfying a frozen boundary condition,
with the total probability of all other droplet states with the same
frozen boundary \cite{Bouchaud-Biroli}.  That is, choosing a state $i$
such that $\Upsilon_{i\alpha}=0$, we define the crossover length
$\xi_*$ by
\begin{equation}
e^{-\beta f_i \xi_*^d} = \sum_{j\neq i} e^{-\beta \left( \xi_*^d f_j +
\xi_*^{\theta}\Upsilon_{j\alpha} \right) } .
\label{equ:mos_len}
\end{equation}
The right hand side is a sum over $e^{s_c \xi_*^d}$ terms, so we can
approximate it by
\begin{equation}
e^{-\beta f_i \xi_*^d} \simeq e^{- \left\{ \xi_*^d \left[ \beta f(T) -
   s_c(T) \right] + \xi_*^{\theta} \beta\Upsilon(T) \right\} } ,
\label{equ:mos_len_typ}
\end{equation}
where the sum over states $j$ was replaced with the typical number of
terms in the sum, $e^{\xi^d s_c(T)}$, multiplied by the typical weight
$e^{-\beta [\xi^d f(T)+ \xi^\theta\Upsilon(T)]}$.  We have included
explicitly the temperature dependence of $f$, $s_c$ and $\Upsilon$, as
they have been averaged over internal configurations of the droplet,
subject to the boundary condition $\alpha$.  As long as the free
energy of the well-matched state is typical, $f_i \simeq f(T)$, the
mosaic lengthscale is
\begin{equation}
\xi_* \sim \left( \frac{\beta \Upsilon}{s_c}
\right)^{\frac{1}{d-\theta}} .
\end{equation}
The configurational energy of the droplet is thought to vanish at some
finite $T_K$, which leads to the divergence of $\xi_*$ at that
temperature \cite{KTW}.

Combining these thermodynamical arguments, with assumption (iii) above
we get the typical relaxation time for a droplet of size $\xi_*$,
\begin{equation}
\tau \sim \exp(\beta \Delta \xi_*^\psi) .
\end{equation}
Droplets of length smaller than $\xi_*$ will not relax at all in this
timescale.  Droplets of length larger than $\xi_*$ relax by
combinations of moves over lengthscales of the order of $\xi_*$ in
times of the order of $\tau$. Hence, $\tau$ gives the typical rate of
rearrangements in the system \cite{xi-variable}.

\subsection{Plaquette models}
\label{sec:models}

The BB procedure outlined in the previous subsection is extremely
general.  However, analysing the partition function of any model in
sufficient detail to evaluate the length $\xi_*$, as defined in
Eq.~(\ref{equ:mos_len}), is rather difficult.  In the rest of this
work we apply the procedure to two models of Ising spins with
plaquette interactions in two dimensions.

We will define the models in the next section, but we first make some
introductory remarks.  The first model is the square plaquette model
(SPM), in which spins are defined on a square lattice, and
interactions involve the quartets of spins that form the square
plaquettes of the lattice \cite{Lipowski,Garrahan}.  In the second
model, the spins occupy the sites of a triangular lattice, and
interactions are between triplets of spins on upward pointing
triangular plaquettes \cite{Newman-Moore,Garrahan-Newman}.  We refer
to it as the triangular plaquette model (TPM).  Both models have dual
representations \cite{Garrahan-Newman,Garrahan}: in a spin
representation they describe interacting spins with simple dynamical
rules (single spin-flips); in a ``defect'' representation they
describe thermodynamically independent excitations with more
complicated dynamical rules.  In this latter description they resemble
the kinetically constrained models advocated as effective models for
the glass transition in the dynamic facilitation approach
\cite{Garrahan-Chandler}.  In this sense, they interpolate between
models with microscopic degrees of freedom, and more phenomenological
models of ``mobility fields''.

The SPM and TPM have trivial thermodynamics: the excitations in the
models are localized and statically non-interacting, so the only
thermodynamic singularity is at zero temperature. In terms of our
assumption (i), above, this means that we must have $T_K=0$. It also
has implications on the assumptions of (ii) and (iii) about the
surface tension and barrier exponents $\theta$ and $\psi$.  Below we
show that the BB procedure can still be applied to extract caging
lengthscales from finite-size static properties.  However, while
relaxation times grow with growing mosaic lengths, the quantitative
dependence of assumption (iii) does not hold in the SPM and TPM, and
these relations need to be generalized.

\section{Square Plaquette Model}
\label{sec:pq}

The SPM is defined by the Hamiltonian \cite{Lipowski,Garrahan}:
\begin{equation}
H = -(1/2)\sum_{xy} \left( \sigma_{xy} \sigma_{x,y+1} \sigma_{x+1,y}
\sigma_{x+1,y+1} -1 \right) ,
\end{equation}
where the $\{\sigma_{xy}\}$ are Ising spins, $\sigma=\pm 1$, and
$(x,y)$ indicates position on a square lattice.  The dynamical moves
are single spin-flips with Glauber rates.  If we write
\begin{equation}
p_{xy} = \sigma_{xy} \sigma_{x,y+1} \sigma_{x+1,y} \sigma_{x+1,y+1} ,
\end{equation}
then the partition function is simply a sum over the non-interacting
variables $p_{xy}$ (up to a non-extensive set of constraints on the
$p_{xy}$ from the boundary conditions on the spins). The variables
$p_{xy}$ are defined on the plaquettes of the square lattice, which
forms a dual square lattice, see Fig.\ \ref{fig:plaq_sketch}.
Plaquettes with $p_{xy}=-1$ cost an energy of unity, and are sparse at
low temperatures: we refer to them as defects.

\begin{figure} \begin{center}
\epsfig{file=sketch.eps,width=0.7\columnwidth}
\end{center}
\caption{Sketch showing spins (up or down) on vertices of a square
lattice, and defect variables (circled or blank) on the dual lattice
formed by the square plaquettes.  }
\label{fig:plaq_sketch}
\end{figure}

The plaquette model has activated dynamics at low temperatures. That
is, there are very few (if any) possible transitions out of a typical
state into states with lower energy. Thus, low temperature states are
very close to inherent structures (on quenching to zero temperature, a
few local relaxations will take place, quickly leading to a state with
no available transitions). Thus it is an ideal model to investigate
the mosaic hypothesis: there are many metastable states, and these can
be easily identified. (In contrast, the statistics of the free energy
minima of an atomistic liquid or a spin glass are much harder to
probe).

The trivial bulk thermodynamics of the SPM are affected by freezing
the boundary spins of a finite droplet.  Consider the partition
function of a finite system of $L^2$ spins (assumed square for
convenience), with $(4L-4)$ frozen spins along the boundary.  The
state of the system is then defined by the configuration of the
$(L-2)^2$ remaining (bulk) spins, and its energy is given by (half of)
the sum of the $(L-1)^2$ defect variables, $p_{xy}$.  We see
immediately that there are more configurations of the plaquettes than
of the bulk spins: a partition sum over the bulk spins does not
contain all configurations of the plaquette variables.  Rather, there
are
\begin{equation}
n_\mathrm{plaqs} - n_\mathrm{spins} = (L-1)^2 - (L-2)^2 = 2L-3 ,
\end{equation}
constraints on the possible arrangements of the defects.

To understand the origin of these constraints, observe that flipping
any of the $(L-2)^2$ free spins preserves the parity of the number of
defects in any row or column of the square (dual) lattice. Therefore
the $2L-3$ constraints select the parity of the $L-1$ rows and $L-1$
columns of the dual lattice.  [Fixing the parities of all the rows
sets the parity of the total number of defects. Thus there are only
$L-2$ independent column parities, the final one being fixed by the
parity of the total number of defects.  The result is that there are
$2L-3$ independent constraints on the plaquettes, as required.]  These
arguments are illustrated in Fig.\ \ref{fig:pq_bulk_boundary}.

\begin{figure} \begin{center}
\epsfig{file=sketch_bdy.eps,width=0.7\columnwidth}
\end{center}
\caption{A droplet of size $L=7$. There are $(4L-4)=24$ frozen boundary
spins (black circles) and $(L-2)^2=25$ free bulk spins (white circles).
The energy is determined by the state of the $36$ defect variables that
sit on the dual lattice. This energy can be divided into a boundary
contribution that comes from the sites of the dual lattice marked
with $+$, and a bulk contribution from the remaining sites.
The partition sum is over the $2^{25}$ configurations of the bulk spins;
these correspond to $2^{25}$ of the $2^{36}$ configurations of the defect
variables. A given configuration of the defect variables contributes to
the partition sum if and only if it satisfies the $9$
independent constraints set by the
frozen boundary spins. For example, one such constraint is that the
parity of the number of defects in the third row of the dual
lattice is the same
as the frozen spin combination $\sigma_1 \sigma_2 \sigma_3 \sigma_4$,
where the spins $\sigma_{1-4}$ are identified in the figure. }
\label{fig:pq_bulk_boundary}
\end{figure}

If we separate the energy of a configuration into a part coming from the
$(L-3)^2$ plaquettes in the bulk, and the $4L-8$ plaquettes along the
boundary then we may write the partition function for all $L^2$ spins
as
\begin{equation}
\mathcal{Z} = \sum_{i\alpha} z_{i\alpha}, \;\;\; \qquad z_{i\alpha} \equiv
e^{-\beta (n_i + n_{i\alpha})} ,
\label{equ:zia}
\end{equation}
where $i$ labels the configuration of the bulk spins and $\alpha$ that
of the boundary. The integer $n_i$ is the number of bulk plaquettes
that are excited, and $n_{i\alpha}$ is the number of boundary
plaquettes that are excited. Making contact with section~\ref{sec:BB},
we identify $n_i$ with the free energy $f_i$ and $n_{i\alpha}$ with
the surface energy $\Upsilon_{i\alpha}$.  Note that we have replaced
the sum in (\ref{equ:mos_len}) over inherent structures by a sum over
states, since the typical droplets under consideration cannot reduce
their energy by any single spin-flip.

The partition function in (\ref{equ:zia}) can be written in terms of
partial sums over boundary conditions $\alpha$,
\begin{equation}
\mathcal{Z} = \sum_{\alpha} Z_\alpha, \;\;\; Z_\alpha \equiv \sum_i
z_{i\alpha} .
\end{equation}
Now imagine choosing a droplet in an infinite system.  In doing so we
find that its boundary state is $\alpha$.  The droplet lengthscale is
calculated by considering the contribution of that droplet to
$Z_\alpha$.   In other words, we need to estimate the expectation value
of $z_{i\alpha}/Z_\alpha$ as a function of droplet size.  This
expectation value is
\begin{equation}
\left\langle \frac{z_{i\alpha}}{Z_\alpha} \right\rangle_L =
 \frac{1}{\mathcal{Z}} \sum_{i\alpha} \frac{z_{i\alpha}^2}{Z_\alpha} =
 \sum_\alpha \frac{Z_\alpha}{\mathcal{Z}} \sum_i \left(
 \frac{z_{i\alpha}}{Z_\alpha} \right)^2 .
\label{equ:z_contrib}
\end{equation}
The second form for the expression above emphasises that it is an
average over the boundary conditions with their thermal weights,
$Z_\alpha$.

We define $\xi_*$ as the system size at which the above expectation
value ceases to be dominated by a single state,
\begin{equation}
\langle z_{i\alpha}/Z_\alpha \rangle_{\xi_*} \equiv (1/2) .
\label{equ:mos_len_Z}
\end{equation}

Since the droplet entropy is
\begin{equation}
S_\alpha = -k_B \sum_{i} (z_{i\alpha}/Z_\alpha) \log
(z_{i\alpha}/Z_\alpha) ,
\label{equ:droplet_entropy}
\end{equation}
Eq.\ (\ref{equ:mos_len_Z}) will be satisfied when $\langle S_\alpha
\rangle$ be of the order of $k_B$. In what follows we set $k_B=1$.
The entropy (\ref{equ:droplet_entropy}) measures the typical number of
states contributing to the partition sum with a fixed boundary.  It
must not be confused with the configurational entropy which measures
the number of states contributing to the overall partition sum.

\subsection{Finite size thermodynamics}

The partition function for a finite SPM with periodic boundaries was
calculated by Espriu and Prats \cite{Espriu-Prats}.  In
appendix~\ref{app:pq_series} we generalise their argument to allow for
a given state of $4(L-1)$ fixed boundary spins. The derivation is in
the spirit of a high temperature series, which can be resumed exactly
for this model since the bulk thermodynamics are those of a trivial
free lattice gas.  As mentioned above, the boundary conditions on the
spins constrain the parity of the number of defects in each row and
column of the dual lattice. If the number of rows with odd parity is
$r$ and the number of columns with odd parity is $r'$, then $Z_\alpha$
depends on the boundary conditions only through $r$ and $r'$.  Since
it is a symmetric function of these two integers, it is convenient to
define
\begin{equation}
m\equiv\mathrm{max}(r,r'), \qquad n\equiv|r-r'| ,
\end{equation}
so that $Z_\alpha = Z_{mn}$.  Note that $n$ is always even since $r$
and $r'$ (and therefore $m$) all have the parity of the total number
of defects in the droplet.  It is also convenient to parameterize
temperature by:
\begin{equation}
c \equiv e^{-\beta} ,
\end{equation}
which is related to the average concentration of excitations by
$\langle (1-p_{xy}) /2 \rangle = c/(1+c) \approx c$, the last
approximate equality being valid at low temperatures.

The low temperature behaviour of the plaquette model obeys scaling
relations \cite{Jack-et-al}.  Length and timescales diverge at low
temperatures as simple powers of $c$.  We therefore work in the
scaling regime, at leading order in $c$. In terms of droplet sizes,
this means that numerical values of $L$ may be large.  The relevant
indicators of droplet size will be $c L$ and $c L^2$. That is, we
assume $c \ll 1 \ll L$, but make no assumption on the absolute sizes
of $c L$ or $c L^2$.

We begin by considering fixed $L$ and very low temperatures: $cL^2 \ll
1$.  On picking a droplet from an infinite system, it is very likely
that it contains no defects.  The boundaries will then be such that
all rows and columns contain even numbers of defects.  There are
$2^{2L-1}$ such boundary conditions (one for each choice of spins,
say, in the bottom and leftmost sides of the box).  Tracing over the
bulk spins with each configuration of the boundary contributes
$Z_\alpha=Z_{00}\equiv Z_{m=0,n=0}$ to $\mathcal{Z}$:
\begin{equation}
\label{Z00cLL}
\mathcal{Z} = 2^{2L-1} [ Z_{00} + \mathcal{O}(cL^2) ] .
\end{equation}

Now, if $m=n=0$ then the boundary of the droplet specifies its unique
ground state.  Further, simple counting arguments show that excited
states contain even numbers of defects $u\geq4$. Their Boltzmann
weights are $c^u$ and the degeneracy of the state with $u$ defects is
proportional to $L^{u}$. This means that:
\begin{equation}
\label{Z00cL}
Z_{00}=1 + \mathcal{O}(cL) .
\label{equ:Z00}
\end{equation}
This partition function is also relevant to periodic boundary
conditions, and is the one calculated in \cite{Espriu-Prats}.  In
Fig.\ \ref{fig:pbc_entropy} we show the droplet entropy $S_{00}=
\partial_T ( T \ln Z_{00})$, as a function of $c$ for different
droplet sizes $L$.  The top panel of Fig.\ \ref{fig:pbc_entropy} shows
that the droplet has an ``entropy crisis'' at $cL \sim 1$.  In the
bottom panel of Fig.\ \ref{fig:pbc_entropy} we see that the droplet
entropy scales with $cL$ when $cL\ll 1$, and is subextensive in this
regime.

\begin{figure}
\epsfig{file=pq_S00_kauz.eps,width=0.85\columnwidth}
\epsfig{file=pq_logS00.eps,width=0.85\columnwidth}
\caption{(Top) Droplet entropy density $S_{00}/L^2$ versus $c \equiv
e^{-\beta}$ for droplets of different sizes $L$, with frozen boundary
conditions enforcing even numbers of defects in every row or column,
$m=n=0$.  The symbols are finite $L$ entropies calculated using the
expressions of appendix \ref{app:pq_series}.  The entropy is extensive
for $cL>1$.  On approaching $cL \sim 1$ from above it deviates from
the bulk entropy (solid line, $L\to \infty$), and becomes subextensive
for $cL<1$.  (Bottom) Plot of total entropy $S_{00}$ against $cL$
showing the (non-extensive) scaling predicted in (\ref{equ:Z00}). }
\label{fig:pbc_entropy}
\end{figure}

From figure~\ref{fig:pbc_entropy}, it is clear that as the temperature
is increased, droplets with $m=n=0$ remain frozen until $cL\sim
1$. However, $Z_{00}$ dominates the droplet partition function only
for $cL^2 \ll 1$.  This means that \emph{typical} droplets are frozen
only for $cL^2\ll1$.  Between $c L \sim 1$ and $c L^2 \sim 1$ we must
consider $Z_{mn}$ with $m,n\neq0$.

\begin{figure}
\epsfig{file=sketch_m2.eps,width=1.0\columnwidth}
\caption{Ground states of droplets ($L=6$) with different boundary
conditions.  Spins are on vertices on the square lattice and are not
shown. Excited plaquettes are marked with $\times$ symbols. (a)
Boundary conditions enforce two rows and two columns with odd numbers
of defects, so that $m=2$, $n=0$. There are two ground states that
differ by flipping the two spins identified with black circles. (b)
Boundary conditions enforce two rows with odd numbers of defects, but
all columns with even numbers: $m=2$, $n=2$. There are $(L-1)=5$
distinct ground ground states of which two are shown: they differ only
at the four marked spins}
\label{fig:bc_effect}
\end{figure}

\begin{figure}
\epsfig{file=pq_Sc.eps,width=0.85\columnwidth}
\caption{Entropy as a function of $cL$ for $L=100$.  The full line is
  the bulk entropy of the SPM.  The open symbols are the entropies
  $S_{m0}$ for frozen boundaries with different $m$, as obtained from
  the finite size partition functions of appendix \ref{app:pq_series}.
  The + symbols give the entropy in the case where boundary conditions
  are generated randomly. }
\label{fig:Zm0_entropy}
\label{fig:entropy_c}
\end{figure}

The behaviour of droplets changes qualitatively in the case of ground
states containing two or more defects. As is clear from
Fig.\ \ref{fig:bc_effect}, if $m\geq2$ then there is more than one
state of the bulk spins that satisfies the boundary condition
perfectly.  Specifically, if $m=2$ then there are two ground states,
each of which contains two defects, and contributes $c^2$ to the
partition function. More generally, we find that
\begin{equation}
Z_{m0} \sim m!\, c^m [ 1 + \mathcal{O}(cL) ] .
\end{equation}
It follows that the zero temperature droplet entropy is now
$S_{m0}(T=0)=\log m!$, and so for these boundary states, we expect
typical values of
\begin{equation}
 z_{i\alpha} / Z_\alpha \sim (1/m!) + \mathcal{O}(cL) .
\end{equation}

While the droplets do have a feature in the entropy at $cL\sim1$, the
degenerate ground states all contribute equally to the partition sum
at low temperature.  Thus, as soon as droplets with $m\geq2$ dominate
$\mathcal{Z}$ then the they are thermodynamically melted according to
condition (\ref{equ:mos_len_Z}). Fig.~\ref{fig:entropy_c} shows that
this happens as soon as $cL^2 \sim 1$, as expected.  The melting
lengthscale is therefore,
\begin{equation}
\xi_* \sim c^{-1/2} .
\label{equ:pq_mos_len}
\end{equation}

It is clear that there are two crossovers in the droplet
thermodynamics as $c$ is increased. For $cL^2 \ll 1$ nearly all
droplets are frozen in true ground states of the system.  For $cL\gg
1$ the system behaves as the bulk.  In the intermediate regime $cL \ll
1 \ll cL^2$, each droplet is almost certainly in a state that
minimises the energy with respect to the boundaries; however, there
are many of these states for a fixed configuration of the boundary.
In this regime $m$ is about $cL^2$.  The low temperature droplet
entropy is then
\begin{equation}
S_{T=0} \simeq cL^2 \log cL^2 ,
\end{equation}
which is large compared to unity, but small compared to its bulk value
$cL^2\log 1/c$ (since $cL \ll 1$ implies $cL^2 \ll 1/c$).  We show
this explicitly in Fig.\ \ref{fig:entropy_c} where we average the
exact results for the entropy at given $m$ and $n$ over a distribution
of $mn$ that is sampled by Monte Carlo.

Thus far we have analyzed only states with $n=0$.  That is, the
boundary condition requires $m$ rows to have odd numbers of defects in
them, and it also requires $m$ columns to have the same property.  The
probability that $n=0$ is the probability that no two defects occupy
the same row or columns as each other,
\begin{equation}
P(n=0) = 1-\mathcal{O}(c^2 L^3) .
\end{equation}
This probability is close to $1$ as long as $c L^2 \ll \sqrt{L}$.

The melting length (\ref{equ:pq_mos_len}) can also be accessed through
static overlap functions, which will be important in our discussion of
dynamics.  Consider the self-overlap for a given boundary $\alpha$,
\begin{equation}
C_{0\alpha} = \frac{1}{(L-2)^{2}} \sum_{xy}
\left\langle  \sigma_{xy} \right\rangle_\alpha^2 ,
\label{equ:c0_replica}
\end{equation}
where we have explicitly summed over the spins of a droplet with given
boundary condition, since for a given realisation of the $\alpha$,
single spin correlations are finite and depend on position.  The
brackets $\langle \dots \rangle_\alpha$ denote a thermal average over
configurations of the bulk spins, with the boundary spins fixed.
$C_{0\alpha}$ should give the long time limit of the spin-spin
autocorrelation for a droplet with given boundary conditions.
Averaging over the boundary spins we get:
\begin{equation}
C_0 = \sum_\alpha Z_\alpha C_{0\alpha} ,
\label{equ:C0_replica}
\end{equation}
which is analogous to a disorder average.

\begin{figure}
\epsfig{file=pq_ss_c.eps,width=0.8\columnwidth}
\epsfig{file=pq_ss_c_coll.eps,width=0.8\columnwidth}
\epsfig{file=pq_ss_x.eps,width=0.8\columnwidth}
\caption{Plots of the overlap function $\langle \sigma_{xy}
\rangle_\alpha^2$, averaged over boundary conditions $\alpha$ at
various temperatures. (Top) Average overlap $C_0$ as a function of
temperature. (Middle) Collapse of this data as a function of $cL^2$
(the dashed line is a guide to the eye). (Bottom) Spatial dependence
along the diagonal of the square droplet, showing strong correlations
near the boundary.  }
\label{fig:replica_correl}
\end{figure}

In Fig.\ \ref{fig:replica_correl} we show these overlaps.  The
correlator $\langle \sigma_{xy} \rangle_\alpha$ can be calculated in a
high temperature series: see appendix \ref{app:pq_series}.  The top
and middle panels of Fig.\ \ref{fig:replica_correl} show that $C_0$ is
a function of $c L^2$, which is close to $1$ for small $c L^2$, and
becomes small when $cL^2$ gets large.  This scaling also indicates
that $C_0$ will be small in the regime $cL \ll 1 \ll cL^2$.  The
bottom panel of Fig.\ \ref{fig:replica_correl} shows the spatial
dependence of $\langle \sigma_{xy} \rangle_\alpha$.  The spins near
the boundaries are strongly fixed by the boundary condition.  This
correlation decays away from the boundary with a lengthscale that
scales as $\xi_4 \sim c^{-1/2}$.  These observations for the overlaps
are consistent with thermodynamic melting of the droplet at $cL^2\sim
1$ found above.

\subsection{SPM and mosaic lengths}

We have established that the melting lengthscale $\xi^*$ scales as
$c^{-1/2}$ in the SPM, but that there is also a second important
thermodynamic crossover at $L\sim c^{-1} > \xi_*$.  We now compare the
results so far with the expectations of BB. The appearance of two
lengthscales was not anticipated in \cite{Bouchaud-Biroli}. We have
shown that the crossover at $cL^2\sim1$ comes from a small number of
states that are have similar weights in the droplet partition
function. We find that distinguishing these few specific states from
typical states with bulk energy $E$ is necessary when discussing the
droplet partition function $Z_\alpha$.

We use (\ref{equ:mos_len}) to define the mosaic lengthscale $\xi_*$.
The assumption of BB is that the lengthscale defined by
(\ref{equ:mos_len_typ}) will scale in a similar way.
Eq.~(\ref{equ:mos_len}) treats all states individually, whereas
(\ref{equ:mos_len_typ}) assumes that we can extrapolate the properties
of the partition sum from typical states at the relevant
temperature. We write the mosaic lengthscale extracted from
(\ref{equ:mos_len_typ}) as $\xi_{*,\mathrm{typ}}$: we find that
$\xi_*$ and $\xi_{*,\mathrm{typ}}$ do not coincide in the SPM.

More precisely, the average bulk energy of a droplet at a given
temperature is $c(L-2)^2$, and the configurational entropy at that
temperature is $c(L-2)^2 \ln(1/c)+(2L-1)\ln2$. If we replace the bulk
of the droplet by a new bulk state with the same energy, then the
typical boundary energy is $2(L-2)$ (half of the boundary plaquettes
will be excited, since there are no two-point spin correlations either
in the bulk or along the boundary). So from
Eq.~(\ref{equ:mos_len_typ}), we have $d=2$, $\theta=1$ and
$$\xi_{*,\mathrm{typ}}\sim c^{-1}$$

On the other hand, we showed in the previous section that the
Eq.~(\ref{equ:mos_len}) leads to
$$\xi_{*}\sim c^{-1/2}$$
The origin of this discrepancy is simply the degenerate ground
states of the droplet, whose boundary energy does not
take the typical value $\Upsilon(T)\sim L$, but rather
have $\Upsilon_{i\alpha}=0$. Thus they must be separated from the
sum before it is approximated:
\begin{eqnarray}
Z_\alpha &=& e^{-\beta f_i L^2} + \sum_{j\neq i, j\in \mathrm{gs}}
e^{-\beta f_j L^2} \nonumber \\ && + \sum_{j\neq i, j \notin
\mathrm{gs}} e^{-\beta f_j L^2 - \beta \Upsilon_{j\alpha}}
\label{equ:Z_decompose}
\end{eqnarray}
where the sum over $j\in\mathrm{gs}$ denotes a sum over states with
$\Upsilon_{j\alpha}=0$.  The final term in (\ref{equ:Z_decompose}) can
then be approximated to give a form resembling
Eq.~(\ref{equ:mos_len_typ}), but that term is irrelevant to droplet
melting in the SPM since the second term dominates the partition
sum for $cL \ll 1 \ll cL^2$.

We interpret this result as evidence that approximations such as those
leading to Eq.~(\ref{equ:mos_len_typ}) are rather dangerous. In
general, systems may possess significant numbers of states in the
partition sum with smaller than average boundary energy, and these may
be sufficient to destroy spin correlations in the infinite time limit
($C_0\to0$ when $L\ll \xi_{*,\mathrm{typ}}$), even if the droplet is
exploring only a small fraction of its metastable states.  Further, we
will show in the next section that if the droplet does melt in stages
then choosing whether to identify the bulk relaxation time as
$\tau(\xi_*)$ or $\tau(\xi_{*,\mathrm{typ}})$ requires investigation
of the specific system under study. In the SPM, we will find that
while complete destruction of of spin correlations at large times does
take place for $L\ll \xi_{*,\mathrm{typ}}$, the timescale for this
relaxation is much longer than the bulk relaxation time. However, this
effect cannot be inferred from thermodynamic arguments: it is a
kinetic effect.

To end this discussion, we note that while the condition $cL^2\sim 1$
leads to rather small droplets, the configurational entropy of a
droplet of this size is $cL^2 \log(1/c)\gg1$. BB comment that the
vanishing of the configurational entropy leads to a trivial lower
bound on the mosaic lengthscale which in this case is: $\xi_*^2 >
[c\log(1/c)]^{-1}$. Thus we see that despite the condition that frozen
droplets contain at most one defect, the freezing lengthscale is still
much larger than its trivial lower bound.

\subsection{Relaxational dynamics}

In this subsection we consider the relaxation of the droplets studied
thermodynamically above.  The relevant correlation function is
\begin{equation}
C_\alpha(t) =
\frac{1}{(L-2)^2}
\sum_{xy\in \hbox{bulk}} \langle \sigma_{xy}(0) \sigma_{xy}(t)
\rangle_\alpha
\end{equation}
where the average is over realisations of the Markov chain that
describes the time evolution of steady state of the system (with fixed
boundary state $\alpha$). We also define the average of $C_\alpha(t)$
over boundary conditions
\begin{equation}
C(t) = \sum_\alpha Z_\alpha C_\alpha(t)
\end{equation}
The long time limit of $c_\alpha(t)$ is given by the overlap
$c_{0\alpha}$.

\begin{figure}
\epsfig{file=pq_ct.eps,width=0.8\columnwidth}
\caption{Simulations of the SPM at $c=0.01$ for various system sizes
and frozen boundary conditions.  The relaxation time increases
significantly as $cL$ decreases through unity, but the long time limit
of $C(t)$ would still appear to vanish since $cL^2 \gg 1$.  We also
show relaxation for $L=1000$ and periodic boundary conditions.  The
relaxation can be well-fitted by an exponential for large $L$.  At
smaller $L$ the decay is slower than exponential.  }
\label{fig:pq_Ct_cL}
\end{figure}

\begin{figure}
\epsfig{file=pq_ct_small.eps,width=0.8\columnwidth}
\caption{Simulations of the SPM at $c=0.01$ for various
system sizes and frozen boundary conditions.  The long time limit of
$C(t)$ increases from zero as $cL^2$ becomes of order unity.}
\label{fig:pq_Ct_cL2}
\end{figure}

>From Monte Carlo simulations, it is clear that the relaxation of the
system slows down as the system size is decreased towards $1/c$: see
Fig.\ \ref{fig:pq_Ct_cL}.  From our discussions of the static single
point correlator [Eq. (\ref{equ:c0_replica}) and
Fig.\ \ref{fig:replica_correl}] we expect the long time limit of this
function to approach zero even for $cL \ll 1 \ll cL^2$, although this
is hard to show in simulations since simulations with large numbers of
defects at small temperatures are very computationally expensive.

Moving towards the regime with $cL^2 \sim 1$, we see that the
relaxation of the system is very slow, and that there are large
correlations that persist to the infinite time limit: see Fig.\
~\ref{fig:pq_Ct_cL2}. We can make a link between static and dynamic
quantities by evaluation of the long time limit of these functions
using the high temperature series (cf.\ Fig.\
~\ref{fig:replica_correl}).

Thus we have shown how the two lengthscales associated with the glassy
mosaic state affect the dynamics of the system.  As the temperature is
reduced, the relaxation slows for $cL\sim1$, before stopping
completely at $cL^2\sim1$. The separation of these lengthscales was
not anticipated in \cite{Bouchaud-Biroli}: we have more to say on this below.

Before ending our discussion of dynamics, we make two further
points. Firstly, we note that it is clear from Ref.~\cite{Jack-et-al}
that the bulk relaxation is caused by correlated motion of defects
over distances $\xi_\mathrm{dyn} \sim c^{-1}$. This coincides with the
system size above which the system relaxes as the bulk.  Secondly, it
is also known \cite{Espriu-Prats,Jack-et-al} that correlated motion
for defects over a distance $l$ takes place on a timescale $\tau \sim
l e^{2\beta}$ so we see that the analogue for equation
(\ref{equ:tau_psi}) is
\begin{equation}
\tau \sim e^{2\beta+\log \xi}
\label{equ:pq_tau_xi}
\end{equation}
The exponent depends logarithmically on $\xi$ with a prefactor that
does not depend on temperature: apart from the constant activation
energy, the free energy barriers to motion are purely entropic in
origin. The entropic nature of the barriers explains why relaxation of
the SPM model diverges only in an Arrhenius fashion ($\tau\sim
e^{3\beta}$) despite the co-operativity of its dynamics.

\subsection{SPM summary}

It is clear from the results of this section that, even in a simple
model like the SPM, the situation regarding caging and mosaic
lengthscales is already more complicated than what the appealingly
simple and general arguments of \cite{Bouchaud-Biroli} would predict.  As we
increase the size of droplets with frozen boundaries, there are two
crossovers.  Static spin-spin correlations vanish at at $c \sim
L^{-2}$, but neither the droplet entropy, nor its dynamical
correlation functions reach their bulk forms until the much higher
temperature $c\sim L^{-1}$.  We note that these two lengthscales were
also identified in the discussion of static and dynamic lengthscales
in Ref.~\cite{Jack-et-al}.  To be precise, the static four-spin
correlator $\langle \sigma_{00} \sigma_{0r} \sigma_{r0} \sigma_{rr}
\rangle$ decays on a lengthscale $\xi_4^\mathrm{stat} \sim c^{-1/2}$,
but the dynamic four point correlator decays on a length that scales
as $\xi_{2,2} \sim c^{-1}$.  There is also a static lengthscale that
mirrors the scaling of $\xi_{2,2}$ which is derived from fluctuations
of two point correlations \cite{Jack-et-al}.

Despite these complications, a general picture like that of
\cite{Bouchaud-Biroli} does seem to be applicable to the SPM, in the
sense that droplet melting does occur, even if it happens in
stages. As the temperature decreases towards the Kauzmann singularity
at $T=0$, the mosaic lengthscale increases, and the relaxation time
increases according to (\ref{equ:pq_tau_xi}).  This increase follows a
simple Arrhenius law because only the entropic barriers depend on
$\xi_*$.  This suggests that one should replace (\ref{equ:tau_psi}) by
the more general parametrization
\begin{equation}
\tau \sim f_s(\xi) e^{\beta f_u(\xi)}
\label{equ:tau_barriers}
\end{equation}
where $f_s(\xi)$ describes the entropic part of the barrier and
$f_u(\xi)$ the energetic part. At small temperatures then we expect
the energetic part to dominate, except in the case where $f_u(\xi)$ is
independent of $\xi$, as is the case for the SPM.

It is at this point at which our discussion of dynamics diverges from
that of KTW and BB: since the (non-perturbative) dynamical moves are
not nucleation events in this model, then the free energy barrier is
not set by a transition state with a critical droplet of the new
phase.  Rather, the relaxation proceeds by less expensive
rearrangements that occur over a lengthscale $\xi_{*,\mathrm{typ}}$,
as was discussed in detail in Ref.\cite{Jack-et-al}.

\section{Triangular plaquette model}
\label{sec:triag}

In this section we compare the results we found above for the SPM,
with a similar model of Ising spins: the triangular plaquette model,
or TPM \cite{Newman-Moore,Garrahan-Newman}. This model has a
super-Arrhenius divergence of the relaxation time at low temperatures,
and is therefore a fragile model, as compared to the SPM, which is
strong.  Moreover, dynamical correlations do not have the strong
anisotropy of the SPM \cite{Jack-et-al}.  The calculation of the
partition function, however, is considerably more involved, so our
results below are less detailed than those for the SPM.

The TPM is defined by the Hamiltonian
\cite{Newman-Moore,Garrahan-Newman}:
\begin{equation}
H = -(1/2) \sum_{xy} \sigma_{xy} \sigma_{x+1,y} \sigma_{x,y+1} .
\end{equation}
where as before $\{\sigma_{xy}\}$ are Ising spins on a square lattice
and the dynamics are is spin-flips with Glauber rates.  If we define
the plaquette variables
\begin{equation}
q_{xy} = \sigma_{xy} \sigma_{x+1,y} \sigma_{x,y+1}
\end{equation}
we again find that the partition sum over the spins reduces to a
product over $q_{xy}$, that are independent up to a non-extensive set of
constraints from the boundary conditions \cite{Newman-Moore,Garrahan}.

We note that defining this model in this way leads to a Hamiltonian
that does not possess the square symmetry group of the lattice: it is
more intuitive to deform the square lattice into a triangular one. The
interactions are then around upward pointing triangular plaquettes
(see Fig.\ \ref{fig:rhomb}). The symmetries of the Hamiltonian are
then translations by lattice vectors, and $120^\circ$ rotations
($60^\circ$ rotations leave the lattice invariant but flip downward
pointing triangles to upward pointing ones, so they change the
Hamiltonian).

\begin{figure}
\epsfig{file=rhomb.eps,width=0.8\columnwidth}
\caption{Sketch showing relation between square and rhomboid droplets
($L=6$). The spins are on the intersections of grids; $\sigma_{11}$ is
marked by a filled circle. The $\times$ symbol marks the plaquette
interaction $q_{24}=\sigma_{24}\sigma_{25}\sigma_{34}$. Rotations of
$120^\circ$ leave both Hamiltonian and triangular lattices
invariant. However, note that the boundaries of the droplets are not
all equivalent, since the rhombus shape is not invariant under this
rotation.}
\label{fig:rhomb}
\end{figure}

The analogy of the square droplet of Fig.\ \ref{fig:plaq_sketch} is a
rhombus shape in the triangular representation. The arguments for
counting bulk and boundary states are the same as those of the
plaquette model.  However, identifying the $2^{2L-1}$ degeneracies of
each state and the nature of the $(2L-3)$ constraints is considerably
more involved.

\subsection{Static spin-spin correlations and constraints}

The key to understanding static properties of the triangular plaquette
model is to realise that
\begin{equation}
\langle q_{i_1,j_1} q_{i_2,j_2} \dots q_{i_n,j_n} \rangle =
	[\tanh(\beta/2)]^n .
\end{equation}
All multi-spin correlators that can be written in this form are
finite. Further, all other multi-spin correlators are zero
\cite{Garrahan}. While one and two point static spin correlations
vanish,
\begin{equation}
\langle \sigma_{xy} \rangle = 0, \;\;\; \langle \sigma_{xy}
\sigma_{x'y'} \rangle=\delta_{x,x'} \delta_{y,y'} ,
\end{equation}
specific three point correlators are non-zero,
\begin{equation}
\langle \sigma_{xy} \sigma_{x+1,y} \sigma_{x,y+1} \rangle =\langle
q_{xy} \rangle = \tanh(\beta/2) .
\end{equation}

\begin{figure}
\epsfig{file=ST.eps,width=0.8\columnwidth}
\caption{Illustration of Eq.\ (\ref{equ:ST_correl}) with $k=6$. The
product of the five spins marked by filled circles is given by the
product of the fifteen plaquette variables marked with crosses. Both
the marked defects and the spins marked with circles (filled
\emph{and} empty) form parts of Sierpinski triangles: they are marked
if and only if the relevant entry of Pascal's triangle is odd.}
\label{fig:ST}
\end{figure}

A useful general relation is that
\begin{equation}
\sigma_{xy} \prod_{r=0}^k \sigma_{x+r,y-k}^{B_{k,r}} =
\prod_{k'=1}^k \prod_{r=0}^{k'-1} q_{x+r,y-k'}^{B_{k',r'}} ,
\label{equ:ST_correl}
\end{equation}
where the $B_{nr}\equiv n!/[r!(n-r)!]$ are binomial coefficients.
Since the $q$'s and the $\sigma$'s are Ising variables, only the
parities of the binomial coefficients are relevant.  In terms of
correlation functions, this relation implies that there is a finite
multi-spin correlator connecting $\sigma_{xy}$ and various spins in
row $x-k$ of the lattice.

This correlator is related to the first $k$ rows of Pascal's triangle
(PT). The correlator connects the spins corresponding to entries of
the $k$th row of the PT that have odd parity. Its value is given by
$\tanh(\beta/2)^p$ where $p$ is the number of entries in the first
$k-1$ rows of the PT that have odd parity. See Fig.\ \ref{fig:ST} and
the associated caption. Now, the typical number of odd entries in a
Pascal's triangle of linear size $l$ is $l^{d_\mathrm{f}}$ with
$d_\mathrm{f}=\log_2(3)$.  It follows that typical correlations over a
lengthscale $r$ scale as
\begin{equation}
\tanh(\beta/2)^{(r^{d_\mathrm{f}})} \sim
\exp\left[-2(rc^{1/d_\mathrm{f}})^{d_\mathrm{f}}\right] ,
\end{equation}
from which we infer that static correlations decay on a lengthscale
\begin{equation}
\xi_\mathrm{stat}\sim c^{-1/d_\mathrm{f}} .
\end{equation}
In the case of finite droplets there must be $(2L-1)$ symmetry
operations that leave the energy of a droplet invariant, but require
the boundary spins to change (recall Fig.\
\ref{fig:pq_bulk_boundary}).  Furthermore, fixing the boundary spins
imposes $(2L-3)$ constraints on the allowed configurations of the
plaquette variables.  The form of these symmetries and constraints are
given explicitly in appendix~\ref{app:triag_constraints}.

The SPM was simple to analyse since the constraints are in a form
where each plaquette appears in exactly two constraints.  In that case
we can make a geometrical interpretation of those constraints.  In the
TPM the consequences of imposing the constraints is less clear.  We
therefore proceed directly to a discussion of dynamic properties, in
an attempt to access the static properties indirectly.

\subsection{Dynamic spin correlations}

Relaxation in the TPM is frustrated by a hierarchy of energy
barriers\cite{Newman-Moore,Garrahan-Newman}: co-operative motion over
a distance $\xi$ requires the crossing of an energy barrier of a
height $\log_2 \xi$.  This behaviour is reminiscent of the East model
\cite{East}.

\begin{figure}
\epsfig{file=triag_ct.eps,width=0.8\columnwidth}
\caption{Relaxation of finite systems at inverse temperature $\beta=4$
($c=0.018$).  Notice that the long-time limit of $C(t)$ is changing,
but that the relaxation time is approximately constant.  Compare to
Fig.\ \ref{fig:pq_Ct_cL2}, where the relaxation time increases
sharply as $C(t \to \infty)$ increases.}
\label{fig:triag_ct_small}
\end{figure}

In order to understand the scaling of the mosaic length, we
investigate the scaling of $C(t)$ in finite droplets: see
figure~\ref{fig:triag_ct_small}. A lower bound on
the freezing lengthscale is set by the elementary condition $cL^2 \sim
1$, since all states with single defects have different boundary
conditions. This would imply that the long time limit of the
autocorrelation, $C_0$ should be a scaling function of $cL^2$, as in
the SPM, Fig.\ \ref{fig:replica_correl}.

However, from the discussions of static correlations above, it is
clear that the physics of the triangular plaquette model is
intrinsically related to the parities of binomial co-efficients, and
hence to Sierpinski's triangle. As mentioned above, this structure has
a fractal dimension of $d_\mathrm{f}=\log_2(3)\simeq1.585$. In Fig.\
\ref{fig:triag_ct_scaling} we show relaxation data at various
temperatures with constant values of $cL^{d_\mathrm{f}}$. We see that
$C_0$ does seem to depend only on this combination. The conclusion is
that the mosaic lengthscale scales as
\begin{equation}
\xi_* \sim c^{-1/d_\mathrm{f}} .
\end{equation}
This relation indicates that typical droplets melt when they contain
of the order of $c^{1-2/d_\mathrm{f}}\sim c^{-0.26}$ defects; below
this number there no co-operative mechanisms allowing the droplet to
rearrange.  This should be compared with the SPM, in which droplets
with $L\sim\xi_*$ contain of the order of one defect, but those at the
dynamical crossover ($cL\sim 1$) contain of the order of $c^{-1}$
defects.

\begin{figure}
\epsfig{file=triag_ct_scal.eps,width=0.8\columnwidth}
\caption{Relaxation at different temperatures showing bulk results
(circles) and two different values of $cL^{d_\mathrm{f}}$.  The actual
system sizes used are (left) $L=13,26$; (middle) $L=16,32$; (right)
$L=26,44$.  The dashed lines are guides to the eye showing that the
long time limit scales as a function of $cL^{d_\mathrm{f}}$ only
(there are weak deviations from scaling which we attribute to
subleading corrections in $c$ and $1/L$).}

\label{fig:triag_ct_scaling}
\end{figure}

An alternative measure of the co-operativity of the dynamics is given
by the bulk ``four-point'' correlation function (see
e.g. \cite{Toninelli-et-al})
\begin{equation}
\tilde G_4(xy;t) = \langle \sigma_{xy}(t) \sigma_{00}(t) \sigma_{xy}(0)
\sigma_{00}(t) \rangle .
\end{equation}
This correlation function measures spatial correlations in the single
site autocorrelation function.  It is convenient here to work with the
connected part of this function, normalised according to
\begin{equation}
G_4(xy;t) = \frac{\tilde G_4(xy;t) -
\langle \sigma_{00}(t) \sigma_{00}(0) \rangle^2}
{1-\langle \sigma_{00}(t) \sigma_{00}(0) \rangle^2} .
\end{equation}
In the triangular plaquette model, the function $G_4$ has a
complicated spatio-temporal structure, again related to Sierpinski's
triangle: we postpone a detailed discussion of this structure to a
later work. However, the circular average of this function, which we
denote by $G_{4r}(r,t)$ decays smoothly with distance (at a fixed
time). This is shown in Fig.\ \ref{fig:triag_g4_scaling}, from which
it is clear that the lengthscale over which dynamical correlations
occur also scales as
\begin{equation}
\xi_\mathrm{dyn} \sim c^{-1/d_\mathrm{f}} .
\end{equation}

\begin{figure}
\epsfig{file=triag_g4.eps,width=0.8\columnwidth}
\caption{Circular average of the normalised four point dynamic
correlator. The system size is $L=128$ with periodic boundaries, which
is large enough to avoid finite size effects.  (Left panel) Three
different temperatures, with times chosen so that $C(t)=0.5$.  (Right
panel) Collapse of the data of the left panel on rescaling of length
as $rc^{1/d_\mathrm{f}}$.}
\label{fig:triag_g4_scaling}
\end{figure}

We have used the dynamics of finite droplets to identify a single
mosaic lengthscale in the triangular plaquette model, that scales as
$c^{-1/d_\mathrm{f}}$. Further, we know \cite{Garrahan-Newman} that
energy barriers to motion over a lengthscale $l$ require an activation
energy of the order of $\log_2 l$. At very low temperatures, we
therefore expect the bulk relaxation time at very low temperatures to
scale as
\begin{equation}
\tau \sim \exp \left( \beta\log_2 \xi \right) \sim \exp \left( \beta^2
/ \ln 3 \right) .
\label{equ:triag_logtau}
\end{equation}
Note that Ref.\ \cite{Garrahan} predicted $\exp(\beta^2/\ln4)$ based
on the assumption that $\xi\sim c^{1/2}$.  Fig.\ \ref{fig:triag_tau}
shows a fit of the relaxation time of $C(t)$ with the function $\ln
\tau = a_0 + a_1\beta + \beta^2/\ln 3$.  The constant and linear terms
come from entropic and dynamical effects which become irrelevant at
very low temperatures where the energy barriers on $\xi$ dominate, see
Eq.~(\ref{equ:tau_barriers}).  The fit in the figure is consistent
with (\ref{equ:triag_logtau}), although the range of $\beta$ that we
consider is too small to confirm it beyond doubt.

\begin{figure}
\epsfig{file=triag_tau.eps,width=0.8\columnwidth}
\caption{Data showing relaxation time for large systems as a function
of inverse temperature, $\beta$. The relaxation time is defined by
$C(\tau)=0.5$. The error bars are smaller than the symbols shown. The
dashed line is a fit to the form $\ln \tau = a_0 + a_1\beta +
q\beta^2/\ln 3$.  Good fits are also possible with different
coefficients for the quadratic term.}
\label{fig:triag_tau}
\end{figure}

\subsection{TPM summary}

The thermodynamics of finite droplets in the triangular plaquette
model are not sufficiently simple to allow an analysis of the droplet
entropy and hence determination of the mosaic lengthscale according to
the criterion (\ref{equ:mos_len}). However, it appears that the long
time limit of the single spin autocorrelation function approaches
unity as $cL^{\log_2 3}$ gets small.  If $C_0$ approaches unity then
it seems reasonable to assume that droplets are effectively trapped in
a single state that dominates the partition sum, and that this should
be an equivalent criterion to (\ref{equ:mos_len}).

We find that the mosaic lengthscale extracted in this way ($\xi_*$),
scales in the same way as the static lengthscale for multi-spin
correlations ($\xi_\mathrm{stat}$), and as the dynamical lengthscale
extracted from the four point correlator ($\xi_\mathrm{dyn}$). This
scaling is evidence for the presence of a single lengthscale in this
model. This is in contrast to the TPM, and more in line with
\cite{Bouchaud-Biroli}.

On the relation of dynamics to statics, the analogue of
(\ref{equ:tau_psi}) seems to be
\begin{equation}
\tau \sim f_s(\xi) e^{\beta \log_2 \xi}
\end{equation}
The increase of the energy barriers with $\xi$ is again slower than
the power law predicted by \cite{Bouchaud-Biroli}, but the assumption of an
increasing function is robust. The absence of a power law dependence
again indicates that the system relaxes by a more efficient mechanism
than nucleation of a new droplet in the background of an uncorrelated
state.

\section{Conclusions}
\label{sec:conc}

We have successfully applied the procedure of
Ref.~\cite{Bouchaud-Biroli} to two simple models of interacting Ising
spins.  We find that the concept of ``caging'' whereby the states of
individual spins are strongly constrained by frozen spins surrounding
them applies well in these models.  We were able to study the
breakdown of this caging effect by considering droplets of increasing
size.

In the square plaquette model (SPM), the caging breaks down in two
stages as the droplet size increases. For $L\ll c^{-1/2}$ the droplet
is frozen. For $c^{-1/2} \ll L \ll c^{-1}$ the droplet will eventually
relax, but this relaxation is slower than that of the bulk
system. Finally, for $L\gg c^{-1}$ the relaxation time coincides with
that of the bulk.  The correlation lengths controlling these two
crossovers have the same scalings as the two separate lengths
identified in this model from four point static and dynamic
correlations\cite{Jack-et-al}. The two stage droplet melting in this
model is more complicated than the simple scenario of BB.
Interestingly, the two crossover temperatures of the SPM, as well as
the structure of the droplet states in the intermediate regime, bear
some resemblance with the situation in mean-field systems like the
$p$-spin model (see e.g. \cite{Castellani-Cavagna} and references
therein).

In the triangular plaquette model (TPM) we find only a single
crossover in the system at $L\sim c^{-1/\log_2 3}$. This is more in
line with the situation anticipated in BB. The same lengthscale is
again present in static multi-spin correlation functions, and in
four-point dynamical correlators.

We draw two conclusions from these results. Firstly, we are able to
verify that the procedure of BB is a suitable way to identify a mosaic
lengthscale.  In the general case it may be difficult to test the
criterion (\ref{equ:mos_len}) directly, although it is possible in the
SPM.  In less favourable situations, it is still possible to use
dynamical studies to extract the mosaic lengthscale, as we showed for
the TPM.

Secondly, we have shown that models without a finite temperature phase
transition can still be described from the mosaic perspective of KTW
and BB.  This indicates that the concept of a glassy mosaic state is
more general than the specific picture of KTW.  While the low
temperature states of the plaquette models are constructed by tiling
the plane with different metastable configurations of the spin system,
there are no well-defined droplets or domain walls in these states.
Rather, there are point-like defects that allow the system to
interpolate smoothly between different metastable configurations.
The crucial
difference with the KTW approach is that, while dynamics does proceed
by non-perturbative events that require co-operation over a
lengthscale that diverges at low temperatures, these events are not
related to nucleation, but to more efficient rearrangements of the
point-like defects.

In summary, typical states in the plaquette models are glassy mosaics
whose dynamics should be understood in terms of point defects with
co-operative dynamical rules. These defects are not locally conserved,
and are not related to any concept of free volume. Rather, they
represent regions in which the free energy barriers to motion are
smaller than average. In this approach, the ``activated processes'' that
dominate glassy $\alpha$-relaxation involve co-operative rearrangements
of defects over a lengthscale $\xi$. The time scales for these processes
increase with $\xi$, with a general relation
similar to~(\ref{equ:tau_barriers}).

The purpose of this work has been to investigate rather general
arguments of KTW and BB explicitly in specific, and rather simple,
finite dimensional models.  We find good evidence for a generalised
caging effect, but our view of static and dynamic properties of the
glassy mosaic is qualitatively different from that of KTW.  The
possible links between spatial variation in the mosaic length and
dynamical heterogeneity remain as an area for further study.

\acknowledgments

We thank G. Biroli and D. Sherrington for discussions.  This work was
supported by EPSRC grants no.\ GR/R83712/01 and GR/S54074/01, and
University of Nottingham grant no.\ FEF 3024.

\begin{appendix}
\section{High temperature series in the SPM}
\label{app:pq_series}

We follow \cite{Espriu-Prats} in making a high temperature expansion
for the square plaquette model.  The partition function for a droplet
of size $L$ with fixed boundary spins is
\begin{equation}
Z_\alpha = e^{-(L-1)^2 \beta/2} \sum_{\hbox{bulk }\sigma_{xy}}
\prod_{1\leq xy \leq L-1} e^{-\beta p_{xy}/2} ,
\end{equation}
with
\begin{equation}
p_{xy} = \sigma_{xy} \sigma_{x+1,y} \sigma_{x,y+1} \sigma_{x+1,y+1} ,
\end{equation}
as above.  Since the $p_{xy}$ are Ising variables then this can be
written
\begin{equation}
Z_\alpha = \left(\frac{1+c}{2}\right)^{(L-1)^2}
\sum_{\hbox{bulk }\sigma_{xy}}
\prod_{1\leq xy \leq L-1} (1+zp_{xy}) ,
\end{equation}
where $z=\tanh(\beta/2)=(1-c)/(1+c)$.

Expanding the product results in a power series in $z$: there are
$2^{(L-1)^2}$ terms corresponding to states of a dual system where the
spins are the $p_{xy}$. Each state is equal to a power of $z$
multiplied by a combination of the $\sigma_{xy}$; the term vanishes on
summation over the $\sigma_{xy}$ if it contains any spin to an odd
power. Otherwise, the coefficient of the power of $z$ depends only on
boundary spins: in that case, the sum over bulk spins just contributes
a trivial factor $2^{(L-2)^2}$.

Terms containing bulk spins to even powers are formed by flipping
whole rows and columns of the dual $p_{xy}$ variables. The result is
that
\begin{eqnarray}
Z_\alpha & = & 2^{-2L+3} (1+c)^{(L-1)^2} (1/2) \times \nonumber \\ & &
\sum_{h_y,v_x\in\{0,1\}} z^{(H+V)(L-1)-2HV} \times \nonumber \\ & &
\prod_{1\leq y\leq L-1} (\sigma_{1y} \sigma_{1,y+1} \sigma_{Ly}
\sigma_{L,y+1})^{h_y} \times \nonumber \\ & & \prod_{1\leq x\leq L-1}
(\sigma_{x1} \sigma_{x+1,1} \sigma_{xL} \sigma_{x+1,L})^{v_x} ,
\end{eqnarray}
where $H=\sum_y h_y$ and $V=\sum_x v_x$; $h_y=1$ means the $p_{xy}$ in
row $y$ have been flipped. We note that the boundary spins enter only
through combinations such as $\sigma_{1y} \sigma_{1,y+1} \sigma_{Ly}
\sigma_{L,y+1}$, which gives the parity of the number of defects in
the $y$th row of the dual lattice. Further, the terms in the sum
depend on only four numbers: $H$; $V$; the number of rows for which
$h_y=1$ and $\sigma_{1y} \sigma_{1,y+1} \sigma_{Ly}
\sigma_{L,y+1}=-1$; and the number of columns for which $v_x=1$ and
$\sigma_{x1} \sigma_{x+1,1} \sigma_{xL} \sigma_{x+1,L}=-1$. Let $r$ be
the number of rows for which $\sigma_{1y} \sigma_{1,y+1} \sigma_{Ly}
\sigma_{L,y+1}=-1$ and $r'$ be the equivalent number of columns. Then
we can write
\begin{eqnarray}
Z_\alpha &=& 2^{-2L+2} (1+c)^{(L-1)^2}
\nonumber \\
&& \times
\sum_{h=0}^r \sum_{v=0}^{r'}
\sum_{h'=0}^{L-1-r} \sum_{v'=0}^{L-1-r'} M^{(1)}_{h,v,h',v'} ,
\end{eqnarray}
with
\begin{eqnarray}
M^{(1)}_{h,v,h',v'} & = & \binc{r}{h} \binc{r'}{v} \binc{L-1-r}{h'}
 \binc{L-1-r'}{v'} \times \nonumber \\ & & (-1)^{h+v}
 z^{(H+V)(L-1)-2HV} ,
\end{eqnarray}
where now $H=h+h'$ and $V=v+v'$ since $h$ and $h'$ are the number of
rows with $h_i=1$ and odd and even defect parities respectively, and
$v$ and $v'$ are similar numbers of columns.  We denote the binomial
coefficients by $B_{n,r}\equiv n! / r!(n-r)!$

We may now sum over $v$ and $v'$, yielding
\begin{eqnarray}
Z_\alpha &=& 2^{-2L+2} (1+c)^{(L-1)^2} \nonumber \\ && \times
\sum_{h=0}^r \sum_{h'=0}^{L-1-r} \binc{r}{h} \binc{L-1-r}{h'}
M^{(2)}_{h,v} ,
\end{eqnarray}
with
\begin{eqnarray}
M^{(2)}_{h,h'} &=& (-1)^{h} (z^H+z^{L-1-H})^{L-r'-1} \nonumber \\ &&
\times (z^H-z^{L-1-H})^{r'} .
\end{eqnarray}

A similar procedure leads to the expression for the expectation value
of any spin, given a boundary condition. We state only the result
which depends on the coordinates of the spin $2\leq xy\leq L-1$, as
well as the total numbers of rows and columns with odd defect parity :
$r$ and $r'$. It also depends on $s$, the number of such rows with $y$
co-ordinate less than $j$ and $s'$: the number of such columns with
$x$ co-ordinate less than $i$.  (Clearly $s\leq r$ and $s'\leq r'$).

The result is that
\begin{equation}
\langle \sigma_{xy} \rangle = \sigma_{x1} \sigma_{11} \sigma_{1y}
Z_\alpha^{-1} M_\alpha ,
\end{equation}
where
\begin{equation}
M_\alpha = \sum_{h=0}^s \sum_{h'=0}^{y-1-s} \sum_{h''=0}^{r-s}
\sum_{h'''=0}^{L-y+s-r} M^{(3)}_{h,h',h'',h'''} ,
\end{equation}
with
\begin{eqnarray}
\lefteqn{ M^{(3)}_{h,h',h'',h'''} =} \nonumber \\ && (-1)^{h+h''}
(z^H+z^{L-1-H})^{L-x+s'-r'} \times \nonumber \\ & &
(z^H-z^{L-1-H})^{r'-s'} \times \nonumber \\ & &
(z^{H-2(h+h')}+z^{L-1-H+2(h+h')})^{x-1-s'} \times \nonumber \\ & &
(z^{H-2(h+h')}-z^{L-1-H+2(h+h')})^{s'} \times \nonumber \\ & &
B_{s',h} B_{y-1-s',h'} B_{r'-s',h''} B_{L-y+s'-r',h'''}
\end{eqnarray}

These results are still a little cumbersome, but they are exact, and
are can be evaluated computationally. The number of terms in the sum
may be as large as $\mathcal{O}(L^4)$, but this is polynomial rather
than exponential in the system size.

\section{TPM symmetries and constraints}
\label{app:triag_constraints}

Here we make some comments about the thermodynamics of finite
($L\times L$) droplets in the triangular plaquette model.  We first
identify the $2^{2L-1}$ transformations that leave the droplet energy
unchanged. Recall that
\begin{equation}
\sigma_{x+1,y} = q_{xy} \sigma_{xy} \sigma_{x,y+1} .
\end{equation}
It follows that if we specify the boundary spins $\sigma_{1y}$ and
$\sigma_{xL}$ (where $x,y=1\dots L$) then the rest of the spins are
determined by the set $\{q_{xy}\}$ (with $x,y=1\dots L-1$). Thus, the
required degenerate states for a given choice of the $\{q_{xy}\}$ can
be generated by cycling through the possible states of these two
boundaries.

To identify the constraints on the plaquette variables arising from
the boundary conditions, observe that for a ground state ($q_{xy}=1
\,\forall\, xy$) with specified spins along two boundaries
($\{\sigma_{1y},\sigma_{xL}\}$) then the state of the remaining
boundary spins are
\begin{eqnarray}
\sigma_{x1}^0 & = & \prod_{y=1}^x (\sigma_{1y})^{B_{x-1,y-1}} , \\
\sigma_{Ly}^0 & = & \prod_{y'=y}^{L-1} (\sigma_{1y'})^{B_{L-1,y'}}
\prod_{x=1}^y (\sigma_{xL})^{B_{L-1-x,L-1-y}} ,
\end{eqnarray}
where $\sigma^{B_{nr}}$ is equal to unity unless $\sigma=-1$ and
$B_{n,r}$ is odd, in which case it takes the value $-1$.

For a general configuration of the $q$'s and the boundary spins
$\sigma_{1y},\sigma_{xL}$, we have
\begin{eqnarray}
\sigma_{x1} & = & \sigma_{x1}^0 \prod_{y=1}^{x-1} \prod_{x'=1}^{x-y}
(q_{x'y})^{B_{x-x'-1,y-1}} , \\ \sigma_{Ly} & = & \sigma_{Ly}^0
\prod_{y'=0}^{L-y-1} \prod_{x=1}^{L-y'-1} (q_{y,y+y'})^{B_{L-x-1,y'}}
.
\label{equ:app_triag_constraint}
\end{eqnarray}

Multiplying both sides of each constraint by the appropriate
$\sigma^{0}$ leaves an equality between a product of boundary spins
and a product of plaquette variable. These are the $2L-3$ constraints
that determine which configurations of the excited plaquettes are
present in the partition sum.

While these expressions appear complicated, for a given droplet
boundary condition they are simply constraints of the form
\begin{equation}
q_{x_1,y_1} q_{x_2,y_2} \dots q_{x_n,y_n} = \pm 1 ,
\end{equation}
where the constrained plaquettes form parts of Sierpinski triangles.
For example, for $L=4$, the constraint coming from the boundary spin
$\sigma_{42}$ is
\begin{equation}
q_{12} q_{22} q_{32} q_{23} =
\sigma_{12} \sigma_{13} \sigma_{24} \sigma_{42} .
\end{equation}
In the SPM the constraints were of the form
\begin{equation}
\prod_{y=1}^{L-1} p_{xy} = \pm 1 .
\label{equ:app_pq_constraint}
\end{equation}
The simplicity of (\ref{equ:app_pq_constraint}) in the SPM, as
compared to (\ref{equ:app_triag_constraint}), is the reason for the
relative intractability of the partition function of the TPM.

\end{appendix}

redpajama_set_name: RedPajamaArXiv


text:
National Transparency CampaignsAIE Editor2017-05-12T14:03:50+02:00
PROMOTING AND DEFENDING TRANSPARENCY
GET THE NEWS IN YOUR E-MAIL
Spain: Access Info calls on Government to use Regulation to strengthen the Transparency Law
Madrid, 5 April 2019 – In its comments submitted to the public consultation on the draft Regulation for Spain's 2013 Transparency Law, Access Info has urged the government to take the opportunity to strengthen the right of access to information, with measures such as simplifying the request procedures. Specifically, Access Info has recommended that the new Regulation permit requests by
Access Info Europe helps create the CONSUL Democracy Foundation
22 March 2019 – Access Info Europe, together with civil society partners from across Europe, meeting today in Amsterdam, created a new foundation to promote citizen engagement in decision making using the CONSUL participation platform. The CONSUL Democracy Foundation will promote citizen participation worldwide and will oversee the use and development of the largest open source digital democracy platform – CONSUL,
Spain: Right to request information limited to citizens with valid ID, email, and a mobile phone
Madrid, 12 April 2019: The Spanish government has announced a new system for identifying people who want to submit a request for information under the 2013 transparency law: you need to give your name, your official Spanish-issued ID number, your email, and your mobile phone number. Access Info Europe, the Madrid-based right to information organisation, today criticised this system as
European Parliament urged to support protections for right of access to information in wake of killing of Slovak journalist
Madrid, 13 March 2018 – In light of the shocking news that Jan Kuciak was likely killed as a result of his work as an investigative journalist and whose freedom of information requests may have been passed to the subject of his inquiry, a total of 61 civil society organizations today wrote to the European Parliament calling for stronger safeguards
Letter to European Parliament with Statement on killing of Jan Kuciak
Madrid/Brussels, 13 March 2018 Dear Member of the European Parliament In advance of tomorrow's parliamentary debate on the killing of Slovak investigative journalist Jan Kuciak, please find below a statement signed by 61 civil society organisations from across Europe and elsewhere expressing their concern about the fact that his investigation may have been exposed after he submitted request for information,
Cyprus government flawed interpretation of European transparency standards
[UPDATE: Madrid, 11 December 2017 - On 25 April, the Cyptriot Government answered the letter sent by Access Info raising questions regarding to Cyprus Access to Information Law. You can find a copy of the letters here : ] Madrid, 15 June 2017 - Access Info Europe and six Cypriot civil society organisations [1] have raised concerns with the Cyprus
Cover photo: Michael Levine-Clark via Flick (CC BY-NC-ND 2.0)

redpajama_set_name: RedPajamaCommonCrawl


The IGA has worked effectively with businesses and Government to raise awareness of glaucoma, to improve the lives of people living with glaucoma and to help with fundraising.
We are very grateful for the support from Santen, as a result of their contribution we have been able to cover the cost for the printing of our Dry eye syndrome booklet. We are also working in collaboration with Santen to produce a series of films about glaucoma, the tests which are used to detect the condition, the support available and some of the issues that people have with the most common treatment which is eye drops.
Since 2015, the IGA has been working with Vision Express to increase professional knowledge of the condition and to increase awareness amongst the general public. We believe that everyone should have an eye health check every two years, as early detection of glaucoma literally saves sight.
From 2016, we have worked in partnership with Specsavers to raise awareness of glaucoma and to help train retail staff about the condition and it's treatment, specifically helping staff to understand the difficulties in taking eye drops and how they can help. The IGA receives £10,000 annually in recognition of this partnership work.
The IGA also works closely with the Drivers and Vehicle Licensing Agency (DVLA) to provide representation for people with glaucoma and to improve the systems, processes and way in which people with glaucoma are managed when reporting glaucoma in both eyes to the DVLA. We have been working with the DVLA since 2013 and have helped to develop an online system for reporting glaucoma, developed systems allowing people to raise concerns individually with the DVLA, effected change with people now being able to request a copy of their visual field test and clarified the testing conditions that should be met when taking a driving visual field test at the approved optometrist.


Advocates fill the news pages and airwaves with passionate statements of justification and condemnation. Their work involves morality: which of the combatants acts with justice, and which acts in evil ways? Those advocates who win this argument shape public opinion and that, in turn, influences or even determines government policies.
Advocacy and strategy each have their role. The advocate speaks of right and wrong, the strategist deals with success and failure. Passion marks the former, ice runs in the latter's veins. The advocate would choke on presenting his adversary's viewpoint but the strategist routinely puts himself in his opponent's place (think of war games). For example, I have imagined myself in charge of the Islamist movement, figuring out what it should do so as to understand how best to stop it.
Like most research institutes, the Middle East Forum focuses primarily on strategic activities. This is, I believe, the best use of our having devoted years or even decades to the study of our topics. To defend our outlook, attack the views of our opponents, and convince the undecided are crucial tasks, but not ours. MEF focuses on helping our side figure out how to win. Not being based in Washington, the Forum's work is directed primarily toward two audiences: (1) the educated public and (2) specialists who both share our outlook and seek information, analysis, and policy recommendations.
For example, when I discuss the media's biased coverage of Israel, I do so not to discredit it but to understand its logic and to suggest ways for those concerned with Israel's security and welfare to deal with it. Likewise, during the Gaza hostilities of 2008-09, I stayed away from justifying Israel's conduct or excoriating Hamas but instead criticized Israeli strategic ineptitude and offered an alternative approach.
"The Solace of Poor U.S.-Israel Relations." National Review Online, April 13, 2010.
"Pollard Pardon? Not Now." National Review Online, Feb 23, 2014.
"The Way to Peace: Israeli Victory, Palestinian Defeat." Commentary, January 2017.


text:
"Adekunle Gold, Viktoh And Chinko Ekun To Leave YBNL"
Adekunle Gold, Viktoh and Chinko Ekun are leaving YBNL following the termination of their two-year recording and management contract each.
Meanwhile, the musicians are still looking to Olamide as they fear for their future.
'Adekunle Gold's contract with YBNL recently expired. 'He's yet to decide on his contract renewal because he's not ready to go unaided.'
Adekunle joined YBNL as an artiste during the first ever OLIC (Olamide Live In Concert) in December 2014.
Before his major contract following the release of his hit, 'Sade', the singer – who designed the famous YBNL logo – used to work with the music label as a graphics designer.
The contracts of Viktoh and Chinko Ekun expired long time ago, but they decided to keep the update on the low as they fear it could make or tarnish careers.
Chinko Ekun was no where to be found at Olamide's concert in Lagos on Monday, where his label mates took turns on stage to thrill the crowd.
'Just like Chinko Ekun, both of them (Adekunle and Viktoh) don comot, but dem no loud am,' another spokesperson told us in pidgin.
On April 26, a month after dropping his debut album (Y.A.G.I), Lil Kesh announced his departure from YBNL following the expiration of his two-year contract to promote his music label.
It only took the label a week to sign new artiste, Temmie Ovwasa. And in November, Olamide contracted fast-rising Mushin-based rapper, Davolee, who's popular for his viral cover of 'Panda' by American artiste, Desiigner.
Three albums were released under YBNL Nation in 2016: Y.A.G.I by Lil Kesh and Adekunle Gold's Gold, which were released in March and July, respectively.
Label boss, Olamide also dropped his sixth studio album on Boxing Day, the same day his OLIC3 concert held at Eko Convention Centre in Lagos.
2017 will certainly be challenging for YBNL (Yahoo Boy No Laptop), which has been one of the prominent Nigerian music labels in the last two years.
Labels: Entertainments


Just last week we were proud to feature a video premiere for the excellent up and coming British band Moulettes and their live video for Patterns. We are back with a remix premiere that we are featuring as the song of the day!
The band plays in Toronto on July 7th at Hugh's Room so check them out!


text:
Arts In Motion
Featured supporter: Michael Giovannetti
Jer XiongAlumni News, Dean's Council Annual Fund, Development, Faculty and Staff, Featured Supporters, VideoAlumni, Alumni News, Faculty, Featured SupportersLeave a comment
Michael Giovannetti is a familiar name to many at Fresno State. Throughout his life, Giovannetti has given back to the university and the Fresno area in multiple ways.
"I have been blessed all of my life with many gifts. Therefore, giving back to the College of Arts and Humanities is an honor and a privilege for me," says Giovannetti.
2018 Outstanding Alumnus – Michael Giovannetti – Kremen School of Education and Human Development
His love for the Arts and Humanities stemmed from his Italian roots, growing up near Lucca, Tuscany, a short distance from Florence, Pisa, and other places of artistic significance. Every town in Italy, he felt, was a work of art. His interest was further nurtured by his parents, family members, and the local Catholic church.
"Michael Giovannetti exemplifies everything we hope for in a graduate from our College: after coming to America from Italy as a boy, he earned a Spanish BA and MA, and then devoted his life to teaching for decades, eventually pursuing a doctorate in educational leadership, too," said Dr. Honora Chapman, interim Dean of the College of Arts and Humanities at Fresno State.
Giovannetti came to Fresno State as an international student in 1968 to study Spanish and literature. He enjoyed his experience so much that after graduating in 1970 he continued his studies to receive his M.A. in Spanish language and literature in 1972 and a secondary education teaching credential.
"I credit my love for teaching primarily to the outstanding educators in the Department of Foreign Languages at Fresno State," says Giovannetti. "They were clearly the mentors and master teachers everyone should experience prior to their first teaching assignment and throughout one's educational journey."
The Department of Foreign Languages is now called the Department of Modern and Classical Languages and Literatures.
Since then, Giovannetti has collaborated with colleagues from the College of Arts and Humanities and the Kremen School of Education and Human Development and has worked for Fresno Unified, Fresno City College, Central Unified, Sanger Unified, and San Joaquin Memorial High School.
In 2001, he completed the Joint Doctorate degree program in educational leadership at Fresno State and UC Davis. After his retirement from the school district in 2004, he taught educational leadership classes at Fresno State and was Executive Director of the Renaissance Group, a national consortium of colleges, universities, schools/districts with a commitment to the preparation of educational professionals.
"I cannot imagine my life without the presence, friendship, and the unconditional support of teachers, administrators, colleagues, and friends from Fresno State."
Michael Giovannetti with his wife Silvana at the 2018 Top Dog Awards Gala
Michael Giovannetti is currently on the committee tasked with the creation of the Italian Studies Program. The ultimate goal is to offer all students exciting opportunities to complete an interdisciplinary minor in Italian studies in conjunction with other disciplines on the Fresno State campus.
"His love of learning and serving his community continues to inspire us as he now participates in our Italian Studies community committee in order to increase excitement and financial support for our new Italian Studies program," said Chapman.
Giovannetti believes the program will have successful outcomes, especially with strong support from the university, the Italian community and friends. He explains how it will offer new educational and cultural opportunities to students in the Central San Joaquin Valley and beyond.
"It will also be a unique model for sharing the rich Italian culture, history, traditions and beautiful language, while honoring the many immigrant Italians, who worked tirelessly, and sacrificed much in order to pave the way for a better life in America. Their families are now able to enjoy the 'American Dream.'"
Giovannetti's reasons for supporting the College of Arts and Humanities and the Dean's Council are simple. He's happy to support the team who puts students first with high quality innovative programs and services. Having worked closely with the leadership and staff for the past two plus years, he appreciates how the college always pays it forward by investing in their students.
He has been on the recipient end for his own education. In 1959, when he had finished his formal schooling in Italy at the age of 12, his aunt and uncle brought him to the United States to further his education.
Michael Giovannetti with Victor E. Bulldog
Now, throughout the years, he has seen how time, energy, talents and resources contributed by generous supporters of the College of Arts and Humanities have realized huge dividends. Because of the additional resources, more students will be able to complete their educational goals. Many more will be able to study abroad and have access to experiences previously only imagined.
"Whether setting up and then joining us for community speaking events or creating a connection with the University of Modena, Michael has been a champion of Fresno State and its commitment to humanistic learning. We are so grateful for his enthusiastic support!" said Chapman.
In the end, Giovannetti reminds us that "[The students] will become outstanding teachers, doctors, artists, musicians, engineers, lawyers, and providers of the many services we need and receive each and every day."
Help support the College of Arts and Humanities today!
Click here, select "College of Arts and Humanities."
When asked how it feels to give back, he says it's the least he could do.
"It's truly a wonderful feeling."
Humbly, he adds that he sees the College of Arts and Humanities as the 'heart' of the university and the community; he is merely one of its many fans.
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2021 Graduate of Fresno State's MFA Creative Writing program, focused in Creative Nonfiction. Hmong American writer.
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All of us in the College of Arts and Humanities hope that you and yours are doing well!
At this moment in time, there is no doubt that the arts and humanities allow us to seek a deeper understanding of ourselves and our world. Our faculty and students have continued this spring semester to grapple with the question of what it means to be human in their mostly virtual classes.
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Fresno State College of Arts and Humanities
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Email: benjaminkirk@csufresno.edu
The College of Arts and Humanities at Fresno State


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Bunker Sand
Originating on the East coast of Scotland, early golfers hit pebbles over sand dunes. These sandy hazards to any golfers game were initially a purely organic landscape feature, created by animals. Over time Bunkers were incorporated and evolved to be the dreaded feature in all golf courses throughout the world.
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Firth of Forth Beach finds
With it's 11,602 miles (18,672 km) of coastline, accessing Scotland's stunning beaches, where the ice cold waters meet the roaming land, is never far out of reach. From popular family-friendly tourist spots with their spectacular shores to remote hidden coastal gems with their nooks and crannies offering peaceful isolation. Scotland's beaches provide many vastly different coastal pictures, yet all are wonderfully evocative in their own rights. From stunning silver sands and crystal clear turquoise water to pebble, driftwood and sea-glass stroon shores with their crashing waves, Scotland's beaches offer a haven of treasures to help you while away the hours and blow the cobwebs away.
Glasgow Cone
An unusual but light hearted addition to your Wee Piece of Scotland collection. Scottish people, have their one of a kind sense of humour. Edinburgh has the Castle and Glasgow has a statue with a traffic cone on its head directly outside the Gallery of Modern Art. For over 140 years the statue of the Duke of Wellingstone was just another statue. Until one evening in 1980's, a traffic cone mysteriously appeared on top of his head. Despite continuous efforts to remove the cone by the local Council, the cone has remained in place for over 30 years. If the cone is removed it's only a matter of days until a new one appears.
Haggis replica
With it's origin's shrouded in mystery and its infamous association with Rabbie Burns, couple with the folklore of wild hairy rodent like creatures, with it's mismatched legs scuttling across the hills in one direction…Haggis is one of Scotland classic symbols. In reality, this rudimental yet practical way of utilising sheep offal the content and casing of haggis may not be everyone's first culinary choice. If the thought of eating the original Haggis recipe leaves you feeling a bit 'peely-wally', the vegetarian variety is a delicious and possibly less formidable gastronomic option, together with Haggis crisps or ice cream. Alternatively yo could always give 'Haggis Hurling' a go!!
Growing freely be it high up on the moorlands or down to the sea cliffs, heather is a sturdy wee ever green tree whose delicate flowers bloom twice a year where the delicate purple flower love to dance in the wind. 'Haeddre', 'Heddir' or 'Hathar' as it was called in times gone casts a glorious purple hue as far as the eye can see. Not just a pretty face on the landscape heather is used in medicines, teas, roof thatching, honey production,scents, jewellery, beers and so much more. Plain and simply put, the much loved heather is 'Braw' (excellent)
Highland Games Caber
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Captured within this Wee Piece of Scotland bottle is a lovingly handmade wee Nessie. This Nessie may not be as terrifying as coming face to face with the real thing…we've added a smile to her face…so's not to scare the bairns!!!
Scottish parritch (porridge) is synonymous with our country and has been for many a century. Since late medieval times oats have been part of the Scot's staple diet. With it's many traditions and superstitions this oaty concoction….. Scottish porridge should be made stirred with a wooden stick (known a Spurtle). It should always be stirred clockwise with your right hand and should be eaten from a wooden bowl whilst standing up. In times gone by, porridge would be cooked up in a big pot then poured into 'drawers' and allowed to cool, then sliced so it could be eaten throught the day. It could be said the Scots and our porridge invented the first take away meal!!. If you disagree 'save yer breath tae cool yer parritch' (hold your tongue or simply say nothing)
Romantic Scotland
Since the days of eloping to Gretna Green, Scotland has been synonymous with romance. Young couples slipping away together, crossing the Scottish border, where the convenient Blacksmith Shop offered the perfect wedding venue for love struck runaways. The hammering of the 'Old Smithy's' anvil romantically signifies the forging of couples together in the heat of the moment, binding them together for eternity. With its historic romantic tales, together with a wealth of ancient castles, remote wilderness, dramatic landscapes and warm welcomes, Scotland truly does capture romance in its very own uniquely magical way.
Shetland Peat
Peat, may look less than exciting, but it's worth it's weight in gold. Formed from organic matter (mostly plant based)which has laid untouched for thousands of years in land water-logged from the melting glaciers of the Ice age, peat truly is a botanical timecapsule. Not only was Peat one of the most commonly used fuels many Scottish island dwellers used to heat their homes, it is also what gives some whiskey a smoky flavour and the reason behind why the water of Loch Ness is so very dark. Not only that…It's an incredibly important natural ally in the fight against climate change. As 20% of Scotland's land is peatland….it truly an often underrated 'Gem of Scotland'
Shetland Sheep's Wool
In Scotland there are 6.6 million sheep and 5.3 million people at the last count (give or take). The most commonly found sheep is the blackfaced, with it's course, tough wool. In Scotland the Ram (male) is called a Tup and the Ewe (female) is called a Girn. Sheep can be found grazing throughout Scotland , often in the highlands and islands these wooly walking clouds can be found strolling in front of cars on unfenced roads and giving you that look that says 'Awa an bile yer heid' (translating into 'get lost')
Before we start..tartan and Plaid are not the same!! :) The criss cross patterned fabric known worldwide as 'Plaid' is only transforms into 'Tartan' when the plaid patterned fabric is associated to a particular name, a specific clan, family, or other community. Without this association the plaid is nothing more than a pretty patterned fabric, maybe used as a shawl or bedcover. Simply put…'No alliance, means, it's No tartan!!'
Unicorn Horn
It true: the unicorn really is the official national animal of Scotland.
This untameable, proud white horse-like creature with it's golden spiralling horn is considered one of the most iconic symbols of Scotland. It even has it's own national day on the 9th of April!! It's a a fiercely independent beast notoriously difficult to seize and conquer. The unicorn, to the Scots, is a symbol of purity, innocence and power. Legend has it, such is the strength of the healing power held within their golden horn, the unicorn can purify even the most poisoned of waters.The unicorn was first used on the Scottish royal coat of arms in the 12th century. The perfect representation of the pride, strength and other-worldly qualtities that the Scots possessed to remain unconquered…or at the least remain that wee big magical and legendary.
Wild Thistle
For over 500 years the thistle has been one of the most important symbols of Scotland. Legend has it this hardy, spike filled plant saved Scots warriors from potential ambuse from an invading Norse army. Many say the magnificent thistle is a weed…we prefer to see it as a strong, free spirited flower which is hardy, resilient and persistent…after all 'beauty is in the eye of the beholder'
Captured within this Wee Piece of Scotland bottle is a hand picked, dried and bottled thistle head.(and yes they do hurt when foraging for them!!) This thistle is a hardy symbol of resilience, devotion, bravery, determination, and strength. Where one person sees a weed , another view a flower. In other words as the Scot's say 'We're a' Jock Tamson's bairns!' (We're all God's creations, nobody is better than anybody else – we're all equal.)
An alternative 'flower of Scotland' can be found in the spring dandelion. The curse for many a keen gardener, but truly a beautiful flowering plant which has three phases to it's growth. The three phases represent the sun, moon and stars. The yellow flower represents the sun, the white puff ball represents the moon and the dispersing seeds represent the stars. So when we blow the seeds to make a wish…we are wish on the stars.Many people believe that dandelion seeds will carry your thoughts and dreams to loved ones when you blow them into the air.
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