Mixing Length Scales of Low Temperature Spin Plaquettes Models

Abstract

Plaquette models are short range ferromagnetic spin models that play a key role in the dynamic facilitation approach to the liquid glass transition. In this paper we perform a rigorous study of the thermodynamic properties of two dimensional plaquette models, the square and triangular plaquette models. We prove that for any positive temperature both models have a unique infinite volume Gibbs measure with exponentially decaying correlations. We analyse the scaling of three a priori different static correlation lengths in the small temperature regime, the mixing, cavity and multispin correlation lengths. Finally, using the symmetries of the model we determine an exact self similarity property for the infinite volume Gibbs measure.

Appendix 1

As anticipated in Sect. 3 we collect here the proof of the ordering of the three length scales $\ell _c^{(\text {mix})},\ell _c^{(\text {cavity})}$ and $\ell _c^{(\text {multispin})}$ and of Proposition 5.2.

1.1 1.1. Proof of (3.1)

We begin by proving that $\ell _c^{(\text {multispin})}=O(\ell _c^{(\text {cavity})})$. Let A be a finite set, with $|A| \ge 2$, such that $\min _{x,y\in A}d(x,y)\ge C\ell _c^{(\text {cavity})}$ and fix $x\in A$. Let also $\Lambda$ be the square centered at x of side $10\ell _c^{(\text {cavity})}$. For C large enough no other points of A belong to $\Lambda$. Using the DLR equations we write

$$\begin{aligned} \mu ^\beta ([\sigma ]_A)= \int d\mu ^\beta (\tau )\, [\tau ]_{A\setminus \{x\}}\,\mu _\Lambda ^{\beta ,\tau }(\sigma _x). \end{aligned}$$

Recall now that $\mu ^\beta (\sigma _x)=0$. Thus

$$\begin{aligned} \big |\mu _\Lambda ^{\beta ,\tau }(\sigma _x)\big |&\le \Big |\int d\mu ^\beta (\tau ')\,\left( \mu _\Lambda ^{\beta ,\tau }(\sigma _x)-\mu _\Lambda ^{\beta ,\tau '}(\sigma _x)\right) \Big | \\&\le \sup _{\tau ,\tau '}\big |\mu _\Lambda ^{\beta ,\tau }(\sigma _x)-\mu _\Lambda ^{\beta ,\tau '}(\sigma _x)\big |\le \frac{1}{5}, \end{aligned}$$

by construction and the definition of $\ell _c^{(\text {cavity})}$. The result follows immediately.

Next we prove that $\ell _c^{(\text {cavity})}=O(\beta \,\ell _c^{(\text {mix})})$. Fix two concentric squares $V\subset \Lambda$ of side $\ell$ and $10\ell$ together with $f:\Omega _V\mapsto {\mathbb R}$ with $\Vert f\Vert _\infty \le 1$. The triangular inequality and (3.4) (notice that $h_x\ge e^{-2\beta \Vert H\Vert }$) imply that

$$\begin{aligned} \big | \mu _\Lambda ^{\beta ,\tau }(f)-\mu _\Lambda ^{\beta ,\tau '}(f)\big |\le e^{2\beta \Vert H\Vert }\sum _{x\in \Lambda ^c}\sup _{\xi }\big |{\text {Cov}}_\Lambda ^{\beta ,\xi }(h_x,f)\big |. \end{aligned}$$

Choose now $\ell = C\beta \,\ell _c^{(\text {mix})}$. The definition of $\ell _c^{(\text {mix})}$ together with (3.2) and Proposition 3.4 implies that there exists a constant c such that the r.h.s. above is not larger than $e^{c\beta }e^{-\ell /\ell _c^{(\text {mix})}}\ll 1$ for $C\gg 1$.

1.2 1.2. Proof of Proposition 5.2

Let $\Lambda = [-\ell ,\ell ]^2\cap {\mathbb Z} ^2$ and recall the definition of the plaquettes family $\mathcal B^+(\Lambda )$ under plus boundary conditions and of the associated space of cycles $\mathcal K^+ (\Lambda )$, given right before Lemma 5.1. In what follows it will useful to think of $\mathcal K^+(\Lambda )$ as ${\mathbb F} _2$–vector space, by taking the symmetric difference as summation. In particular, if $\mathcal A$ is a collections of cycles in $\mathcal K^+(\Lambda )$ then a plaquette $B^+$ will belong to $\sum _{\alpha \in \mathcal A}\alpha$ iff it belongs to an odd number of cycles in $\mathcal A$.

In $\mathcal K^+(\Lambda )$ we consider the following collection $\mathcal G$ of special cycles $\{R_i,C_i\}_{i=0}^{2\ell +1}$, called row and column cycles respectively. Order the rows and columns of $\Lambda$ from bottom to top and left to right. For $i,j\notin \{0,2\ell +1\}$, the cycle $R_i(C_j)$ consists of all the plaquettes in $\mathcal B^+(\Lambda )\cap \mathcal B(\Lambda )$ whose lowermost (leftmost) vertex lies in the $i^{th}$-row ($j^{th}$-column) of $\Lambda$. For $i=0$ ($i=2\ell +1$) $R_i$ consists of all the plaquettes in $\mathcal B^+(\Lambda )$ lying on the first (last) row. Similarly for $C_0,C_\ell$.

The following are few elementary properties of $\mathcal G$ which we collect for convenience in a lemma whose proof is omitted.

.

Lemma A.1

• (a) Let $\mathcal A\subset \mathcal G$ be nonempty and suppose that $\sum _{\alpha \in \mathcal A} \alpha =\emptyset$. Then $\mathcal A=\mathcal G$.

• (b) Let $\mathcal A,\mathcal B\subset \mathcal G$. Suppose that $\sum _{\alpha \in \mathcal A}\alpha =\sum _{\alpha \in \mathcal B}\alpha$. Then $\mathcal A=\mathcal B$ or $\mathcal B=\mathcal G\setminus \mathcal A$.

• (c) Given $\alpha ' \in \mathcal K(\Lambda )$ there exists $\mathcal A\subset \mathcal G$ such that $\alpha '=\sum _{\alpha \in \mathcal A} \alpha =\sum _{\alpha \in \mathcal G\setminus \mathcal A} \alpha$.

Corollary A.2

Given $f: \mathcal K^+(\Lambda )\rightarrow {\mathbb R}$, we have the identity

$$\begin{aligned} \sum _{\alpha \in \mathcal K^+(\Lambda )} f(\alpha ) = \frac{1}{2} \sum _{ W \subset \mathcal G} f( \alpha (W) ), \end{aligned}$$

where $\alpha (W):= \sum _{\alpha \in W} \alpha$.

We are now ready to write a workable formula for $\mu _\Lambda ^+(\sigma _0)$. Let $\alpha _*$ denotes the family of plaquettes in $\mathcal B^+(\Lambda )$ contained in $[0,\ell ]\times [0,-\ell ]\cap {\mathbb Z} ^2$ so that $\sum _{B^+\in \alpha _*}B^+=\{0\}$. Using [8, eq. (2.12)] and Corollary A.2 we have .

$$\begin{aligned} \mu ^+_\Lambda ( \sigma _0) = \frac{ \sum _{W \subset \mathcal G} \tanh (\beta /2)^{| \alpha (W) \triangle \alpha _*|}}{ \sum _{W \subset \mathcal G} \tanh (\beta /2)^{| \alpha (W) |} }\equiv \frac{N(\beta )}{D(\beta )}. \end{aligned}$$

(A.1)

Lemma A.3

If $W\subset \mathcal G$ contains i row cycles and j column cycles then $|\alpha (W)|=(i+j)(2\ell +2) -2 ij$. Moreover, if $W\subset \mathcal G$ consists of u rows among $R_0, R_1, \dots , R_\ell$, v rows among $R_{\ell +1}, R_{\ell +2}, \dots , R_{2\ell +1}$, j columns among $C_0, C_1, \dots , C_\ell$ and k columns among $C_{\ell +1}, C_{\ell +2}, \dots , C_{2\ell +1}$, then

$$\begin{aligned} |\alpha (W) \Delta \alpha _*|= jL+vL-2vj-2uj-2vk+2uk\,.\end{aligned}$$

Proof

For the first assertion we observe that each $\mathcal B^+(\Lambda )$–plaquette that appears in $\alpha (W)$ must be exactly either in a row cycle or in a column cycle of W. For the second assertion, we note that $|\alpha (W) \Delta \alpha _*|$ can be obtained from $|\alpha (W)|=(u+v+j+k)(2\ell +2) -2 (u+v)(j+k)$ (as in the first assertion) by subtracting twice the number of $\mathcal B^+(\Lambda )$–plaquettes of $\alpha (W)$ inside $[0,\ell ]\times [0,-\ell ]$ (this number is $(u+k)(\ell +1)- 2 uk$) and adding the number all $\mathcal B^+(\Lambda )$–plaquettes in $[0,\ell ]\times [0,-\ell ]$ (this number is $(\ell +1)^2$). $\square$

For notation convenience let $L=2\ell +2$ and let $t=\tanh (\beta /2)$. As a consequence of the above lemma we get immediately .

$$\begin{aligned} D(\beta )&= \sum _{i=0}^L \sum _{j=0}^L \left( {\begin{array}{c}L\\ i\end{array}}\right) \left( {\begin{array}{c}L\\ j\end{array}}\right) t^{iL +jL - 2 ij}= \sum _{i=0}^L \left( {\begin{array}{c}L\\ i\end{array}}\right) ( t^i+ t^{L-i})^L\end{aligned}$$

(A.2)

$$\begin{aligned} N(\beta )&= t^{\frac{L^2}{4}}\sum _{u,v,j,k} \left( {\begin{array}{c}L/2\\ u\end{array}}\right) \left( {\begin{array}{c}L/2\\ v\end{array}}\right) \left( {\begin{array}{c}L/2\\ j\end{array}}\right) \left( {\begin{array}{c}L/2\\ k\end{array}}\right) t^{ jL+vL-2vj-2uj-2vk+2uk}\end{aligned}$$

(A.3)

$$\begin{aligned}&= t^{\frac{L^2}{4} } \sum _{u=0} ^{L/2}\sum _{v=0}^{L/2} \left( {\begin{array}{c}L/2\\ u\end{array}}\right) \left( {\begin{array}{c}L/2\\ v\end{array}}\right) ( t^{2v}+ t^{L-2v} + t^{2u} + t^{L-2u} ) ^{L/2}. \end{aligned}$$

(A.4)

In order to bound from below the ratio $N(\beta )/D(\beta )$ it is convenient to rewrite both expressions in a probabilistic fashion. Let X be a centered Bin(L, 1 / 2) random variable and let Y, Z be i.i.d. centered Bin(L / 2, 1 / 2) random variables. Then (A.2) and a little algebra give

$$\begin{aligned} D(\beta )&=2^{2L}t^{L^2/2}\ {\mathbb E} \left( \left[ \frac{t^{X}+t^{- X}}{2}\right] ^L\right) ,\\ N(\beta )&=t^{\frac{L^2}{2} } 2^{2L}{\mathbb E} \left( \left[ \frac{t^{2Y}+t^{-2 Y}+t^{2Z}+t^{-2Z}}{4}\right] ^{L/2}\right) . \end{aligned}$$

Trivially $N(\beta )\ge 2^{2L}t^{\frac{L^2}{2} }$. Moreover

$\cosh (x)\le e^{x^2/2}$ implies that

$$\begin{aligned} D(\beta )\le 2^{2L}t^{L^2/2}\ {\mathbb E} (e^{ (\log t)^2 L X^2/2}).\end{aligned}$$

Thus

$$\begin{aligned} \frac{N(\beta )}{D(\beta )}\ge \frac{1}{{\mathbb E} (e^{(\log t)^2 L X^2/2})}\,. \end{aligned}$$

Next we bound from above the above expectation value. For any $\lambda >0$ we write

$$\begin{aligned} {\mathbb E} (e^{ (\log t)^2 L X^2/2}) \le \lambda + \int _{\lambda }^\infty da \ {\mathbb P} (e^{ (\log t)^2 L X^2/2}\ge a). \end{aligned}$$

The Azuma–Hoeffding’s inequality (cf. e.g. [19]) implies that ${\mathbb P} (| X|\ge \kappa )\le 2e^{-2\kappa ^2/L}$, so that

$$\begin{aligned} {\mathbb P} (e^{(\log t)^2 L X^2/2}\ge a)\le 2\exp \left( -4 \,\log a /(\log t)^2 L^2\right) = 2a^{-4/((\log t)^2 L^2)}. \end{aligned}$$

If $\gamma :=\frac{4}{(\log t)^2 L^2}>1$, then

$$\begin{aligned} {\mathbb E} (e^{\log ( t)^2 L X^2/2})\le \lambda + \frac{2}{(\gamma -1)\lambda ^{\gamma -1}}, \end{aligned}$$

(A.5)

which, after optimising over the free parameter $\lambda$, becomes

$$\begin{aligned} {\mathbb E} (e^{(\log t)^2 L X^2/2})\le 2^{1/\gamma } +\frac{2}{(\gamma -1)2^{(\gamma -1)/\gamma }}. \end{aligned}$$

Notice that $\gamma >1$ is equivalent to $L< 2/|\log t| =e^\beta (1+o(1))$.

Recalling that $L=2\ell +2$ we conclude that for any $\delta \in (0,1)$ and any $\ell < (1-\delta )e^\beta /2$

$$\begin{aligned} \liminf _{\beta \rightarrow \infty } \mu ^+_\Lambda ( \sigma _0) =\liminf _{\beta \rightarrow \infty }\frac{N(\beta )}{D(\beta )}>0 \end{aligned}$$

as required.$\square$

1.3 1.3. Proof of Lemma 4.8

To prove the first assertion it is enough to show that if $\alpha ,\gamma \in \mathcal K( T_*^{(n)})$ then $\alpha + \gamma \in \mathcal K( T_*^{(n)})$. To this aim take $v \in T_*^{(n)}$ and call $\Delta _1, \Delta _2, \Delta _3$ the three triangular plaquettes containing v. We set $n_i := \mathbb {1} ( \Delta _i \in \alpha )$ and $m_i:= \mathbb {1} ( \Delta _i \in \gamma )$. Since $\alpha$ and $\gamma$ are cycles of $T_n^*$ we have $n_1+n_2+n_3\equiv 0$ mod 2, and $m_1+m_2+m_3\equiv 0$ mod 2. If we call $k_i := \mathbb {1} ( \Delta _i \in \alpha +\gamma )$, by definition of $\alpha +\gamma$ we have $k_i = n_i +m_i$ mod 2. We conclude that $k_1+k_2+k_3\equiv 0$ mod 2, hence v belongs to an even number of plaquettes in $\alpha +\gamma$. By the arbitrariness of $v \in T_*^{(n)}$ we conclude that $\alpha +\gamma$ is a cycle.

We now prove that $P_{-1},P_0,\ldots ,P_n$ is a basis of $\mathcal K(T^{(n)}_*)$. We observe that the Pascal’s triangle $\mathcal P_0$ (rooted at the origin 0 of ${\mathbb Z} ^2$) is a cycle in ${\mathbb Z} ^2 \setminus \{0\}$, that is every site in ${\mathbb Z} ^2$ is contained in an even number of plaquettes of $\mathcal P_0$ except for the origin. It follows immediately that $P_i$ is a cycle of $T_*^{(n)}$ (see Fig. 4). To prove that $P_{-1}, \dots , P_n$ are linearly independent, suppose that $a_{-1} P_{-1}+ \dots + a_n P_n = \emptyset$ with $a_i \in {\mathbb F} _2$. By construction we have $(i,-1)+B_* \in P_j$ if and only if $i=j$. Hence $(i,-1)+B_*$ appears $a_i$ times in the cycle $a_{-1} P_{-1}+ \dots + a_n P_n = \emptyset$, and therefore $a_i=0$ for each $i=-1,0,\ldots ,n$.

It remains to show that $P_{-1},P_0,\ldots ,P_n$ generates $\mathcal K( T_*^{(n)})$. For this we will use the following result:

Claim A.4

Every non-empty cycle in $\mathcal K( T_*^{(n)})$ contains a plaquette of the form $(i,-1)+B_*$ for some $i\in \{-1,0,\ldots ,n\}$.

Before proving our claim, we conclude the proof of Lemma 4.8. Fix $\alpha \in \mathcal K( T_*^{(n)})$, let $I(\alpha ) = \{i \in \{-1,0,\ldots ,n\}\,:\, (i,-1)+B_* \in \alpha \}$, and consider the cycle $\gamma := \alpha + \sum _{ i\in I(\alpha ) }P_i\, .$ By construction $(i,-1)+B_* \not \in \gamma$ for any $i=-1,0, \dots , n$. Therefore, by Claim A.4, $\gamma = \emptyset$. Equivalently, we have $\alpha =\sum _{ i\in I(\alpha ) }P_i$ as required.

Proof

(Proof of Claim A.4) Suppose, for contradiction, that $\alpha$ is a non-empty cycle and $(i,-1)+B_* \not \in \alpha$ for any $i=0, \dots , n$. Let $\mathcal R$ be the horizontal line passing through the lowest vertices contained in any plaquette of the cycle $\alpha$. By assumption $\mathcal R$ lies on or above the line $\{ k\mathbf {e}_1: k \in {\mathbb Z} \}$, and therefore $\mathcal R\cap \left( \cup _{B \in \alpha }B \right) \subset T^{(n)}_*$ and is non empty. Fix $v\in \mathcal R\cap \left( \cup _{B \in \alpha }B \right)$, by the definition of a cycle v must belong to an even number of plaquettes in $\alpha$. Since v belongs to exactly one plaquette rooted on $\mathcal R$, and two plaquette rooted below $\mathcal R$, $\alpha$ must contain at least one plaquette rooted below $\mathcal R$, which contradicts the minimality of $\mathcal R$. $\square$