Cluster Expansion for the Ising Model in the Canonical Ensemble

Abstract

We show the validity of the cluster expansion in the canonical ensemble for the Ising model. We compare the lower bound of its radius of convergence with the one computed by the virial expansion working in the grand-canonical ensemble. Using the cluster expansion we give direct proofs with quantification of the higher order error terms for the decay of correlations, central limit theorem and large deviations.

Appendix: Proofs of Theorem 3, Theorem 4 and Corollary 5

We define the following objects:

$$ J^{C}_{\mu}(N,N^{\prime}):=\frac{e^{\beta\mu N}Z^{\mathbf{0}}_{{\varLambda},\beta}(N)}{e^{\beta\mu N^{\prime}}Z^{\mathbf{0}}_{{\varLambda},\beta}(N^{\prime})}, $$

(A.1)

and

$$ K(\mu,N):=\left( \frac{{\varXi}^{\mathbf{0}}_{{\varLambda},\beta}(\mu)}{e^{\beta\mu N}Z^{\mathbf{0}}_{{\varLambda},\beta}(N)}\right)^{-1}. $$

(A.2)

Let us note that from (2.13) the term $J^{C}_{\mu _{0}}(\tilde {N},\bar {N}_{{\varLambda }})$ can be written as:

$$ J^{C}_{\mu_{0}}(\tilde{N},\bar{N}_{{\varLambda}})=\exp\left\{ \beta\mu_{0}(\tilde{N}-\bar{N}_{{\varLambda}})+|{\varLambda}|\beta f_{{\varLambda},\beta,\mathbf{0}}(\bar{N}_{{\varLambda}})-|{\varLambda}|\beta f_{{\varLambda},\beta,\mathbf{0}}(\tilde{N})\right\}, $$

(A.3)

which is the finite volume version of (6.7) viewed in the canonical ensemble. Moreover, we can also write

$$ [K(\mu_{0},\bar N_{{\varLambda}})]^{-1}=\sum\limits_{N\ge0}J^{C}_{\mu_{0}}(N,\bar{N}_{{\varLambda}}). $$

(A.4)

Finally, before giving the proofs of the theorems, we remark that the object defined in (6.1), can be written as follows:

$$ L^{\mathbf{0}}_{{\varLambda},\beta,\mu_{0}}(\mu)=\beta|{\varLambda}|\left[p_{\beta,{\varLambda},\mathbf{0}}(\mu+\mu_{0})-p_{\beta,{\varLambda},\mathbf{0}}(\mu_{0})\right]. $$

(A.5)

Proof of Theorem 3

We rewrite $\mathbb {P}^{\mathbf {0}}_{{\varLambda },\mu _0}(A_{\tilde {N}})$ as follows:

$$ \mathbb{P}^{\mathbf{0}}_{{\varLambda},\mu_0}(A_{\tilde{N}})=\frac{{\varXi}^{\mathbf{0}}_{{\varLambda},\beta}(\tilde{\mu}_{{\varLambda}})e^{\beta\mu_{0}\tilde{N}}}{{\varXi}^{\mathbf{0}}_{{\varLambda},\beta}(\mu_{0})e^{\beta\tilde{\mu}_{{\varLambda}}\tilde{N}}}\mathbb{P}^{\mathbf{0}}_{{\varLambda},\tilde{\mu}_{{\varLambda}}}(A_{\tilde{N}}). $$

(A.6)

In the previous one we did the Radon-Nikodým derivative of our probability measure with respect to the one with $\tilde {\mu }_{{\varLambda }}$ instead of μ₀. Note that the definition of $\tilde {\mu }_{{\varLambda }}$ given via (2.43), i.e., such that

$$ \beta f^{GC}_{{\varLambda},\beta,\mathbf{0}}(\tilde{\rho}_{{\varLambda}})=\beta \tilde{\mu}_{{\varLambda}}\tilde{\rho}_{{\varLambda}}-\beta p_{{\varLambda},\beta,\mathbf{0}}(\tilde{\mu}_{{\varLambda}}), $$

(A.7)

is equivalent to define implicitly $\tilde {\mu }_{{\varLambda }}$ as the chemical potential such that

$$ \frac{\tilde{N}}{|{\varLambda}|}=\mathbb{E}^{\mathbf{0}}_{{\varLambda},\tilde{\mu}_{{\varLambda}}}\left[\frac{N}{|{\varLambda}|}\right]=\frac{\partial}{\partial\mu}p_{{\varLambda},\beta,\mathbf{0}}(\mu)\bigg|_{\mu=\tilde{\mu}_{{\varLambda}}}. $$

(A.8)

Moreover, from (A.5) and (6.7) we have that this $\tilde {\mu }_{{\varLambda }}$ is equal to the one which satisfies (6.6).

From (2.44), (2.43), (2.41) and (A.7) we get

$$ \begin{array}{@{}rcl@{}} \frac{{\varXi}^{\mathbf{0}}_{{\varLambda},\beta}(\tilde{\mu}_{{\varLambda}})e^{\beta\mu_{0}\tilde{N}}}{{\varXi}^{\mathbf{0}}_{{\varLambda},\beta}(\mu_{0})e^{\beta\tilde{\mu}_{{\varLambda}}\tilde{N}}} &= &\exp\left\{|{\varLambda}|\left[\beta\mu_{0}\tilde{\rho}_{{\varLambda}}-\beta\tilde{\mu}_{{\varLambda}}\tilde{\rho}_{{\varLambda}}+\beta p_{{\varLambda},\beta,\mathbf{0}}(\tilde{\mu}_{{\varLambda}})-\beta p_{{\varLambda},\beta,\mathbf{0}}(\mu_{0})\pm\beta\mu_{0}\bar{\rho}_{{\varLambda}}\right]\right\} \\ &=&\exp\left\{|{\varLambda}|\left[\beta f^{GC}_{{\varLambda},\beta,\mathbf{0}}(\bar{\rho}_{{\varLambda}})-\beta f^{GC}_{{\varLambda},\beta,\mathbf{0}}(\tilde{\rho}_{{\varLambda}})+\beta\mu_{0}(\tilde{\rho}_{{\varLambda}}-\bar{\rho}_{{\varLambda}})\right]\right\} \\ &=& \exp\left\{- |{\varLambda}| I^{GC}_{{\varLambda},\beta,\mathbf{0}}(\tilde{\rho}_{{\varLambda}};\bar{\rho}_{{\varLambda}})\right\}. \end{array} $$

(A.9)

On the other hand, denoting with $\tilde {N}^{*}$ the number of particles such that

$$ \sup_{N}\left\{e^{\beta\tilde{\mu}_{{\varLambda}}N}Z^{\mathbf{0}}_{{\varLambda},\beta}(N)\right\}=e^{\beta\tilde{\mu}_{{\varLambda}}\tilde{N}^{*}}Z^{\mathbf{0}}_{{\varLambda},\beta}(\tilde{N}^{*}), $$

(A.10)

using (A.1) and (A.2) we have

$$ \mathbb{P}^{\mathbf{0}}_{{\varLambda},\tilde{\mu}_{{\varLambda}}}(A_{\tilde{N}})=J^{C}_{\tilde{\mu}_{{\varLambda}}}(\tilde{N},\tilde{N}^{*})K(\tilde{\mu}_{{\varLambda}},\tilde{N}^{*}). $$

(A.11)

The novelty here is that we compute the above term using cluster expansions instead of inverting the characteristic function. First, we recall that we have

$$ |\tilde{N}-\tilde{N}^{*}|\le C, $$

(A.12)

for some C > 0 which does not depend on Λ. Then we find

$$ \begin{array}{@{}rcl@{}} J^{C}_{\tilde{\mu}_{{\varLambda}}}(\tilde{N},\tilde{N}^{*})&=&\exp\left\{ S^{\prime}_{|{\varLambda}|}(\tilde{\rho}_{{\varLambda}}^{*})(\tilde{N}-\tilde{N}^{*})-\sum\limits_{m\ge 2}\frac{(\tilde{N}-\tilde{N}^{*})^{m}}{|{\varLambda}|^{m-1}}\frac{\mathcal{F}_{{\varLambda},\beta,\mathbf{0}}^{(m)}(\tilde{\rho}_{{\varLambda}}^{*})}{m!}+ |{\varLambda}|S_{|{\varLambda}|}(\tilde{\rho}_{{\varLambda}}^{*})\right\} \\ &\lesssim &\exp\left\{|{\varLambda}|S_{|{\varLambda}|}(\tilde{\rho}_{{\varLambda}}^{*})\right\}\left( 1+\frac{1}{|{\varLambda}|}\right), \end{array} $$

(A.13)

since (A.12) and where $S_{|{\varLambda }|}(\tilde {\rho }^{*}_{{\varLambda }})$ is a term of order $\log {\sqrt {|{\varLambda }|}}/|{\varLambda }|$ (see Appendix B in [16]).

The study of $K(\tilde {\mu }_{{\varLambda }},\tilde {N}^{*})$ is the same as the one done in Lemma 6.3 of [16] where now we consider $\tilde {N}^{*}$ as center of fluctuations of order 1/2. Hence the conclusion follows from

$$ \begin{array}{@{}rcl@{}} K(\tilde{\mu}_{{\varLambda}},\tilde{N}^{*})\le e^{-|{\varLambda}|S_{|{\varLambda}|}(\tilde{\rho}_{{\varLambda}}^{*})}\left[\sqrt{2\pi D_{{\varLambda},\mathbf{0}}(\tilde{\rho}_{{\varLambda}}^{*})|{\varLambda}|}\left( 1-\frac{C}{\sqrt{|{\varLambda}|}}\right)\right]^{-1} \end{array} $$

(A.14)

and

$$ K(\tilde{\mu}_{{\varLambda}},\tilde{N}^{*})\ge e^{-|{\varLambda}|S_{|{\varLambda}|}(\tilde{\rho}_{{\varLambda}}^{*})}\left[\sqrt{2\pi D_{{\varLambda},\mathbf{0}}(\tilde{\rho}_{{\varLambda}}^{*})|{\varLambda}|}\left( 1+\frac{C}{\sqrt{|{\varLambda}|}}\right)\right]^{-1} $$

(A.15)

for some $C\in \mathbb {R}^{+}$ independent on Λ. □

Proof of Theorem 4

From (A.1), (A.2) we have

$$ \mathbb{P}^{\mathbf{0}}_{{\varLambda},\mu_0}(A_{\tilde{N}})=J^{C}_{\mu_{0}}(\tilde{N},N^{*})K(\mu_{0},N^{*}). $$

(A.16)

Then using Lemma 6.2 in [16] we have

$$ J^{C}_{\mu_{0}}(\tilde{N},N^{*})\ge\exp\left\{-\frac{(u^{\prime})^{2}|{\varLambda}|^{2\alpha-1}}{2D^{\alpha}_{{\varLambda},\beta,\mathbf{0}}(\rho^{*}_{{\varLambda}})}+|{\varLambda}|S_{|{\varLambda}|}(\rho^{*}_{{\varLambda}})-E_{|{\varLambda}|}(\alpha,u^{\prime},\rho^{*}_{{\varLambda}})\right\} $$

(A.17)

and

$$ J^{C}_{\mu_{0}}(\tilde{N},N^{*})\le \exp\left\{-\frac{(u^{\prime})^{2}|{\varLambda}|^{2\alpha-1}}{2D^{\alpha}_{{\varLambda},\beta,\mathbf{0}}(\rho^{*}_{{\varLambda}})}+|{\varLambda}|S_{|{\varLambda}|}(\rho^{*}_{{\varLambda}})+E_{|{\varLambda}|}(\alpha,u^{\prime},\rho^{*}_{{\varLambda}})\right\} $$

(A.18)

where $S_{|{\varLambda }|}(\rho ^{*}_{{\varLambda }})$ is a term of order $\log {\sqrt {|{\varLambda }|}}/|{\varLambda }|$ (see Appendix B in [16]).

The conclusion follows from Lemma 6.3 in [16] which gives us

$$ \begin{array}{@{}rcl@{}} K(\mu_{0},N^{*})\le e^{-|{\varLambda}|S_{|{\varLambda}|}(\rho^{*}_{{\varLambda}})}\left[\sqrt{2\pi D^{\alpha,+}_{{\varLambda},\mathbf{0}}(\rho^{*}_{{\varLambda}})|{\varLambda}|}\left( 1-E_{|{\varLambda}|}(\alpha,u^{\prime},\rho^{*}_{{\varLambda}})\right)\right]^{-1} \end{array} $$

(A.19)

and

$$ K(\mu_{0},N^{*})\ge e^{-|{\varLambda}|S_{|{\varLambda}|}(\rho^{*}_{{\varLambda}})}\left[\sqrt{2\pi D^{\alpha,+}_{{\varLambda},\mathbf{0}}(\rho^{*}_{{\varLambda}})|{\varLambda}|}\left( 1+E_{|{\varLambda}|}(\alpha,u^{\prime},\rho^{*}_{{\varLambda}})\right)\right]^{-1}. $$

(A.20)

□