Product type potential on the one-dimensional lattice systems: selection of maximizing probability and a large deviation principle
One of the main results that the author proves is the existence of an equilibrium probability measure for a one-dimensional lattice system. More precisely, consider the set of functions \(\Omega=[0,1]^{\mathbb{N}}\) from the natural numbers to the closed interval \([0,1]\). Here \(\mathbb{N}=\{1,2,3,\dots\}\). A real-valued function \(f:\Omega\to\mathbb{R}\) is assumed to take the form \(f(x_1,x_2,\dots)=\sum^\infty_{j=1}f_j(x_j)\), where each real-valued function \(f_j:[0,1]\to\mathbb{R}\) is Lipschitz continuous with Lipschitz constant being less than \(\frac{1}{2^j}\) and periodic, \(f_j(0)=f_j(1)\). Consider the real-valued function defined as \(F(a)=f(a,a,a,\dots)\) with some differentiability conditions (please see the conditions in Theorem 7). This is assumed to achieve maximum value at one point \(a_1\). Define the probability measure \[\tilde{\mu}_{0,\beta}=\frac{e^{\beta F(a)}}{\int_0^1e^{\beta F(a)}da}da\] on the closed interval \([0,1]\) and the probability measure \(\tilde{\mu}_\beta=\bigotimes^\infty_{i=1}\tilde{\mu}_{0,\beta}\) on \(\Omega\). Then the author proves the following two results: (1) There holds: \[\lim\limits_{\beta\to+\infty}\tilde{\mu}_\beta\bigoplus^\infty_{i=1}\delta_{a_1};\] (2) There holds: \[\sum^\infty_{j=1}f_j(a_1)=\max\limits_{\mu\in\mathcal{M}_\sigma}\int_{\Omega}fd\mu,\] where \(\mathcal{M}_\sigma\) is the set of invariant probability measures with respect to \(\sigma:\Omega\to\Omega\), \(\sigma(x_1,x_2,\dots)=(x_2,x_3,\dots)\). The ideas of the proofs are as follows. Regarding statement (1), since the function \(F(a)\) achieves its maximum at one point \(a_1\) as \(\beta\to+\infty\), the support for the probability measure \(\tilde{\mu}_{0,\beta}\) gets concentrated near \(a_1\), and in the limit \(\beta\to+\infty\), one gets the Dirac delta function \(\delta_{a_1}\). Regarding statement (2), the assumptions on the function \(f(x_1,x_2,\dots)\) and statement (1) imply the conditions of Theorem 5 in [\textit{A. O. Lopes} et al., Ergodic Theory Dyn. Syst. 35, No. 6, 1925--1961 (2015; Zbl 1352.37090)], that in turn implies statement (2).
scientific article; zbMATH DE number 7489709
scientific article