An infinite-dimensional non-linear equation related to gibbs measures of a sos model

IF 0.6 4区数学 Q4 MATHEMATICS, APPLIED

Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2023-11-04 DOI:10.1142/s0219025723500261

U. A. Rozikov

引用次数: 0

Abstract

For the solid-on-solid (SOS) model with an external field and with spin values from the set of all integers on a Cayley tree, each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a nonlinear functional equation. Recently some translation-invariant and height-periodic (non-normalizable) solutions to the equation are found. Here, our aim is to find non-height-periodic and non-normalizable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalizable boundary laws. Moreover, we reduce the problem to solving of a nonlinear, second-order difference equation. We give analytic and numerical analyses of the difference equation.

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一个与sos模型的吉布斯测度有关的无限维非线性方程

对于具有外场且自旋值来自Cayley树上所有整数集合的solid-on-solid (SOS)模型，每个(梯度)Gibbs测度对应于满足非线性泛函数方程的边界律(在Cayley树的顶点上定义的无限维向量函数)。最近发现了该方程的平移不变解和高度周期(不可归一化)解。在这里，我们的目标是找到非高度周期和不可归一化的SOS模型的边界律。通过这样的解，我们可以构造一个非概率吉布斯测度。我们明确地发现了几个不可归一化的边界律。此外，我们将问题简化为求解一个非线性二阶差分方程。对差分方程进行了解析和数值分析。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Infinite Dimensional Analysis Quantum Probability and Related Topics 数学-统计学与概率论

CiteScore

1.50

自引率

11.10%

发文量

审稿时长

>12 weeks

期刊介绍： In the past few years the fields of infinite dimensional analysis and quantum probability have undergone increasingly significant developments and have found many new applications, in particular, to classical probability and to different branches of physics. The number of first-class papers in these fields has grown at the same rate. This is currently the only journal which is devoted to these fields. It constitutes an essential and central point of reference for the large number of mathematicians, mathematical physicists and other scientists who have been drawn into these areas. Both fields have strong interdisciplinary nature, with deep connection to, for example, classical probability, stochastic analysis, mathematical physics, operator algebras, irreversibility, ergodic theory and dynamical systems, quantum groups, classical and quantum stochastic geometry, quantum chaos, Dirichlet forms, harmonic analysis, quantum measurement, quantum computer, etc. The journal reflects this interdisciplinarity and welcomes high quality papers in all such related fields, particularly those which reveal connections with the main fields of this journal.