在高維度下受波氏分配自我相斥隨機漫步的均場行為

Abstract

self-avoiding walk是線性聚合物的模型。它是機率和統計力學中一個重要而有趣的模型。一些重要問題已經解決(c.f.[5]). 然而，許多重要問題仍未解決，特別是涉及關鍵指數的問題，尤其是遠程模型的關鍵指數。\n在本文中，我們獲得了對於一個特殊的長域模型，其單步分佈是波松分佈的特殊敏感度模型，其敏感性指數滿足均值場行為，且其值大於上臨界值d(c) = 4 。參數 lambda > lambda(d) 的類型分佈，其中lambda(d)取決於維度。\n為此，我們選擇一組特殊的 bootstrapping functions，它們類似於[4]，並使用lace expansion分析有關bootstrapping functions的複雜部分。 此外，對於d>4，我們得到lambda(d)的確切值。

Self-avoiding walk is a model for linear polymers.\nIt is an important and interesting model in Probability and Statistical mechanics.\nSome of the important problems had been solved (c.f.[5]). However,\nmany of the important problems remain unsolved, particularly those involving critical exponents, especially the critical exponents for long-range models.\nIn this thesis, we see Lace expansion to obtain that the critical exponent of the susceptibility satisfies the mean-field behavior with the dimensions above the upper critical dimension (d(c) = 4) for a special loge-range model in which each one-step distribution is the Poisson-type distribution with parameter lambda > lambda(d) where lambda(d) depends on the dimensions. To achieve this, we choose a particular set of bootstrapping functions which is similar as [4] and using a notoriously complicated part of the lace expansion analysis. Moreover we get the exactly value of lambda(d) for d > 4.