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Title: 在高維度下受波氏分配自我相斥隨機漫步的均場行為
Mean-field behavior for self-avoiding walks with Poisson interactions in high dimensions
Authors: 王守朋
Wang, Shou-Peng
Contributors: 陳隆奇
Wang, Shou-Peng
Keywords: 雖機漫步
self-avoiding walk
Date: 2020
Issue Date: 2020-08-03 17:57:38 (UTC+8)
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。
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。
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.
It is an important and interesting model in Probability and Statistical mechanics.
Some of the important problems had been solved (c.f.[5]). However,
many of the important problems remain unsolved, particularly those involving critical exponents, especially the critical exponents for long-range models.
In 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.
Reference: [1] Roland Bauerschmidt, Hugo DuminilCopin,
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[3] LungChi
Chen and Akira Sakai. Critical twopoint
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[4] Satoshi Handa, Yoshinori Kamijima, and Akira Sakai. A survey on the lace expansion
for the nearestneighbor
models on the bcc lattice. To appear in Taiwanese Journal of
Mathematics, 2019.

[5] Takashi Hara and Gordon Slade. Selfavoiding
walk in five or more dimensions. i. the
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[6] Takashi Hara, Remco van der Hofstad, and Gordon Slade. Critical twopoint
functions and
the lace expansion for spreadout
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Probab., 31(1):349–408, 01 2003.

[7] Markus Heydenreich, Remco van der Hofstad, and Akira Sakai. Meanfield
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[8] N. Madras and G. Slade. The SelfAvoiding
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[9] Yuri Mejia Miranda and Gordon Slade. The growth constants of lattice trees and lattice
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[10] A Sakai. Lace expansion for the Ising model. Technical Report mathph/
0510093, Oct

[11] Akira Sakai. Meanfield
critical behavior for the contact process. Journal of Statistical
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[12] Gordon Slade. The lace expansion and its applications, 2005.

[13] Remco van der Hofstad, Frank den Hollander, and Gordon Slade. The survival probability
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[14] Doron Zeilberger. The abstract lace expansion, 1998.
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