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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CHEN, LUNG-CHI王守朋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,Jesse Goodman, and Gordon Slade. Lectureson selfavoidingwalks, 2012.[2] David Brydges and Thomas Spencer. Selfavoidingwalk in 5 or more dimensions.Communications in Mathematical Physics, 97(1):125–148, Mar 1985.[3] LungChiChen and Akira Sakai. Critical twopointfunction for longrangemodelswith powerlawcouplings: The marginal case for $${d\ge d_{\rm c}}$$d≥dc.Communications in Mathematical Physics, 372(2):543–572, 2019.[4] Satoshi Handa, Yoshinori Kamijima, and Akira Sakai. A survey on the lace expansionfor the nearestneighbormodels on the bcc lattice. To appear in Taiwanese Journal ofMathematics, 2019.[5] Takashi Hara and Gordon Slade. Selfavoidingwalk in five or more dimensions. i. thecritical behaviour. Comm. Math. Phys., 147(1):101–136, 1992.[6] Takashi Hara, Remco van der Hofstad, and Gordon Slade. Critical twopointfunctions andthe lace expansion for spreadouthighdimensionalpercolation and related models. Ann.Probab., 31(1):349–408, 01 2003.[7] Markus Heydenreich, Remco van der Hofstad, and Akira Sakai. Meanfieldbehaviorfor longandfinite range ising model, percolation and selfavoidingwalk. Journal ofStatistical Physics, 132(6):1001–1049, 2008.[8] N. Madras and G. Slade. The SelfAvoidingWalk. Probability and Its Applications.Birkhäuser Boston, 1996.[9] Yuri Mejia Miranda and Gordon Slade. The growth constants of lattice trees and latticeanimals in high dimensions, 2011.[10] A Sakai. Lace expansion for the Ising model. Technical Report mathph/0510093, Oct2005.[11] Akira Sakai. Meanfieldcritical behavior for the contact process. Journal of StatisticalPhysics, 104(1):111–143, Jul 2001.[12] Gordon Slade. The lace expansion and its applications, 2005.[13] Remco van der Hofstad, Frank den Hollander, and Gordon Slade. The survival probabilityfor critical spreadoutoriented percolation above 4+1 dimensions. ii. expansion. Annalesde l’Institut Henri Poincare (B) Probability and Statistics, 43(5):509 – 570, 2007.[14] Doron Zeilberger. The abstract lace expansion, 1998. Description: 碩士國立政治大學應用數學系106751002 Source URI: http://thesis.lib.nccu.edu.tw/record/#G0106751002 Data Type: thesis Appears in Collections: [應用數學系] 學位論文

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