Please use this identifier to cite or link to this item: https://ah.lib.nccu.edu.tw/handle/140.119/131107
題名: 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
作者: 王守朋
Wang, Shou-Peng
貢獻者: 陳隆奇
CHEN, LUNG-CHI
王守朋
Wang, Shou-Peng
關鍵詞: 雖機漫步
self-avoiding walk
日期: 2020
上傳時間: 3-Aug-2020
摘要: 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.
參考文獻: [1] Roland Bauerschmidt, Hugo DuminilCopin,\nJesse Goodman, and Gordon Slade. Lectures\non selfavoiding\nwalks, 2012.\n\n[2] David Brydges and Thomas Spencer. Selfavoiding\nwalk in 5 or more dimensions.\nCommunications in Mathematical Physics, 97(1):125–148, Mar 1985.\n\n[3] LungChi\nChen and Akira Sakai. Critical twopoint\nfunction for longrange\nmodels\nwith powerlaw\ncouplings: The marginal case for $${d\\ge d_{\\rm c}}$$d≥dc.\nCommunications in Mathematical Physics, 372(2):543–572, 2019.\n\n[4] Satoshi Handa, Yoshinori Kamijima, and Akira Sakai. A survey on the lace expansion\nfor the nearestneighbor\nmodels on the bcc lattice. To appear in Taiwanese Journal of\nMathematics, 2019.\n\n[5] Takashi Hara and Gordon Slade. Selfavoiding\nwalk in five or more dimensions. i. the\ncritical behaviour. Comm. Math. Phys., 147(1):101–136, 1992.\n\n[6] Takashi Hara, Remco van der Hofstad, and Gordon Slade. Critical twopoint\nfunctions and\nthe lace expansion for spreadout\nhighdimensional\npercolation and related models. Ann.\nProbab., 31(1):349–408, 01 2003.\n\n[7] Markus Heydenreich, Remco van der Hofstad, and Akira Sakai. Meanfield\nbehavior\nfor longand\nfinite range ising model, percolation and selfavoiding\nwalk. Journal of\nStatistical Physics, 132(6):1001–1049, 2008.\n\n[8] N. Madras and G. Slade. The SelfAvoiding\nWalk. Probability and Its Applications.\nBirkhäuser Boston, 1996.\n\n[9] Yuri Mejia Miranda and Gordon Slade. The growth constants of lattice trees and lattice\nanimals in high dimensions, 2011.\n\n[10] A Sakai. Lace expansion for the Ising model. Technical Report mathph/\n0510093, Oct\n2005.\n\n[11] Akira Sakai. Meanfield\ncritical behavior for the contact process. Journal of Statistical\nPhysics, 104(1):111–143, Jul 2001.\n\n[12] Gordon Slade. The lace expansion and its applications, 2005.\n\n[13] Remco van der Hofstad, Frank den Hollander, and Gordon Slade. The survival probability\nfor critical spreadout\noriented percolation above 4+1 dimensions. ii. expansion. Annales\nde l’Institut Henri Poincare (B) Probability and Statistics, 43(5):509 – 570, 2007.\n\n[14] Doron Zeilberger. The abstract lace expansion, 1998.
描述: 碩士
國立政治大學
應用數學系
106751002
資料來源: http://thesis.lib.nccu.edu.tw/record/#G0106751002
資料類型: thesis
Appears in Collections:學位論文

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