| dc.contributor.advisor | 陳隆奇 | zh_TW |
| dc.contributor.advisor | CHEN, LUNG-CHI | en_US |
| dc.contributor.author (Authors) | 王守朋 | zh_TW |
| dc.contributor.author (Authors) | Wang, Shou-Peng | en_US |
| dc.creator (作者) | 王守朋 | zh_TW |
| dc.creator (作者) | Wang, Shou-Peng | en_US |
| dc.date (日期) | 2020 | en_US |
| dc.date.accessioned | 3-Aug-2020 17:57:38 (UTC+8) | - |
| dc.date.available | 3-Aug-2020 17:57:38 (UTC+8) | - |
| dc.date.issued (上傳時間) | 3-Aug-2020 17:57:38 (UTC+8) | - |
| dc.identifier (Other Identifiers) | G0106751002 | en_US |
| dc.identifier.uri (URI) | https://ah.lib.nccu.edu.tw/item?item_id=150076 | - |
| dc.description (描述) | 碩士 | zh_TW |
| dc.description (描述) | 國立政治大學 | zh_TW |
| dc.description (描述) | 應用數學系 | zh_TW |
| dc.description (描述) | 106751002 | zh_TW |
| dc.description.abstract (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。 | zh_TW |
| dc.description.abstract (摘要) | 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. | en_US |
| dc.description.tableofcontents | 1 Introduction 1\n\n2 Models and Main Results 3\n\n2.1 Notations and Definitions 3\n\n2.2 Main results and their proofs 6\n\n3 The lace expansion for selfavoiding walk 10\n\n4 Diagrammatic bounds estimate 14\n\n4.1 Diagrammatic bounds on the lace expansion coefficients 14\n\n4.2 Diagramatic bounds on the bootstrapping argument 25\n\n5 Random walk estimate 29\n\n5.1 The diagrams bound of randomwalk quantities for p = 1\n29\n5.2 The diagrams bound of randomwalk quantities for p > 1\n35\n6 Proof of Proposition 2.2.7 2.2.9 40\n\n6.1 Proof of Proposition 2.2.7 40\n\n6.2 Proof of Proposition 2.2.8 - 2.2.9 and Lemma 4.1.1 44\n\nAppendix A 48\n\nBibliography 50 | zh_TW |
| dc.format.extent | 572132 bytes | - |
| dc.format.mimetype | application/pdf | - |
| dc.source.uri (資料來源) | http://thesis.lib.nccu.edu.tw/record/#G0106751002 | en_US |
| dc.subject (關鍵詞) | 雖機漫步 | zh_TW |
| dc.subject (關鍵詞) | self-avoiding walk | en_US |
| dc.title (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 | zh_TW |
| dc.title (題名) | Mean-field behavior for self-avoiding walks with Poisson interactions in high dimensions | en_US |
| dc.type (資料類型) | thesis | en_US |
| dc.relation.reference (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. | zh_TW |
| dc.identifier.doi (DOI) | 10.6814/NCCU202000775 | en_US |
