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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. Lectures
on selfavoiding
walks, 2012.
[2] David Brydges and Thomas Spencer. Selfavoiding
walk in 5 or more dimensions.
Communications in Mathematical Physics, 97(1):125–148, Mar 1985.
[3] LungChi
Chen and Akira Sakai. Critical twopoint
function for longrange
models
with powerlaw
couplings: 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 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
critical behaviour. Comm. Math. Phys., 147(1):101–136, 1992.
[6] Takashi Hara, Remco van der Hofstad, and Gordon Slade. Critical twopoint
functions and
the lace expansion for spreadout
highdimensional
percolation and related models. Ann.
Probab., 31(1):349–408, 01 2003.
[7] Markus Heydenreich, Remco van der Hofstad, and Akira Sakai. Meanfield
behavior
for longand
finite range ising model, percolation and selfavoiding
walk. Journal of
Statistical Physics, 132(6):1001–1049, 2008.
[8] N. Madras and G. Slade. The SelfAvoiding
Walk. Probability and Its Applications.
Birkhäuser Boston, 1996.
[9] Yuri Mejia Miranda and Gordon Slade. The growth constants of lattice trees and lattice
animals in high dimensions, 2011.
[10] A Sakai. Lace expansion for the Ising model. Technical Report mathph/
0510093, Oct
2005.
[11] Akira Sakai. Meanfield
critical behavior for the contact process. Journal of Statistical
Physics, 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 probability
for critical spreadout
oriented percolation above 4+1 dimensions. ii. expansion. Annales
de 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 |
| DOI: | 10.6814/NCCU202000775 |
| Appears in Collections: | [應用數學系] 學位論文
|
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| 100201.pdf | | 558Kb | Adobe PDF2 | 117 | View/Open |
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