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    Title: 在高維度下滲流的均場行為
    Mean-Field Behavior for Percolation in High Dimensions
    Authors: 陳隆奇
    Contributors: 應數系
    Keywords: 自我相斥隨機漫步;渗流;定向渗流;Ising模型;兩點函數;臨界行為;漸 進行為;迴轉半徑;誤差估計;收斂速度;body-centered cubic晶格;大分差;Berry-Essen 定理;Domany Kinzel 模型
    Self-Avoiding Walk;Percolation;Oriented Percolation;Ising model;two point function;critical behavior;asymptotic behavior;gyration radius;error estimate;rate of convergence;body-centered cubic lattice;Lace expansion;Berry-
    Date: 2018-09
    Issue Date: 2025-04-16 14:28:03 (UTC+8)
    Abstract: 我此次申請的研究計劃主要想做四個問題。第一個問題是和北海道大學Akira Sakai 教授共同合作的問題，在這問題我們有興趣探討自我相斥隨機漫步，滲流與Ising模 型兩點函數的臨界行為，在2013 - 2014的主要研究計劃我們已經成功證明當維度大 於上臨界為維度時，長域兩點函數趙近于Cbr2-d其中C是常數且a不等於2。此問 題想解決長域兩點函數的臨界行為當a等於2且維度大於上臨界維度。第二個問題是 和Akira Sakai教授和荷蘭萊登大學Markus Heydenreich教授共同合作的問題，我們 想要用數學嚴謹證明渗流的mean field行為成立在維度大於六維之body-centered cubic 晶格。第三個問題是想和成功大學張書銓教授與中研院數學所黃建豪博士合作的問題， 在這個問題中，我們定義一個二維特殊的定向滲流，此模型推廣了 DomanyKinzel模 型，我們進而分析此模型的兩點函數之收斂速度。最後一個主題，我想要推廣之前和 Akira Sakai教授探討長域自我相斥隨機漫步與長域定向渗流的迴轉半徑的結果，進而 分析其收斂速度。
    The proposal of my research that I would like to apply from August 1st 2015 to July 31th 2018 consists of 4 programs. The first program is a joint work with professor Akira Sakai at Hokkaido university. In this program we are interested in the critical behavior of two-point function for self-avoiding walk, percolation and Ising model above their upper critical dimensions. In the proposal from 2013 to 2015 we have proved that, when the dimensions larger than their upper critical dimensions and the spread-out parameter sufficiently large, the critical two-point function for each model is asymptotically C|x|aA2-d, where the constant C 2 (0,1) is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between a < 2 and a > 2. In this program we want to analyze the critical two point function for three aforementioned long-range models for a = 2 and L sufficiently large and the dimension when the dimensions larger than their upper critical dimensions. The second program is a joint work with professor Akira Sakai and professor Markus Heydenreich at University Leiden. We would like to prove the mean field behavior holds for nearest-neighbor percolation on d-dimensional body-centered cubic lattices where d> 6. The third program is a joint work with professor Shu-Chiuan Chang at national Cheng-Kung university and Chien-hao Huang at the institute of Mathematics, Academia Sinica. In this program we consider a version of directed bond percolation on the square lattice such that vertical edges are directed upward with probability pi and p in alternate rows, and horizontal edges are directed rightward with probabilities one. If pi = p2, our model corresponds to a Domany-Kinzel model. We have to estimate the rate of convergence for two-point function of this model. The last program, I would like to extend the precious work with professor Sakai for the gyration radius of long-range oriented percolation and long-range self-avoiding walk. I would like to investigate the main terms and error estimates of the gyration radius, with index r 2 (0, a), for the two aforementioned long-range models.
    Relation: 科技部, MOST104-2115-M004-003-MY3, 104.08-107.07
    Data Type: report
    Appears in Collections:[Department of Mathematical Sciences] NSC Projects

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