Mixing Time for Ising Model (On Two Special Graphs: the Line and the Circle)

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题名:	Mixing Time for Ising Model (On Two Special Graphs: the Line and the Circle)
作者:	伊凱馬;Elkamash, Badr Saleh Badr Mohamed
贡献者:	數學系
关键词:	伊辛模型;格勞伯動力;混合時間;馬爾可夫鏈;路徑耦合;Ising Model;Glauber Dynamics;Mixing Time;Markov Chains;Path Coupling
日期:	2020-07-07
上传时间:	2020-09-02 18:44:52 (UTC+8)
出版者:	國立中央大學
摘要:	在本論文中，我們研究了Ising模型的Glauber動力學。基於[10]的專著，我們提供了馬爾可夫鏈混合時間一般理論的詳細介紹，尤其是收斂到平穩測度的速率。然後，我們計算出Ising模型的兩個特殊（也許是最重要）案例的細節：直線和圓。我們的貢獻是（1）我們針對這兩種特殊情況獲得了改進的估計；和（2）我們提供了許多細節和例子和圖片示例來說明該理論。更詳細地，我們證明在高溫下快速混合。我們確定混合時間是log(n)和log( 1/e)的多項式。或者，顯示tmix在log(n)也足以進行快速混合。我們證明了Glauber動力學的混合時間為在高溫下具有n個頂點的直線和圓上的（鐵磁）伊辛模型的上限為n log n/e 。;In this thesis we study Glauber dynamics of one dimensional Ising models. We provide a detailed presentation of the general theory of the mixing times of Markov chains, especially the rate of convergence to stationary measures, based on the monograph of [10]. Then we work out the details of two special (and perhaps the most important) cases of Ising models: the line and the circle. Our contribution is that (1) we obtain improved estimates for these two special cases; and (2) we provide many examples with details and pictures to illustrate the theory. In more details, we prove a fast mixing at high temperature. We establish that the mixing time is a polynomial in log(n) and log( 1/e). Alternatively, we show that tmix is a polynomial in log(n). It is also enough for fast mixing. We show that the mixing time of Glauber dynamics for the (ferromagnetic) Ising model on a line and a circle with n vertices at high temperature has an upper bound of n log n/e.
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