arXiv 1502.009522015-02math.PRmath-phmath.MP

圆上简单排它过程的截断轮廓

The cutoff profile for the simple exclusion process on the circle

Hubert Lacoin

备注：: Published at http://dx.doi.org/10.1214/15-AOP1053 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：: Annals of Probability 2016, Vol. 44, No. 5, 3399-3430
DOI：: 10.1214/15-AOP1053

摘要：

本文对简单排它过程松弛到平衡态的方式给出了非常精确的描述。令$P_t$表示与具有$2N$位点和$N$粒子的圆上的排斥相关联的半群。对于任何初始条件$χ$，和任何$t\ge\frac{4N^2}{9π^2}\log N$，我们表明，概率密度$P_t(χ,\cdot)$是由粒子系统的主要本征函数的平衡测量的指数倾斜。由于$\frac{4N^2}{9π^2}\log N$小于混合时间$\frac{N^2}{2π^2}\log N$，这允许给出截止曲线的清晰描述：如果$d_N(t)$表示从最差初始条件开始的总变化距离，则我们有\[\lim_{N\to\infty}d_N\biggl(\frac{N^2}{2π^2}\log N+\frac{N^2}{π^2}s\biggr)=\operatorname {erf}\biggl(\frac{\sqrt{2}}πe^{-s}\biggr),\]，其中$\operatorname {erf}$是高斯误差函数。

英文摘要：

In this paper, we give a very accurate description of the way the simple exclusion process relaxes to equilibrium. Let $P_t$ denote the semi-group associated the exclusion on the circle with $2N$ sites and $N$ particles. For any initial condition $χ$, and for any $t\ge\frac{4N^2}{9π^2}\log N$, we show that the probability density $P_t(χ,\cdot)$ is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system. As $\frac{4N^2}{9π^2}\log N$ is smaller than the mixing time which is $\frac{N^2}{2π^2}\log N$, this allows to give a sharp description of the cutoff profile: if $d_N(t)$ denote the total-variation distance starting from the worse initial condition we have \[\lim_{N\to\infty}d_N\biggl(\frac{N^2}{2π^2}\log N+\frac{N^2}{π^2}s\biggr)=\operatorname {erf}\biggl(\frac{\sqrt{2}}πe^{-s}\biggr),\] where $\operatorname {erf}$ is the Gauss error function.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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