Critical behavior of the number of minima of a random landscape at the glass transition point and the Tracy-Widom distribution.
Critical behavior of the number of minima of a random landscape at the glass transition point and the Tracy-Widom distribution.
复制标题
玻璃化转变点处随机景观的最小值数量和 Tracy-Widom 分布的临界行为。
DOI：
10.1103/physrevlett.109.167203
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发表时间：
2012
影响因子：
8.6
通讯作者：
C'eline Nadal
中科院分区：
文献类型：
--
作者：
Y. Fyodorov;C'eline Nadal
我们利用随机高斯曲面最小值的平均值N(m)与随机矩阵的极值特征值之间的关系，来理解N维随机环境中单个粒子的玩具模型中最简单的类玻璃跃迁中N(m)的临界行为，其中N为>>1。将控制参数μ改变到临界值μ(c)，详细分析了N(m)（μ）如何从玻璃相的指数大下降到跃迁另一侧的N(m)(μ)~1。我们还在玻璃相和简单相中提取了N(m)（μ）的次超前行为。发现临界区域的宽度δμ/μ(c)按N（-1/3）缩放，并且在该区域内N(m)（μ）收敛到用Tracy-Widom分布表示的极限形状。
We exploit a relation between the mean number N(m) of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behavior of N(m) in the simplest glasslike transition occuring in a toy model of a single particle in an N-dimensional random environment, with N>>1. Varying the control parameter μ through the critical value μ(c) we analyze in detail how N(m)(μ) drops from being exponentially large in the glassy phase to N(m)(μ)~1 on the other side of the transition. We also extract a subleading behavior of N(m)(μ) in both glassy and simple phases. The width δμ/μ(c) of the critical region is found to scale as N(-1/3) and inside that region N(m)(μ) converges to a limiting shape expressed in terms of the Tracy-Widom distribution.