Hydrodynamic Limit of Multiple SLE
Hydrodynamic Limit of Multiple SLE
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DOI：
10.1007/s10955-018-1996-y
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发表时间：
2018-04-01
影响因子：
1.6
通讯作者：
Katori, Makoto
中科院分区：
文献类型：
--
作者：
Hotta, Ikkei;Katori, Makoto
最近，摩纳哥和施莱辛格解决了一个有趣的问题，人们是否可以采取限制多个施拉姆-洛纳演化（SLE）的狭缝数N趋于无穷大。当N个狭缝从真实的直线上的点同时增长并在上半平面内到达无穷远时，在极限下得到了描述共形映射时间演化的常微分方程，在无粘极限下耦合了复Burgers方程.众所周知，在随机矩阵理论中，复Burgers方程决定了Dyson模型的流体力学极限，当所有粒子从原点出发时，该Burgers方程的解由测度的Stieltjes变换给出，该变换遵循Wigner周期律的含时形式。在本文中，我们首先研究了所有狭缝都从原点开始的多重SLE的流体动力学极限。我们表明，依赖于时间的版本的维格纳的波动定律确定的SLE船体的时间演变，在这个流体动力学的限制。其次，我们考虑的情况下，一半的狭缝开始，另一半的狭缝开始，并确定多个SLE在流体动力学极限。在报告这些精确解之后，我们将讨论多重SLE及其船体在水动力极限下的普遍长期行为。
Recently del Monaco and Schleiinger addressed an interesting problem whether one can take the limit of multiple Schramm-Loewner evolution (SLE) as the number of slits N goes to infinity. When the N slits grow from points on the real line in a simultaneous way and go to infinity within the upper half plane , an ordinary differential equation describing time evolution of the conformal map was derived in the limit, which is coupled with a complex Burgers equation in the inviscid limit. It is well known that the complex Burgers equation governs the hydrodynamic limit of the Dyson model defined on studied in random matrix theory, and when all particles start from the origin, the solution of this Burgers equation is given by the Stieltjes transformation of the measure which follows a time-dependent version of Wigner's semicircle law. In the present paper, first we study the hydrodynamic limit of the multiple SLE in the case that all slits start from the origin. We show that the time-dependent version of Wigner's semicircle law determines the time evolution of the SLE hull, , in this hydrodynamic limit. Next we consider the situation such that a half number of the slits start from and another half of slits start from , and determine the multiple SLE in the hydrodynamic limit. After reporting these exact solutions, we will discuss the universal long-term behavior of the multiple SLE and its hull in the hydrodynamic limit.