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Hydrodynamic Limit of Multiple SLE. (arXiv:1712.02049v2 [math-ph] UPDATED)
来源于:arXiv
Recently del Monaco and Schlei{\ss}inger 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 ${\mathbb{R}}$ in a simultaneous way and go to infinity within
the upper half plane ${\mathbb{H}}$, an ordinary differential equation
describing time evolution of the conformal map $g_t(z)$ was derived in the $N
\to \infty$ 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 ${\mathbb{R}}$ 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
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