"Scaling limit of vicious walks and two-matrix model",
( with Makoto Katori)
Physical Review E66 (2002)
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We consider the diffusion scaling limit of the one-dimensional vicious
walker model of Fisher and derive a system of nonintersecting Brownian
motions. The spatial distribution of $N$ particles is studied and it is
described by use of the probability density function of eigenvalues of $N
\times N$ Gaussian random matrices. The particle distribution depends on the
ratio of the observation time $t$ and the time interval $T$ in which the
nonintersecting condition is imposed. As $t/T$ is going on from 0 to 1,
there occurs a transition of distribution, which is identified with the
transition observed in the two-matrix model of Pandey and Mehta. Despite of
the absence of matrix structure in the original vicious walker model, in the
diffusion scaling limit accumulation of contact repulsive interactions
realizes the correlated distribution of eigenvalues in the multimatrix model
as the particle distribution.