Abstract
We consider an ensemble of $N$ discrete nonintersecting paths starting from equidistant points and ending at consecutive integers. Our first result is an explicit formula for the correlation kernel that allows us to analyze the process as $N\to \infty$. In that limit we obtain a new general class of kernels describing the local correlations close to the equidistant starting points. As the distance between the starting points goes to infinity, the correlation kernel converges to that of a single random walker. As the distance to the starting line increases, however, the local correlations converge to the sine kernel. Thus, this class interpolates between the sine kernel and an ensemble of independent particles. We also compute the scaled simultaneous limit, with both the distance between particles and the distance to the starting line going to infinity, and obtain a process with number variance saturation, previously studied by Johansson.
Highlights
Nonintersecting path ensembles are a natural arena for generating and studying a variety of mathematical and physical phenomena
An ensemble of nonintersecting paths consists of a number of random walkers with prescribed initial and final positions that do not collide while performing the walk
We shall show that eventually, as we move away from the starting points, the local correlations converge to the usual sine kernel limit
Summary
Nonintersecting path ensembles are a natural arena for generating and studying a variety of mathematical and physical phenomena. We shall show that eventually, as we move away from the starting points, the local correlations converge to the usual sine kernel limit This takes some time, and our main interest is in describing the process before universality is reached. It is interesting to consider the continuum limit which places our model in the framework of Dyson Brownian motion [6, 9] or GUE with external source [5] This continuous analogue played an important role in Johansson’s proof of universality for the local correlation for GUE divisible Wigner matrices [12].
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