TY - JOUR

T1 - Nonintersecting paths with a staircase initial condition

AU - Breuer, Jonathan

AU - Duits, Maurice

PY - 2012

Y1 - 2012

N2 - 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 → ∞. 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.

AB - 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 → ∞. 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.

KW - Determinantal point processes

KW - Random non-intersecting paths

KW - Random tilings

UR - http://www.scopus.com/inward/record.url?scp=84864839472&partnerID=8YFLogxK

U2 - 10.1214/EJP.v17-1902

DO - 10.1214/EJP.v17-1902

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AN - SCOPUS:84864839472

SN - 1083-6489

VL - 17

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

ER -