We study the following interacting particle system. There are ρn particles, ρ<1, moving clockwise (“right”), in discrete time, on n sites arranged a circle. Each site may contain at most one particle. At each move to right-neighbor according rules. If its is occupied by another particle, does not move. has unoccupied (“holes”) as neighbors both sides, it moves right with probability 1. hole and...

Let π be the occupancy density of an obliquely reflected Brownian motion in half plane and let (ρ,α) polar coordinates a point upper plane. This work determines exact asymptotic behavior π(ρ,α) as ρ→∞ with α∈(0,π). We find explicit functions a, b, c such that π(ρ,α)∼ρ→∞a(α)ρb(α)e−c(α)ρ. closes open problem first stated by Professor J. Michael Harrison August 2013. also compute asymptotics for t...

A previous paper studied the so-called borderline curves of Kac–Murdock–Szegő matrix Kn(ρ)=[ρ|j−k|]j,k=1n, where ρ∈C. These are level (contour lines) in complex-ρ plane on which Kn(ρ) has a type-1 or type-2 eigenvalue modulus n, n is dimension. Those have cusps at all critical points ρ=ρc multiple (double) eigenvalues occur. The present determines corresponding pertaining to ν≠n. We find that t...