We study the boundary of the liquid region L in large random lozenge tiling models defined by uniform random interlacing particle systems with general initial configuration, which lies on the line px, 1q, x P R " BH. We assume that the initial particle configuration converges weakly to a limiting density φpxq, 0 ď φ ď 1. The liquid region is given by a homeomorphism WL : L Ñ H, the upper half plane, and we consider the extension of W´1 L to H. Part of BL is given by a curve, the edge E , parametrized by intervals in BH, and this corresponds to points where φ is identical to 0 or 1. If 0 ă φ ă 1, the non-trivial support, there are two cases. Either W´1 L pwq has the limit px, 1q as w Ñ x non-tangentially and we have a regular point, or we have what we call a singular point. In this case W´1 L does not extend continuously to H. Singular points give rise to parts of BL not given by E and which can border a frozen region, or be "inside" the liquid region. This shows that in general the boundary of BL can be very complicated. We expect that on the singular parts of BL we do not get a universal point process like the Airy or the extended Sine kernel point processes. Furthermore, E and the singular parts of BL are shocks of the complex Burgers equation.
Given a sequence of bounded random variables that satisfies a well-known mixing condition, it is shown that empirical estimates of the rate function for the partial sums process satisfy the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queue length distribution at a single-server queue with infinite waiting space is proved.
A standard Gelfand-Tsetlin pattern of depth n is a configuration of particles in {1, . . . , n} × R. For each r ∈ {1, . . . , n}, {r} × R is referred to as the r th level of the pattern. A standard Gelfand-Tsetlin pattern has exactly r particles on each level r, and particles on adjacent levels satisfy an interlacing constraint.Probability distributions on the set of Gelfand-Tsetlin patterns of depth n arise naturally as distributions of eigenvalue minor processes of random Hermitian matrices of size n. We consider such probability spaces when the distribution of the matrix is unitarily invariant, prove a determinantal structure for a broad subclass, and calculate the correlation kernel.In particular we consider the case where the eigenvalues of the random matrix are fixed. This corresponds to choosing uniformly from the set of Gelfand-Tsetlin patterns whose n th level is fixed at the eigenvalues of the matrix. Fixing qn ∈ {1, . . . , n}, and letting n → ∞ under the assumption that qn n → α ∈ (0, 1) and the empirical distribution of the particles on the n th level converges weakly, the asymptotic behaviour of particles on level qn is relevant to free probability theory. Saddle point analysis is used to identify the set in which these particles behave asymptotically like a determinantal random point field with the Sine kernel.
Given a sequence of bounded random variables that satisfies a well known mixing condition, it is shown that empirical estimates of the rate-function for the partial sums process satisfies the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queuelength distribution at a single server queue with infinite waiting space is proved.
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