We study determinantal point processes arising in random domino tilings of a double Aztec diamond, a region consisting of two overlapping Aztec diamonds. At a turning point in a single Aztec diamond where the disordered region touches the boundary, the natural limiting process is the GUE-minor process. Increasing the size of a double Aztec diamond while keeping the overlap between the two Aztec diamonds finite, we obtain a new determinantal point process which we call the tacnode GUE-minor process. This process can be thought of as two colliding GUE-minor processes. As part of the derivation of the particle kernel whose scaling limit naturally gives the tacnode GUE-minor process, we find the inverse Kasteleyn matrix for the dimer model version of the Double Aztec diamond.

A family of random matrix ensembles interpolating between the Ginibre ensemble of n x n matrices with iid centered complex Gaussian entries and the Gaussian unitary ensemble (GUE) is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n (-1/3). In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy-Widom distributions.

We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green’s function is the probability that the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the (Formula presented.) Green’s function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities: the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a “spinor” observable for random walk started from one of the vertices of the marked edge.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

A finitization of the bead process2012In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 152, no 1-2, p. 321-356Article in journal (Refereed)

Abstract [en]

The bead process is the particle system defined on parallel lines, with underlying measure giving constant weight to all configurations in which particles on neighbouring lines interlace, and zero weight otherwise. Motivated by the statistical mechanical model of the tiling of an abc-hexagon by three species of rhombi, a finitized version of the bead process is defined. The corresponding joint distribution can be realized as an eigenvalue probability density function for a sequence of random matrices. The finitized bead process is determinantal, and we give the correlation kernel in terms of Jacobi polynomials. Two scaling limits are considered: a global limit in which the spacing between lines goes to zero, and a certain bulk scaling limit. In the global limit the shape of the support of the particles is determined, while in the bulk scaling limit the bead process kernel of Boutillier is reclaimed, after appropriate identification of the anisotropy parameter therein.

We relate the Gaussian free field on a planar domain to the variational formula of Hadamard which explains the change of the Green function under a perturbation of the domain. This is accomplished by means of a natural integral operator-called the Hadamard operator-associated with a given flow of growing domains.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

From Gumbel to Tracy-Widom2007In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 138, no 2-Jan, p. 75-112Article in journal (Refereed)

We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the ErdAs-R,nyi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the ErdAs-R,nyi graph this establishes the Tracy-Widom fluctuations of the second largest eigenvalue when p is much larger than wth a deterministic shift of order (Np)(-1)..

An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the entries are zero, and all other entries are independent exponentially distributed random variables with mean 1. Thereby we prove the formula 1+1/4+1/9+...+1/k(2) conjectured by G. Parisi for the case k=m=n, and the generalized conjecture of D. Coppersmith and G. B. Sorkin for arbitrary k, m and n.

11.

Metcalfe, Anthony P.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

A standard Gelfand-Tsetlin pattern of depth n is a configuration of particles in . For each is referred to as the rth 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 nth level is fixed at the eigenvalues of the matrix. Fixing , and letting n -> a under the assumption that and the empirical distribution of the particles on the nth level converges weakly, the asymptotic behaviour of particles on level q (n) 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.