Related papers: Large gap asymptotics on annuli in the random norm…

We consider the set M_n of all n-truncated power moment sequences of probability measures on [0,1]. We endow this set with the uniform probability. Picking randomly a point in M_n, we show that the upper canonical measure associated with…

We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm…

Closed-system quantum annealing is expected to sometimes fail spectacularly in solving simple problems for which the gap becomes exponentially small in the problem size. Much less is known about whether this gap scaling also impedes…

We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees).…

An infinite log-gas formalism, due to Dyson, and independently Fogler and Shklovskii, is applied to the computation of conditioned gap probabilities at the hard and soft edges of random matrix $\beta$-ensembles. The conditioning is that…

Let X be a translation invariant point process on the complex plane and let D be a bounded open set whose boundary has zero Lebesgue measure. We ask what does the point configuration obtained by taking the points of X outside D tell us…

Asymptotic normality is frequently observed in large combinatorial structures, rigorously established for many quantities such as cycles or inversions in random permutations, the number of prime factors of random integers, and various…

In this paper we study a family of non-classical Jacobi polynomials with varying parameters of the form $\alpha_n=n+1/2$ and $\beta_n=-n-1/2$. We obtain global asymptotics for these polynomials, and use this to establish results on the…

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$…

If $p:\mathbb{C} \to \mathbb{C}$ is a non-constant polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when $p$ is a random polynomial of degree $n$ with…

This thesis lies in the field of constant mean curvature (cmc) hypersurfaces and specifically cmc $1/2$ surfaces in the three-manifold $\mathbb{H}^2\times\mathbb{R}$. The value $1/2$ is the critical mean curvature for…

In an earlier work we had considered a Gaussian ensemble of random matrices in the presence of a given external matrix source. The measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials, or on the…

Consider a system \Psi of non-constant affine-linear forms \psi_1,...,\psi_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of…

This is the third in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. In this…

There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on $\mathbb{R}$ and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for…

We use a semi-Markov process method to calculate large deviations of counting statistics for three open quantum systems, including a resonant two-level system and resonant three-level systems in the $\Lambda$- and $V$-configurations. In the…

In this paper we study the critical behavior of an $N$-component ${\phi}^{4}$-model in hyperbolic space, which serves as a model of uniform frustration. We find that this model exhibits a second-order phase transition with an unusual…

In the present paper and the companion paper [9] a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on a complex algebraic varieties X is introduced, by sampling "temperature deformed" determinantal…

For a zero-temperature Landau symmetry breaking transition in $n$-dimensional space that completely breaks a finite symmetry $G$, the critical point at the transition has the symmetry $G$. In this paper, we show that the critical point also…

We prove a law of large numbers in terms of complete convergence of independent random variables taking values in increments of monotone functions, with convergence uniform both in the initial and the final time. The result holds also for…