Related papers: Tracy-Widom asymptotics for a random polymer model…
The Hardy-Ramanujan partition function asymptotics is a famous result in the asymptotics of combinatorial sequences. It was originally derived using complex analysis and number-theoretic ideas by Hardy and Ramanujan. It was later re-derived…
We study the distribution of partition parts in arithmetic progressions and find asymptotic results that capture all exponentially growing terms. This is accomplished by studying the behavior of non-modular Eisenstein series that appear in…
The Ewens-Pitman model refers to a distribution for random partitions of $[n]=\{1,\ldots,n\}$, which is indexed by a pair of parameters $\alpha \in [0,1)$ and $\theta>-\alpha$, with $\alpha=0$ corresponding to the Ewens model in population…
We establish necessary and sufficient conditions for convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. We show that this convergence is equivalent to asymptotic independence…
Building on work of Hardy and Ramanujan, Rademacher proved a well-known formula for the values of the ordinary partition function $p(n)$. More recently, Bruinier and Ono obtained an algebraic formula for these values. Here we study the…
We consider the solution of the stochastic heat equation \partial_T \mathcal{Z} = 1/2 \partial_X^2 \mathcal{Z} - \mathcal{Z} \dot{\mathscr{W}} with delta function initial condition \mathcal{Z} (T=0)= \delta_0 whose logarithm, with…
Let A be a p-variate real Wishart matrix on n degrees of freedom with identity covariance. The distribution of the largest eigenvalue in A has important applications in multivariate statistics. Consider the asymptotics when p grows in…
We revisit the transfer-matrix approach to directed polymers in random media and show that a single ensemble of random transfer-matrix products provides a unified realization of the canonical one-point fluctuation laws in $(1+1)$…
We study the Tracy-Widom (TW) distribution $f_\beta(a)$ in the limit of large Dyson index $\beta \to +\infty$. This distribution describes the fluctuations of the rescaled largest eigenvalue $a_1$ of the Gaussian (alias Hermite) ensemble…
The well known Erdos-Turan law states that the logarithm of an order of a random permutation is asymptotically normally distributed. The aim of this work is to estimate convergence rate in this theorem and also to prove analogous result for…
We analyze the tomographic representation for the Friedmann-Robertson-Walker (FRW) model within the Loop Quantum Cosmology framework. We focus on the Wigner quasi-probability distributions associated with Gaussian and Schr\"odinger cat…
Based on the multivariate saddle point method we study the asymptotic behavior of the characteristic polynomials associated to Wishart type random matrices that are formed as products consisting of independent standard complex Gaussian and…
We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems…
This paper develops asymptotic theory of integrals of empirical quantile functions with respect to random weight functions, which is an extension of classical $L$-statistics. They appear when sample trimming or Winsorization is applied to…
We consider the q-TASEP that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on Z for q in [0,1) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the…
Representation theory of finite groups portrays a marvelous crossroad of group theory, algebraic combinatorics, and probability. In particular the Plancherel measure is a probability that arises naturally from representation theory, and in…
We study the joint distribution of SYK Hamiltonians for different systems with specified overlaps. We show that, in the large-system limit, their joint distribution converges in distribution to a mixed $q$-Gaussian system. We explain that…
We investigate the asymptotic behavior of the empirical eigenvalues distribution of the partial transpose of a random quantum state. The limiting distribution was previously investigated via Wishart random matrices indirectly (by…
We consider the fluctuations of the largest eigenvalue of sparse random matrices, the class of random matrices that includes the normalized adjacency matrices of the Erd\H{o}s-R\'enyi graph $G(N, p)$. We show that the fluctuations of the…
In a recent study of large non-null sample covariance matrices, a new sequence of functions generalizing the GUE Tracy-Widom distribution of random matrix theory was obtained. This paper derives Painlev\'e formulas of these functions and…