Related papers: On an average over the Gaussian Unitary Ensemble
Let $N(L)$ be the number of eigenvalues, in an interval of length $L$, of a matrix chosen at random from the Gaussian Orthogonal, Unitary or Symplectic ensembles of ${\cal N}$ by ${\cal N}$ matrices, in the limit ${\cal…
We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble (IUE) is equal to the eigenvalue density of n-th power of a single matrix from this ensemble, in the…
In this work, we provide the first strong convergence result of numerical approximation of a general second order semilinear stochastic fractional order evolution equation involving a Caputo derivative in time of order $\alpha\in(\frac 34,…
Until now only for specific crossovers between Poissonian statistics (P), the statistics of a Gaussian orthogonal ensemble (GOE), or the statistics of a Gaussian unitary ensemble (GUE) analytical formulas for the level spacing distribution…
I present an unbiased method of mapping particles to distribution functions and vice versa. This method alone defines the canonical formulation of statistical mechanics, since it can be used to derive the principle of maximum entropy in…
We study the fluctuations of the largest eigenvalue $\lambda_{\max}$ of $N \times N$ random matrices in the limit of large $N$. The main focus is on Gaussian $\beta$-ensembles, including in particular the Gaussian orthogonal ($\beta=1$),…
The new Theorem on location of maximum of probability density functions of dimensionless second difference of the three adjacent energy levels for $N$-dimensional Gaussian orthogonal ensemble GOE($N$), $N$-dimensional Gaussian unitary…
The complex elliptic Ginibre ensemble with coupling $\tau$ is a complex Gaussian matrix interpolating between the Gaussian Unitary Ensemble (GUE) and the Ginibre ensemble. It has been known for some time that its eigenvalues form a…
In this paper we establish new integral representations for the remainder term of the known asymptotic expansion of the logarithm of the Barnes $G$-function. Using these representations, we obtain explicit and numerically computable error…
The eigenvalue correlations of random matrices from the Jacobi Unitary Ensemble have a known asymptotic behavior as their size tends to infinity. In the bulk of the spectrum the behavior is described in terms of the sine kernel, and at the…
Consider an ensemble of $N\times N$ non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded…
We study rates of convergence in central limit theorems for the partial sum of squares of general Gaussian sequences, using tools from analysis on Wiener space. No assumption of stationarity, asymptotically or otherwise, is made. The main…
We study global fluctuations for singular values of $M$-fold products of several right-unitarily invariant $N \times N$ random matrix ensembles. As $N \to \infty$, we show the fluctuations of their height functions converge to an explicit…
The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct aymptotics of the variance of the eigenvalue counting function of GUE matrices…
We consider a class of matrix integrals over the unitary group $U(N)$ with an infinite set of couplings characterized by a series $f(q)=\sum_{n \ge 1} a_n q^n$, with $a_n \in \mathbb{Z}$. Such integrals arise in physics as the partition…
We calculate the time-evolution of a discrete-time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size-dependent rates. In the continuous-time limit the process turns into the…
In the current paper we consider a Wigner matrix and consider an analytic function of polynomial growth on a set containing the support of the semicircular law in its interior. We prove that the linear spectral statistics corresponding to…
We compute the partition functions of $\mathcal{N} = 1$ gauge theories on $S^2 \times \mathbb{R}^2_\varepsilon$ using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of $S^2$ and at the origin of…
The starting point of this work is a theorem due to Maxwell characterizing the distribution of a Gaussian vector with at least two coordinates. We define the Gaussian orthogonal, unitary and symplectic tensor ensembles for notions of real…
The stochastic partial differential equation approach to Gaussian processes (GPs) represents Mat\'ern GP priors in terms of $n$ finite element basis functions and Gaussian coefficients with sparse precision matrix. Such representations…