We compute the large size limit of the moment formula derived in [14] for the Hermitian Jacobi process at fixed time. Our computations rely on the polynomial division algorithm which allows to obtain cancellations similar to those obtained in [3, Lemma 3]. In particular, we identify the terms contributing to the limit and show they satisfy a double recurrence relation. We also determine explicitly some of them and revisit a special case relying on Carlitz summation identity for terminating $1$-balanced ${}_4F_3$ functions taken at unity.

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime p and a positive integer n, consider a random $n \times n$ matrix $X_n$ over the ring $\mathbb{Z}_p$ of p-adic integers whose entries are independent. Previously, Wood showed that as long as each entry of $X_n$ is not too concentrated on a single residue modulo p, regardless of its distribution, the distribution of the cokernel $\mathrm{cok}(X_n)$ of $X_n$, up to isomorphism, weakly converges to the Cohen–Lenstra distribution, as $n \rightarrow \infty$. Here on the contrary, we consider the case when $X_n$ has a concentrated residue $A_n$ so that $X_n = A_n + pB_n$. When $B_n$ is a Haar-random $n \times n$ matrix over $\mathbb{Z}_p$, we explicitly compute the distribution of $\mathrm{cok}(P(X_n))$ for every fixed n and a non-constant monic polynomial $P(t) \in \mathbb{Z}_p[t]$. We deduce our result from an interesting equidistribution result for matrices over $\mathbb{Z}_p[t]/(P(t))$, which we prove by establishing a version of the Weierstrass preparation theorem for the noncommutative ring $\mathrm{M}_n(\mathbb{Z}_p)$ of $n \times n$ matrices over $\mathbb{Z}_p$. We also show through cases the subtlety of the “universality” behavior when $B_n$ is not Haar-random.

The walk matrix associated to an $n\times n$ integer matrix $\mathbf{X}$ and an integer vector $b$ is defined by ${\mathbf{W}} \,:\!=\, (b,{\mathbf{X}} b,\ldots, {\mathbf{X}}^{n-1}b)$. We study limiting laws for the cokernel of $\mathbf{W}$ in the scenario where $\mathbf{X}$ is a random matrix with independent entries and $b$ is deterministic. Our first main result provides a formula for the distribution of the $p^m$-torsion part of the cokernel, as a group, when $\mathbf{X}$ has independent entries from a specific distribution. The second main result relaxes the distributional assumption and concerns the ${\mathbb{Z}}[x]$-module structure.

The motivation for this work arises from an open problem in spectral graph theory, which asks to show that random graphs are often determined up to isomorphism by their (generalised) spectrum. Sufficient conditions for generalised spectral determinacy can, namely, be stated in terms of the cokernel of a walk matrix. Extensions of our results could potentially be used to determine how often those conditions are satisfied. Some remaining challenges for such extensions are outlined in the paper.

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well.

We obtain asymptotic formulas for the number of matrices in the congruence subgroup

$$\begin{align*}\Gamma_0(Q) = \left\{ A\in\operatorname{SL}_2({\mathbb Z}):~c \equiv 0 \quad\pmod Q\right\}, \end{align*}$$

We establish the asymptotic expansion in $\beta $ matrix models with a confining, off-critical potential in the regime where the support of the equilibrium measure is a finite union of segments. We first address the case where the filling fractions of these segments are fixed and show the existence of a $\frac {1}{N}$ expansion. We then study the asymptotics of the sum over the filling fractions to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ($\beta = 2$) as well as orthogonal ($\beta = 1$) and skew-orthogonal ($\beta = 4$) polynomials outside the bulk.

Let A be an $n \times n$ symmetric matrix with $(A_{i,j})_{i\leqslant j}$ independent and identically distributed according to a subgaussian distribution. We show that

$$ \begin{align*}\mathbb{P}(\sigma_{\min}(A) \leqslant \varepsilon n^{-1/2} ) \leqslant C \varepsilon + e^{-cn},\end{align*} $$

where $\sigma _{\min }(A)$ denotes the least singular value of A and the constants $C,c>0 $ depend only on the distribution of the entries of A. This result confirms the folklore conjecture on the lower tail of the least singular value of such matrices and is best possible up to the dependence of the constants on the distribution of $A_{i,j}$. Along the way, we prove that the probability that A has a repeated eigenvalue is $e^{-\Omega (n)}$, thus confirming a conjecture of Nguyen, Tao and Vu [Probab. Theory Relat. Fields 167 (2017), 777–816].

We find closed formulas for arbitrarily high mixed moments of characteristic polynomials of the Alternative Circular Unitary Ensemble, as well as closed formulas for the averages of ratios of characteristic polynomials in this ensemble. A comparison is made to analogous results for the Circular Unitary Ensemble. Both moments and ratios are studied via symmetric function theory and a general formula of Borodin-Olshanski-Strahov.

For a $k$-uniform hypergraph $\mathcal{H}$ on vertex set $\{1, \ldots, n\}$ we associate a particular signed incidence matrix $M(\mathcal{H})$ over the integers. For $\mathcal{H} \sim \mathcal{H}_k(n, p)$ an Erdős–Rényi random $k$-uniform hypergraph, ${\mathrm{coker}}(M(\mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for $p = \omega (1/n^{k - 1})$, ${\mathrm{coker}}(M(\mathcal{H}))$ is torsion-free.

We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.

We calculate the moments of the characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, at a position t in the bulk of the spectrum, as a series expansion in powers of t. We focus in particular on the Gaussian Unitary Ensemble. We employ a novel approach to calculate the coefficients in this series expansion of the moments, appropriately scaled. These coefficients are polynomials in N. They therefore grow as $N\to\infty$, meaning that in this limit the radius of convergence of the series expansion tends to zero. This is related to oscillations as t varies that are increasingly rapid as N grows. We show that the $N\to\infty$ asymptotics of the moments can be derived from this expansion when $t=0$. When $t\ne 0$ we observe a surprising cancellation when the expansion coefficients for N and $N+1$ are formally averaged: this procedure removes all of the N-dependent terms leading to values that coincide with those expected on the basis of previously established asymptotic formulae for the moments. We obtain as well formulae for the expectation values of products of the secular coefficients.

Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$-matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random $\{\pm 1\}$-matrices over $\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$. Previously, such estimates were available only for $p = o(n^{1/8})$. At the heart of our proof is a way to combine multiple inverse Littlewood–Offord-type results to control the contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$ with anticoncentration at least $1/p + \Omega(1/p^2)$. Previously, inverse Littlewood–Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C > 1$.

We investigate the concept of orbital free entropy from the viewpoint of the matrix liberation process. We will show that many basic questions around the definition of orbital free entropy are reduced to the question of full large deviation principle for the matrix liberation process. We will also obtain a large deviation upper bound for a certain family of random matrices that is essential to define the orbital free entropy. The resulting rate function is made up into a new approach to free mutual information.

We show that a nearly square independent and identically distributed random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz [6]. Our result extends to sparse matrices as well as to matrices of dependent entries.

We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.

We generalize the Cohen–Lenstra heuristics over function fields to étale group schemes $G$ (with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg–Venkatesh–Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of $\wedge ^{2}G(1)$, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen–Lenstra heuristics for abelian $\ell$-groups in cases where $\ell \mid q-1$; the nature of this failure was suggested already in the works of Malle, Garton, Ellenberg–Venkatesh–Westerland, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.

We consider a class of sample covariance matrices of the form Q = TXX*T*, where X = (xij) is an M×N rectangular matrix consisting of independent and identically distributed entries, and T is a deterministic matrix such that T*T is diagonal. Assuming that M is comparable to N, we prove that the distribution of the components of the right singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of xij coincide with the Gaussian random variables. For the right singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of xij match those of the Gaussian random variables. Similar results hold for the left singular vectors if we further assume that T is diagonal.

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let $M_{N}$ be a deterministic $N\times N$ matrix, and let $G_{N}$ be a complex Ginibre matrix. We consider the matrix ${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$, where $\unicode[STIX]{x1D6FE}>1/2$. With $L_{N}$ the empirical measure of eigenvalues of ${\mathcal{M}}_{N}$, we provide a general deterministic equivalence theorem that ties $L_{N}$ to the singular values of $z-M_{N}$, with $z\in \mathbb{C}$. We then compute the limit of $L_{N}$ when $M_{N}$ is an upper-triangular Toeplitz matrix of finite symbol: if $M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$ where $\mathfrak{d}$ is fixed, $a_{i}\in \mathbb{C}$ are deterministic scalars and $J$ is the nilpotent matrix $J(i,j)=\mathbf{1}_{j=i+1}$, then $L_{N}$ converges, as $N\rightarrow \infty$, to the law of $\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$ where $U$ is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when $\mathfrak{d}=1$, also of independent and identically distributed entries on the diagonals in $M_{N}$.

We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives an example where the partial transpose produces freeness at the operator level. Finally, we investigate the case of real Wishart matrices.