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It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a $Ntimes N$ random unitary matrix sampled from the Haar measure grows like $CN/(log N)^{3/4}$ for some random variable $C$. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the range $[N^{1 - varepsilon},N^{1 + varepsilon}]$, for arbitrarily small $varepsilon$. The method is based on identifying an approximate branching random walk in the Fourier decomposition of the characteristic polynomial, and uses techniques developed to describe the extremes of branching random walks and of other log-correlated random fields. A key technical input is the asymptotic analysis of Toeplitz determinants with dimension-dependent symbols. The original argument for these asymptotics followed the general idea that the statistical mechanics of $1/f$-noise random energy models is governed by a freezing transition. We also prove the conjectured freezing of the free energy for random unitary matrices.
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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 consider the empirical eigenvalue distribution of an $mtimes m$ principle submatrix of an $ntimes n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and Reffy identified the limiting spectral measure if $frac{m}{n}toalpha$, as $ntoinfty$; under suitable scaling, the family ${mu_alpha}_{alphain(0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $alpha$) and uniform measure on the unit circle (as $alphato1$). In this note, we prove an explicit concentration inequality which shows that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $mu_alpha$ is typically of order $sqrt{frac{log(m)}{m}}$ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new Coulomb transport inequality due to Chafai, Hardy, and Maida.
We consider the empirical eigenvalue distribution of an $mtimes m$ principal submatrix of an $ntimes n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $frac{m}{n}=alpha$, the empirical spectral measure is well-approximated by a deterministic measure $mu_alpha$ supported on the unit disc. In earlier work, we showed that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $mu_alpha$ is typically of order $sqrt{frac{log(m)}{m}}$ or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.
We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As applications, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph.
This is a brief survey of classical and recent results about the typical behavior of eigenvalues of large random matrices, written for mathematicians and others who study and use matrices but may not be accustomed to thinking about randomness.
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