arXiv subjects

Bertrand Stone

Publications and source records attributed to Bertrand Stone.

3 recordsLinked to original sources

arXiv2025

Localization-delocalization transition for a random block matrix model at the edge

Consider a random block matrix model consisting of $D$ random systems arranged along a circle, where each system is modeled by an independent $N\times N$ complex Hermitian Wigner matrix. Neighboring systems interact via an arbitrary deterministic $N\times N$ matrix $A$. In this paper, we extend the localization-delocalization transition previously established in arxiv:2312.07297 for the bulk eigenvalue spectrum to the entire spectrum, including the spectral edges. Let $[E^-,E^+]$ denote the support of the limiting spectral density, and define $\kappa_E:=|E-E^+|\wedge |E-E^-|$ as the distance from a given energy $E \in [E^-, E^+]$ to the spectral edges. We show that for eigenvalues near $E$, the corresponding eigenvectors undergo a localization-delocalization transition when $\|A\|_{\mathrm{HS}}$ crosses the critical threshold $(\kappa_E + N^{-2/3})^{-1/2}$. In the delocalized phase, the extreme eigenvalues asymptotically follow the Tracy-Widom distribution, while in the localized phase, the edge eigenvalue statistics asymptotically match those of $D$ independent GUE ensembles, up to a deterministic shift. Our results recover those of arxiv:2312.07297 in the bulk regime, where $\kappa_E \asymp 1$, and further reveal the presence of mobility edges near $E^\pm$ when $1 \ll \|A\|_{\mathrm{HS}} \ll N^{1/3}$. Specifically, bulk eigenvectors corresponding to energies $E$ with $\kappa_E \gg \|A\|_{\mathrm{HS}}^{-2}$ are delocalized, while those with $\kappa_E \ll \|A\|_{\mathrm{HS}}^{-2}$ are localized.

math.PR↗

arXiv2023

A random matrix model towards the quantum chaos transition conjecture

Consider $D$ random systems that are modeled by independent $N\times N$ complex Hermitian Wigner matrices. Suppose they are lying on a circle and the neighboring systems interact with each other through a deterministic matrix $A$. We prove that in the asymptotic limit $N\to \infty$, the whole system exhibits a quantum chaos transition when the interaction strength $\|A\|_{HS}$ varies. Specifically, when $\|A\|_{HS}\ge N^{\varepsilon}$, we prove that the bulk eigenvalue statistics match those of a $DN\times DN$ GUE asymptotically and each bulk eigenvector is approximately equally distributed among the $D$ subsystems with probability $1-o(1)$. These phenomena indicate quantum chaos of the whole system. In contrast, when $\|A\|_{HS}\le N^{-\varepsilon}$, we show that the system is integrable: the bulk eigenvalue statistics behave like $D$ independent copies of GUE statistics asymptotically and each bulk eigenvector is localized on only one subsystem. In particular, if we take $D\to \infty$ after the $N\to \infty$ limit, the bulk statistics converge to a Poisson point process under the $DN$ scaling.

math.PR↗

arXiv2016

Line Complexity Asymptotics of Polynomial Cellular Automata

Cellular automata are discrete dynamical systems that consist of patterns of symbols on a grid, which change according to a locally determined transition rule. In this paper, we will consider cellular automata that arise from polynomial transition rules, where the symbols in the automaton are integers modulo some prime $p$. We are principally concerned with the asymptotic behavior of the line complexity sequence $a_T(k)$, which counts, for each $k$, the number of coefficient strings of length $k$ that occur in the automaton. We begin with the modulo $2$ case. For a given polynomial $T(x) = c_0 + c_1x + ... + c_nx^n$ with $c_0,c_n\neq 0$, we construct odd and even parts of the polynomial from the strings $0c_1c_3c_5...$ and $c_0c_2c_4...$, respectively. We prove that for polynomials for which the odd and even parts are relatively prime, $a_T(k)$ satisfies recursions of a specific form. We also consider powers of transition rules modulo $p$, introducing a notion of the order of a recursion. We show that the property of "having a recursion of some order" is preserved when the transition rule is raised to a positive integer power. Extending to a more general setting, we investigate the asymptotics of $a_T(k)$ by considering an abstract generating function $ϕ(z)=\sum_{k=1}^\inftyα(k)z^k$ which satisfies a general functional equation relating $ϕ(z)$ and $ϕ(z^p)$ for some prime $p$. We show that there is a continuous, piecewise quadratic function $f$ on $[1/p, 1]$ for which $\lim_{k\to\infty}(α(k)/k^2 - f(p^{-\langle\log_p k\rangle})) = 0$, where $\langle y\rangle$ denotes the fractional part of $y$. We use this result to show that for certain positive integer sequences $s_k\to\infty$ with a parameter $x\in [1/p,1]$, the ratio $α(s_k(x))/s_k(x)^2$ tends to $f(x)$, and that the limit superior and inferior of $α(k)/k^2$ are given by the extremal values of $f$.

math.CO↗