arXiv subjects
Publications and source records attributed to Benjamin Landon.
We consider the largest eigenvalues of minors of the classical Gaussian random matrices with a finite rank spike in the critical BBP regime. We show that they converge to a multi-level interlacing particle system whose lowest level is the Airy$_\beta$ point process. Our construction and scaling limits extend in a natural way to non-classical $\beta >0$.
We consider $N\times N$ matrices $X$ with i.i.d. entries, and prove that the random measure $\mu_N(z):=\frac{ | \det (X-z)|^\gamma}{\mathbb{E}[ | \det (X-z)|^\gamma]} \mathrm{d} z\mathrm{d} \overline{z}$ converges to the Gaussian Multiplicative Chaos (GMC) on the unit disc in the full subcritical regime $\gamma \in (0, 2 \sqrt{2})$ as $N \to \infty$. Our result holds for both symmetry classes and in particular is new even for real Ginibre matrices. This result is the first of its kind for any non-invariant ensemble of random matrices. In the case of real matrices, we further prove that the restriction of $\mu_N(z)$ to the unit interval converges to a one-dimensional GMC. We establish the asymptotics for the $K$-point function of $| \det (X-z)|$ at any collection of mesoscopically separated points $z_i$. Our methods are analytic and probabilistic in nature, relying in part on the dynamical approach based on Dyson Brownian motion.
We prove an optimal order delocalization estimate for the eigenvectors of general $N \times N$ non-Hermitian matrices $X$: $\| {\bf v } \|_\infty \leq C \sqrt{\frac{\log N}{N}}$ with very high probability, for any right or left eigenvector ${\bf v}$ of $X$. This improves upon the previous tightest bound of Rudelson and Vershynin [arXiv:1306.2887] of $\mathcal{O}( ( \log N)^{9/2}N^{-1/2})$, and holds under weaker assumptions on the tail of the matrix elements. In addition to the coordinate basis, our bound holds for the $\ell^\infty$ norm in any deterministic orthonormal basis. Our result is proven via a dynamical method, by studying the flow of the resolvent of the Hermitization of $X$ and proving local laws on short scales.
We prove a homogenization result for the difference of two coupled Dyson Brownian motions started from generalized Wigner matrix initial data. We prove an optimal order, high probability estimate that is valid throughout the spectrum, including up to the spectral edges. Prior homogenization results concerned only the bulk of the spectrum. We apply our estimate to address the question of quantifying edge universality. Here, we have two results. We show that the Kolmogorov-Smirnov distance of the distribution of the gap between the largest two eigenvalues of a generalized Wigner matrix (with smooth entry distribution) and its GOE/GUE counterpart is $\mathcal{O}(N^{-1+\varepsilon})$. On the other hand, we show that, for the distribution of the largest eigenvalue, there are Wigner matrices so that the analogous Kolmogorov-Smirnov distance is bounded below by $N^{-1/3-\varepsilon}$.
We consider the characteristic function of linear spectral statistics of generalized Wigner matrices. We provide an expansion of the characteristic function with error $\mathcal{O} ( N^{-1})$ around its limiting Gaussian form, and identify sub-leading non-Gaussian corrections of size $\mathcal{O} (N^{-1/2})$. Prior expansions with this error rate held only for Wigner matrices; only a weaker error rate was available for more general matrix ensembles. We provide some applications.
We compute the leading order asymptotic of the maximum of the characteristic polynomial for i.i.d. matrices with real or complex entries. In particular, this result is new even for real Ginibre matrices, which was left as an open problem in [arXiv:2303.09912]; the complex Ginibre case was covered in [arXiv:1902.01983]. These are the first universality results for the non--Hermitian analog of the first order term of the Fyodorov--Hiary--Keating conjecture. Our methods are based on constructing a coupling to the branching random walk via Dyson Brownian motion. In particular, we find a new connection between real i.i.d. matrices and inhomogeneous branching random walk.
This work studies the tail exponents for the height function of the stationary stochastic six vertex model in the moderate deviations regime. For the upper tail of the height function we find upper and lower bounds of matching order, with a tail exponent of $\frac{3}{2}$, characteristic of KPZ distributions. We also obtain an upper bound for the lower tail of the same order. Our results for the stochastic six vertex model hold under a restriction on the model parameters for which a certain "microscopic concavity" condition holds. Nevertheless, our estimates are sufficiently strong to pass through the degeneration of the stochastic six vertex model to the ASEP. We therefore obtain tail estimates for both the current as well as the location of a second class particle in the ASEP with stationary (Bernoulli) initial data. Our estimates complement the variance bounds obtained in the seminal work of Bal\'azs and Sepp\"al\"ainen.}
We derive upper and lower bounds for the upper and lower tails of the O'Connell-Yor polymer of the correct order of magnitude via probabilistic and geometric techniques in the moderate deviations regime. The inputs of our work are an identity for the generating function of a two-parameter model of Rains and Emrah-Janjigian-Sepp\"al\"ainen, and the geometric techniques of Ganguly-Hegde and Basu-Ganguly-Hammond-Hegde. As an intermediate result we obtain strong tail estimates for the transversal fluctuation of the polymer path from the diagonal.
We present a proof of an upper tail bound of the correct order (up to a constant factor in the exponent) in two classes of stationary models in the KPZ universality class. The proof is based on an exponential identity due to Rains in the case of Last Passage Percolation with exponential weights, and recently re-derived by Emrah-Jianjigian-Sepp\"ail\"ainen (EJS). Our proof follows very similar lines for the two classes of models we consider, using only general monotonocity and convexity properties, and can thus be expected to apply to many other stationary models.
We consider linear spectral statistics of the form $\mathrm{tr} ( \varphi (H))$ for test functions $\varphi$ of low regularity and Wigner matrices $H$ with smooth entry distribution. We show that for functions $\varphi$ in the Sobolev space $H^{1/2+\varepsilon}$ or the space $C^{1/2+\varepsilon}$, that are supported within the spectral bulk of the semicircle distribution, these linear spectral statistics have asymptotic Gaussian fluctuations with the same variance as in the CLT for functions of higher regularity, for any $\varepsilon >0$.
We establish high probability estimates on the eigenvalue locations of Brownian motion on the $N$-dimensional unitary group, as well as estimates on the number of eigenvalues lying in any interval on the unit circle. These estimates are optimal up to arbitrarily small polynomial factors in $N$. Our results hold at the spectral edges (showing that the extremal eigenvalues are within $\mathcal{O} (N^{-2/3+})$ of the edges of the limiting spectral measure), in the spectral bulk, as well as for times near $4$ at which point the limiting spectral measure forms a cusp. Our methods are dynamical and are based on analyzing the evolution of the Borel transform of the empirical spectral measure along the characteristics of the PDE satisfied by the limiting spectral measure, that of the free unitary Brownian motion.
We consider the single eigenvalue fluctuations of random matrices of general Wigner-type, under a one-cut assumption on the density of states. For eigenvalues in the bulk, we prove that the asymptotic fluctuations of a single eigenvalue around its classical location are Gaussian with a universal variance. Our method is based on a dynamical approach to mesoscopic linear spectral statistics which reduces their behavior on short scales to that on larger scales. We prove a central limit theorem for linear spectral statistics on larger scales via resolvent techniques and show that for certain classes of test functions, the leading-order contribution to the variance agrees with the GOE/GUE cases.
We consider systems of $N$ diffusions in equilibrium interacting through a potential $V$. We study a "height function" which for the special choice $V(x) = \e^{-x}$, coincides with the partition function of a stationary semidiscrete polymer, also known as the (stationary) O'Connell-Yor polymer. For a general class of smooth convex potentials (generalizing the O'Connell-Yor case), we obtain the order of fluctuations of the height function by proving matching upper and lower bounds for the variance of order $N^{2/3}$, the expected scaling for models lying in the KPZ universality class. The models we study are not expected to be integrable and our methods are analytic and non-perturbative, making no use of explicit formulas or any results for the O'Connell-Yor polymer.
We investigate the fluctuations of the free energy of the $2$-spin spherical Sherrington-Kirkpatrick model at critical temperature $\beta_c = 1$. When $\beta = 1$ we find asymptotic Gaussian fluctuations with variance $\frac{1}{6N^2} \log(N)$, confirming in the spherical case a physics prediction for the SK model with Ising spins. We furthermore prove the existence of a critical window on the scale $\beta = 1 +\alpha \sqrt{ \log(N) } N^{-1/3}$. For any $\alpha \in \mathbb{R}$ we show that the fluctuations are at most order $\sqrt{ \log(N) } / N$, in the sense of tightness. If $ \alpha \to \infty$ at any rate as $N \to \infty$ then, properly normalized, the fluctuations converge to the Tracy-Widom$_1$ distribution. If $ \alpha \to 0$ at any rate as $N \to \infty$ or $ \alpha <0$ is fixed, the fluctuations are asymptotically Gaussian as in the $\alpha=0$ case. In determining the fluctuations, we apply a recent result of Lambert and Paquette on the behavior of the Gaussian-$\beta$-ensemble at the spectral edge.
We analyze the fluctuations of the free energy, replica overlaps, and overlap with the external field in the quadratic spherical SK model with a magnetic field. We identify several different behaviors for these quantities depending on the size of the magnetic field, confirming predictions by Fyodorov-Le Doussal and recent work of Baik, Collins-Wildman, Le Doussal and Wu.
For general $\beta \geq 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N \to \infty$. For each fixed time, this ensemble is distributed as the Airy$_\beta$ random point field. We prove that the increments of the limiting process are locally Brownian. When $\beta >1$ we prove that after subtracting a Brownian motion, the sample paths are almost surely locally $r$-H{\"o}lder for any $r<1-(1+\beta)^{-1}$. Furthermore for all $\beta \geq 1$ we show that the limiting process solves an SDE in a weak sense. When $\beta=2$ this limiting process is the Airy line ensemble.
We describe the fluctuations of the overlap between two replicas in the 2-spin spherical SK model about its limiting value in the low temperature phase. We show that the fluctuations are of order $N^{-1/3}$ and are given by a simple, explicit function of the eigenvalues of a matrix from the Gaussian Orthogonal Ensemble. We show that this quantity converges and describe its limiting distribution in terms of the Airy1random point field (i.e., the joint limit of the extremal eigenvalues of the GOE) from random matrix theory.
We introduce a method for the comparison of some extremal eigenvalue statistics of random matrices. For example, it allows one to compare the maximal eigenvalue gap in the bulk of two generalized Wigner ensembles, provided that the first four moments of their matrix entries match. As an application, we extend results of Bourgade--Ben Arous and Feng--Wei that identify the limit of the maximal eigenvalue gap in the bulk of the GUE to all complex Hermitian generalized Wigner matrices.