P. Lopatto

activity

20142026

most citedFluctuations in local quantum unique ergodicity for generalized Wigner matrices

22 citations · 30 across the 31 of their papers we have counts for

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27 papers · 1 filter

math.PR2026

Overlap structure of mixed even $p$ -spin models

P. M. Aronow, Patrick Lopatto

We identify the limiting overlap structure of mixed even $p$ -spin models, including the Sherrington-Kirkpatrick model, at every fixed deterministic external field. In particular, w…

math.PR2026

Parisi measure of the Sherrington-Kirkpatrick model with external field

Patrick Lopatto

We prove that the Parisi measure of the Sherrington-Kirkpatrick model has connected support at every positive temperature and every positive external field. When the model breaks r…

math.PR2026

Quantitative Parisi formulas and fluctuations in the Sherrington-Kirkpatrick model

P. M. Aronow, Patrick Lopatto

We establish quantitative Parisi formulas and fluctuation bounds for the Sherrington-Kirkpatrick model at zero external field. In particular, using that the Parisi measure is suppo…

math.PR2026

Optimal central limit theorem for bounded random variables in high dimensions

P. M. Aronow, Patrick Lopatto

Let $W = n^{- 1/2} \sum_{i = 1}^{n} X_{i}$ , where the $X_{i}$ are independent centered random vectors in $R^{p}$ with $∣ X_{ij} ∣ \leq B$ almost surely. Suppose that $Cov (W)$ has un…

math.PR2026

Replica symmetry breaking at arbitrary depth

P. M. Aronow, Patrick Lopatto

We study finite-step replica symmetry breaking (RSB) in mixed $p$ -spin Ising spin glasses at positive temperature and zero external field. For every integer $k \geq 1$ , we construct…

math.PR2026

Full replica symmetry breaking in the Sherrington-Kirkpatrick model

Patrick Lopatto

We prove that, at zero external field and for every inverse temperature $β > 1$ , the Parisi measure of the Sherrington-Kirkpatrick model is supported on an interval $[0, q_{β}]$ , with a…

math.PR2026

Sobolev convergence of log-determinants for smooth Wigner matrices

Giorgio Cipolloni, Patrick Lopatto

We show that the fields emerging from the log-determinant and the eigenvalue counting function of smooth Wigner matrices converge in law to centered Gaussian, logarithmically corre…

math.PR2026

Replica symmetry up to the de Almeida-Thouless line in the Sherrington-Kirkpatrick model

Patrick Lopatto

We show that in the Sherrington-Kirkpatrick model at inverse temperature $β$ with uniform external field $h > 0$ , replica symmetry holds for the set of pairs $(β, h)$ predicted by de…

math.PR2026

Mobility Edge for the Anderson Model on Random Regular Graphs

Suhan Liu, Patrick Lopatto

We determine the phase diagram of the Anderson tight-binding model on random regular graphs with Gaussian disorder and sufficiently large degree. In particular, we prove that if th…

math.PR2025

Mobility Edge for the Anderson Model on the Bethe Lattice

Amol Aggarwal, Patrick Lopatto

We pinpoint the spectral decomposition for the Anderson tight-binding model with an unbounded random potential on the Bethe lattice of sufficiently large degree. We prove that ther…

math.PR2025

Maximum gap in complex Ginibre matrices

Patrick Lopatto, Moritz Otto

We determine the asymptotic size of the largest gap between bulk eigenvalues in complex Ginibre matrices.

math.PR2024

Smallest gaps between eigenvalues of real Gaussian matrices

Patrick Lopatto, Matthew Meeker

We consider an $n \times n$ matrix of independent real Gaussian random variables and determine the asymptotic distribution of the smallest gaps between complex eigenvalues.

math.PR2023★ 1 cited

Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

Paul Bourgade, Patrick Lopatto, Ofer Zeitouni

We determine to leading order the maximum of the characteristic polynomial for Wigner matrices and $β$ -ensembles. In the special case of Gaussian-divisible Wigner matrices, our met…

math.PR2023

Quenched large deviation principles for random projections of $ℓ_{p}^{n}$ balls

Patrick Lopatto, Kavita Ramanan, Xiaoyu Xie

Let $(k_{n})_{n \in N}$ be a sequence of positive integers growing to infinity at a sublinear rate, $k_{n} \to \infty$ and $k_{n} / n \to 0$ as $n \rightarrow \inf…

math.PR2023

Central limit theorem for the complex eigenvalues of Gaussian random matrices

Advay Goel, Patrick Lopatto, Xiaoyu Xie

We establish a central limit theorem for the eigenvalue counting function of a matrix of real Gaussian random variables.

math.PR2023

Fluctuations in Quantum Unique Ergodicity at the Spectral Edge

Lucas Benigni, Nixia Chen, Patrick Lopatto +1

We study the eigenvector mass distribution of an $N \times N$ Wigner matrix on a set of coordinates $I$ satisfying $∣ I ∣ \geq c N$ for some constant $c > 0$ . For eigenvectors corresp…

math.PR2022★ 1 cited

Existence of the free energy for heavy-tailed spin glasses

Aukosh Jagannath, Patrick Lopatto

We study the free energy of a mean-field spin glass whose coupling distribution has power law tails. Under the assumption that the couplings have infinite variance and finite mean,…

math.PR2022

Mobility Edge for Lévy Matrices

Amol Aggarwal, Charles Bordenave, Patrick Lopatto

Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an $α$ -stable law. For $α < 1$ , predictions from the physics literature sugg…

math.PR2021★ 22 cited

Fluctuations in local quantum unique ergodicity for generalized Wigner matrices

Lucas Benigni, Patrick Lopatto

We study the eigenvector mass distribution for generalized Wigner matrices on a set of coordinates $I$ , where $N^{ε} \leq ∣ I ∣ \leq N^{1 - ε}$ , and prove it converg…

math.PR2020

Optimal Delocalization for Generalized Wigner Matrices

Lucas Benigni, Patrick Lopatto

We study the eigenvectors of generalized Wigner matrices with subexponential entries and prove that they delocalize at the optimal rate with overwhelming probability. We also prove…

math.PR2020

Eigenvector Statistics of Lévy Matrices

Amol Aggarwal, Patrick Lopatto, Jake Marcinek

We analyze statistics for eigenvector entries of heavy-tailed random symmetric matrices (also called Lévy matrices) whose associated eigenvalues are sufficiently small. We show tha…

math.PR2019★ 3 cited

Universality of the least singular value for the sum of random matrices

Ziliang Che, Patrick Lopatto

We consider the least singular value of $M = R^{*} X T + U^{*} Y V$ , where $R, T, U, V$ are independent Haar-distributed unitary matrices and $X, Y$ are deterministic diagonal matrices. Un…

math.PR2019

Tail bounds for gaps between eigenvalues of sparse random matrices

Patrick Lopatto, Kyle Luh

We prove the first eigenvalue repulsion bound for sparse random matrices. As a consequence, we show that these matrices have simple spectrum, improving the range of sparsity and er…

math.PR2018

Comparison theorem for some extremal eigenvalue statistics

Benjamin Landon, Patrick Lopatto, Jake Marcinek

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…

Overlap structure of mixed even p-spin models

Parisi measure of the Sherrington-Kirkpatrick model with external field

Quantitative Parisi formulas and fluctuations in the Sherrington-Kirkpatrick model

Optimal central limit theorem for bounded random variables in high dimensions

Replica symmetry breaking at arbitrary depth

Full replica symmetry breaking in the Sherrington-Kirkpatrick model

Sobolev convergence of log-determinants for smooth Wigner matrices

Replica symmetry up to the de Almeida-Thouless line in the Sherrington-Kirkpatrick model

Mobility Edge for the Anderson Model on Random Regular Graphs

Mobility Edge for the Anderson Model on the Bethe Lattice

Maximum gap in complex Ginibre matrices

Smallest gaps between eigenvalues of real Gaussian matrices

Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

Quenched large deviation principles for random projections of ℓpn​ balls

Central limit theorem for the complex eigenvalues of Gaussian random matrices

Fluctuations in Quantum Unique Ergodicity at the Spectral Edge

Existence of the free energy for heavy-tailed spin glasses

Mobility Edge for Lévy Matrices

Fluctuations in local quantum unique ergodicity for generalized Wigner matrices

Optimal Delocalization for Generalized Wigner Matrices

Eigenvector Statistics of Lévy Matrices

Universality of the least singular value for the sum of random matrices

Tail bounds for gaps between eigenvalues of sparse random matrices

Comparison theorem for some extremal eigenvalue statistics

Overlap structure of mixed even $p$ -spin models

Quenched large deviation principles for random projections of $ℓ_{p}^{n}$ balls