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

collaborators

39 papers

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…

cs.LG2026

A positive resolution of the gap-entropy conjecture

P. M. Aronow, Nathan Kallus, Patrick Lopatto

We prove the gap-entropy conjecture for fixed-confidence best-arm identification with independent unit-variance Gaussian arms, means in $[0, 1]$ , and a unique optimal arm. For each…

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.ST2026

Adaptive Confidence Sets for Binary Regression without Design Smoothness

P. M. Aronow, Patrick Lopatto

We study honest adaptive confidence sets for the regression function in random-design binary regression under $L^{2} (d x)$ loss. Assuming only known bounds $0 < c \leq g \leq C < \infty$ on…

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.ST2026

Nearly-minimax variance estimation under rough random design

P. M. Aronow, Patrick Lopatto

We identify the minimax exponent for constant conditional variance estimation under rough random design. The unknown design density is bounded above and away from zero, with no smo…

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.ST2026

Minimax unbiased estimation for finite populations with bounded outcomes

P. M. Aronow, Patrick Lopatto

We study design-unbiased estimation of the finite-population total $\sum_{i = 1}^{N} y_{i}$ when each outcome satisfies known bounds $y_{i} \in [a_{i}, b_{i}]$ . For any sampling design with inclu…

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.ST2026

One-step TMLE for weighted average treatment effects

Yang Liu, Patrick Lopatto, Ivana Malenica

We consider Targeted Maximum Likelihood Estimation (TMLE) of weighted average treatment effects (WATEs), a class of causal estimands that reweight the covariate distribution using…

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.NT2024

Formulas for odd zeta values and powers of $π$

Marc Chamberland, Patrick Lopatto

Plouffe conjectured rapidly converging series formulas for $π^{2 n + 1}$ and $ζ (2 n + 1)$ for small values of $n$ . We find the general pattern for all nonnegative integer values of $n$ a…

math.ST2024★ 1 cited

Randomization-based confidence sets for the local average treatment effect

P. M. Aronow, Haoge Chang, Patrick Lopatto

We consider the problem of generating confidence sets in randomized experiments with noncompliance. We show that a refinement of a randomization-based procedure proposed by Imbens…

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.

stat.ME2023

Fast computation of exact confidence intervals for randomized experiments with binary outcomes

P. M. Aronow, Haoge Chang, Patrick Lopatto

Given a randomized experiment with binary outcomes, exact confidence intervals for the average causal effect of the treatment can be computed through a series of permutation tests.…

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…

Overlap structure of mixed even p-spin models

Parisi measure of the Sherrington-Kirkpatrick model with external field

A positive resolution of the gap-entropy conjecture

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

Adaptive Confidence Sets for Binary Regression without Design Smoothness

Full replica symmetry breaking in the Sherrington-Kirkpatrick model

Nearly-minimax variance estimation under rough random design

Sobolev convergence of log-determinants for smooth Wigner matrices

Minimax unbiased estimation for finite populations with bounded outcomes

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

One-step TMLE for weighted average treatment effects

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

Formulas for odd zeta values and powers of π

Randomization-based confidence sets for the local average treatment effect

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

Fast computation of exact confidence intervals for randomized experiments with binary outcomes

Fluctuations in Quantum Unique Ergodicity at the Spectral Edge

Overlap structure of mixed even $p$ -spin models

Formulas for odd zeta values and powers of $π$

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