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Patrick Lopatto

Publications and source records attributed to Patrick Lopatto.

At least 19 recordsLinked to original sources
arXiv2026

A positive resolution of the gap-entropy conjecture

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 suboptimal arm $i$, let $\Delta_i=\mu_*-\mu_i$ be its gap from the optimal mean, and write $H=\sum_{i\ne *}\Delta_i^{-2}$. Let $p_r$ be the fraction of $H$ contributed by arms with $2^{-(r+1)}<\Delta_i\le2^{-r}$, and let $\mathrm{Ent}(I)=\sum_{r:p_r>0} p_r\log(1/p_r)$. Among all algorithms that identify the optimal arm with probability at least $1-\delta$ on every Gaussian instance, the optimal expected number of samples on a given instance, averaged over all permutations of the arm labels, is within absolute constant factors of $H(\log(1/\delta)+\mathrm{Ent}(I))$. Moreover, there is an algorithm, independent of the instance, whose expected number of samples is bounded by a constant multiple of this quantity plus $g^{-2}\log\log(e^e/g)$, where $g=\min_{i\ne *}\Delta_i$ is the gap to the closest competitor.

cs.LG↗
arXiv2026

Quantitative Parisi formulas and fluctuations in the Sherrington-Kirkpatrick model

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 supported on an interval, we show that for every fixed inverse temperature greater than one, the variance of the logarithmic partition function lies between orders $N^{4/15}$ and $N^{7/15}$. The same exponents hold for the ground-state variance. We also obtain upper and lower bounds on the finite-size bias of the mean free energy and ground-state energy, together with two-sided upper-tail estimates with exponent $6/5$ at both positive and zero temperature.

math.PR↗
arXiv2026

Optimal central limit theorem for bounded random variables in high dimensions

Let $W=n^{-1/2}\sum_{i=1}^n X_i$, where the $X_i$ are independent centered random vectors in ${\mathbb R}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance, uniformly over axis-aligned rectangles, is at most $C\min\{1,b^{-2}Bn^{-1/2}\log^{3/2}(ep)\}$. For fixed $b$, the dependence on summand size and dimension matches known lower bounds in growing-dimensional regimes. The proof combines a concentration estimate near rectangle boundaries with a carefully chosen Gaussian comparison.

math.PR↗
arXiv2026

Replica symmetry breaking at arbitrary depth

We study finite-step replica symmetry breaking (RSB) in mixed even $p$-spin Ising spin glasses at positive temperature and zero external field. For every integer $k\geq1$, we construct finite polynomial mixtures whose Parisi measures are exactly $k$-RSB, with each such phase persisting on an open set of coefficients. More generally, for any admissible finite polynomial covariance $\Xi$ with a nondegenerate $r$-RSB Parisi measure, adding $\lambda q^p$ for sufficiently large even $p$ produces, as $\lambda$ varies near a critical value, a unique $r$-RSB-to-$(r+1)$-RSB transition. Since such covariances exist at every finite RSB depth, this yields transitions of arbitrary depth. The main idea of the proof is that for large $p$, the perturbation $\lambda q^p$ is negligible for overlaps bounded away from $1$, while its effect is concentrated at overlaps very close to $1$.

math.PR↗
arXiv2026

Adaptive Confidence Sets for Binary Regression without Design Smoothness

We study honest adaptive confidence sets for the regression function in random-design binary regression under $L^2(dx)$ loss. Assuming only known bounds $0<c\leq g\leq C<\infty$ on the unknown design density, we construct asymptotically honest, rate-adaptive confidence sets without requiring $g$ to be smooth. Full adaptation is possible when the range of regression-function smoothness spans at most a factor of two. Over wider smoothness ranges, adaptation is achieved on the usual separated classes at the corresponding testing rates $n^{-2s/(4s+d)}$. A lower bound under the uniform design shows that these separation rates are rate-optimal. This answers a question raised by Mukherjee and Sen (2018).

math.ST↗
arXiv2026

Nearly-minimax variance estimation under rough random design

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 smoothness assumption, and the conditional error laws may depend on the covariates and have uniformly bounded fourth moments. For an $s$-H\"older regression function with $s>1$ in dimension $d>4s$, the minimax root-mean-square risk lies between $cn^{-2(s+1)/(d+4)}e^{-C\sqrt{\log n}}$ and $Cn^{-2(s+1)/(d+4)}$. These bounds show that the rate proposed by Robins is not uniformly attainable over this model class. For $0 1$ and $d\le4s$, it is $n^{-1/2}$.

math.ST↗
arXiv2026

Full replica symmetry breaking in the Sherrington-Kirkpatrick model

We prove that, at zero external field and for every inverse temperature $\beta>1$, the Parisi measure of the Sherrington-Kirkpatrick model is supported on a single interval $[0,q_\beta]$, has a smooth density on $[0,q_\beta)$, and has a single atom at $q_\beta$. The proof first establishes that zero is an accumulation point of the support. By direct analysis of the Gaussian Cole-Hopf solutions associated with finitely supported measures, followed by an approximation argument, we show that a positive gap separating zero from the rest of the support contradicts the variational characterization of the Parisi measure established by Jagannath and Tobasco (2017). We then derive a monotonicity constraint on a quantity appearing in the self-consistency conditions arising from the Parisi variational criteria, which is incompatible with the existence of any gap in the support. Finally, log-concavity of a transformed probability density associated with the optimal diffusion implies that the maximal support point carries an atom, while the regularity result of Auffinger and Chen (2015) excludes the possibility of other singular components.

math.PR↗
arXiv2026

Minimax unbiased estimation for finite populations with bounded outcomes

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 inclusion probabilities $\pi_i>0$, we prove a sharp lower bound on the worst-case squared error over the rectangular parameter space. This bound is attained if and only if the unit inclusion indicators are pairwise independent, in which case the minimax estimator is the midpoint-differenced Horvitz-Thompson estimator $\sum_{i=1}^N m_i+\sum_{i\in S}(y_i-m_i)/\pi_i$, with $m_i=(a_i+b_i)/{2}$. We then solve the joint design-and-estimation problem under the constraint $\sum_i \pi_i\le n$. We find that a minimax strategy samples units independently with probabilities $\pi_i^\ast=\min(1,c (b_i-a_i))$ where $c>0$ is chosen so that $\sum_i \pi_i^\ast=n$, and uses the midpoint-differenced estimator. This extends Gabler (1990)'s linear minimax result to the full class of design-unbiased estimators. We also show that the estimator is admissible among unbiased estimators and affine equivariant.

math.ST↗
arXiv2026

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

We show that in the Sherrington-Kirkpatrick model at inverse temperature $\beta$ with uniform external field $h>0$, replica symmetry holds in the regime $ \beta^2\mathrm{E}[ \mathrm{sech}^4(\beta\sqrt{q}Z+h)] \le 1$, where $Z$ is a standard Gaussian random variable. This confirms a prediction of de Almeida and Thouless (1978). The proof proceeds by a direct analysis of the Parisi measure using the characterization provided by Jagannath and Tobasco (2017).

math.PR↗
arXiv2026

One-step TMLE for weighted average treatment effects

We consider Targeted Maximum Likelihood Estimation (TMLE) of weighted average treatment effects (WATEs), a class of causal estimands that reweight the covariate distribution using a specified function of the propensity score. This class includes the average treatment effect and average treatment effect on the treated, as well as various overlap-based targets. We provide a comprehensive analysis of the one-step TMLE along the universal least favorable path for such parameters. Under explicit regularity conditions on the weight function and initialization, we show that the targeting procedure is well-defined, reaches a solution of the estimating equation in finite time, and yields an asymptotically efficient estimator. In particular, convergence of the targeting dynamics and control of the second-order remainder are derived from these conditions rather than imposed as separate assumptions on the output of the algorithm.

math.ST↗
arXiv2026

Mobility Edge for the Anderson Model on Random Regular Graphs

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 the degree is fixed and the number of vertices goes to infinity, the spectrum asymptotically consists of a finite delocalized interval surrounded by two unbounded localized components. Our argument uses a recent description of the spectrum of the tight-binding model on the Bethe lattice (Aggarwal--Lopatto, 2025). By viewing the Bethe lattice as the local limit of a random regular graph, and establishing suitable concentration, eigenvalue-counting, and resolvent estimates, we transfer this characterization of the spectrum of the limiting model to the finite-volume setting.

math.PR↗
arXiv2025

Mobility Edge for the Anderson Model on the Bethe Lattice

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 there exist a finite number of mobility edges separating intervals of pure-point spectrum from intervals of absolutely continuous spectrum, confirming a prediction of Abou-Chacra, Thouless, and Anderson. A central component of our proof is a monotonicity result for the leading eigenvalue of a certain transfer operator, which governs the decay rate of fractional moments for the tight-binding model's off-diagonal resolvent entries.

math.PR↗
arXiv2024

Formulas for odd zeta values and powers of $\pi$

Plouffe conjectured rapidly converging series formulas for $\pi^{2n+1}$ and $\zeta (2n+1)$ for small values of $n$. We find the general pattern for all nonnegative integer values of $n$ and offer a proof.

math.NT↗
arXiv2024

Randomization-based confidence sets for the local average treatment effect

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 and Rosenbaum (2005) has desirable properties. Namely, we show that using a studentized Anderson--Rubin-type statistic as a test statistic yields confidence sets that are finite-sample exact under treatment effect homogeneity, and remain asymptotically valid for the Local Average Treatment Effect when the treatment effect is heterogeneous. We provide a uniform analysis of this procedure and efficient algorithms to construct the confidence set.

math.ST↗
arXiv2023

Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

We determine to leading order the maximum of the characteristic polynomial for Wigner matrices and $\beta$-ensembles. In the special case of Gaussian-divisible Wigner matrices, our method provides universality of the maximum up to tightness. These are the first universal results on the Fyodorov--Hiary--Keating conjectures for these models, and in particular answer the question of optimal rigidity for the spectrum of Wigner matrices. Our proofs combine dynamical techniques for universality of eigenvalue statistics with ideas surrounding the maxima of log-correlated fields and Gaussian multiplicative chaos.

math.PR↗