About

Paul Bourgade is a Professor of Mathematics at NYU with research interests that include probability, random matrices, statistical physics, and stochastic processes. His work focuses on understanding complex systems through probabilistic models and mathematical physics, contributing to the theoretical foundations of these areas. As a faculty member at the Courant Institute, he engages in advanced research that intersects various domains of mathematics and physics, emphasizing the development of rigorous mathematical frameworks for phenomena involving randomness and statistical behavior.

Research topics

• Combinatorics
• Mathematics
• Physics
• Mathematical analysis
• Quantum mechanics
• Computer Science
• Pure mathematics
• Statistical physics
• Mathematical physics
• Statistics

Selected publications

• Optimal Rigidity and Maximum of the Characteristic Polynomial of Wigner Matrices

  Geometric and Functional Analysis · 2025-02-01 · 1 citations

  articleOpen access1st authorCorresponding

  Abstract 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 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.

  PublisherOA PDFDOI

• Fisher-Hartwig asymptotics for non-Hermitian random matrices

  ArXiv.org · 2025-12-09

  preprintOpen access1st authorCorresponding

  We prove the two-dimensional analogue of the asymptotics for Toeplitz determinants with Fisher-Hartwig singularities, for general real symbols. This formula has applications to random normal matrices with complex spectra: (i) the characteristic polynomial converges to a Gaussian multiplicative chaos random measure on the limiting droplet, in the subcritical phase; (ii) the electric potential converges pointwise to a logarithmically correlated field; (iii) the measure of its level sets (i.e. thick points) is identified; (iv) the associated free energy undergoes a freezing transition. This establishes emergence of the Liouville quantum gravity measure from free fermions in 2d, and universality with respect to the external potential.

  PublisherOA PDFDOI

• Fluctuations for non-Hermitian dynamics

  arXiv (Cornell University) · 2024-09-04

  preprintOpen access1st authorCorresponding

  We prove that under the Brownian evolution on large non-Hermitian matrices the log-determinant converges in distribution to a 2+1 dimensional Gaussian field in the Edwards-Wilkinson regularity class, namely it is logarithmically correlated for the parabolic distance. This dynamically extends a seminal result by Rider and Virág about convergence to the Gaussian free field. The convergence holds out of equilibrium for centered, i.i.d. matrix entries as an initial condition. A remarkable aspect of the limiting field is its non-Markovianity, due to long range correlations of the eigenvector overlaps, for which we identify the exact space-time polynomial decay. In the proof, we obtain a quantitative, optimal relaxation at the hard edge, for a broad extension of the Dyson Brownian motion, with a driving noise arbitrarily correlated in space.

  PublisherOA PDFDOI

• Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

  arXiv (Cornell University) · 2023-12-20 · 1 citations

  preprintOpen access1st authorCorresponding

  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 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.

  PublisherOA PDFDOI

• The Fyodorov-Hiary-Keating Conjecture. II

  arXiv (Cornell University) · 2023-07-03 · 3 citations

  preprintOpen access

  We prove a lower bound on the maximum of the Riemann zeta function in a typical short interval on the critical line. Together with the upper bound from the previous work of the authors, this implies tightness of $$ \max_{|h|\leq 1}|ζ(\tfrac 12+{\rm i} τ+{\rm i} h)|\cdot \frac{(\log\log T)^{3/4}}{\log T}, $$ for large $T$, where $τ$ is uniformly distributed on $[T,2T]$. The techniques are also applied to bound the right tail of the maximum, proving the distributional decay $\asymp y e^{-2y}$ for $y$ positive. This confirms the Fyodorov-Hiary-Keating conjecture, which states that the maximum of $ζ$ in short intervals lies in the universality class of logarithmically correlated fields.

  PublisherOA PDFDOI

• Landscape complexity beyond invariance and the elastic manifold

  Communications on Pure and Applied Mathematics · 2023-09-14 · 16 citations

  article

  Abstract This paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self‐interactions in a random medium. We establish the simple versus glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as the Larkin mass, confirming formulas of Fyodorov and Le Doussal. One essential, dynamical, step of the proof also applies to a general signal‐to‐noise model of soft spins in an anisotropic well, for which we prove a negative‐second‐moment threshold distinguishing positive from zero complexity. A universal near‐critical behavior appears within this phase portrait, namely quadratic near‐critical vanishing of the complexity of total critical points, and cubic near‐critical vanishing of the complexity of local minima. These two models serve as a paradigm of complexity calculations for Gaussian landscapes exhibiting few distributional symmetries, that is, beyond the invariant setting. The two main inputs for the proof are determinant asymptotics for non‐invariant random matrices from our companion paper (Ben Arous, Bourgade, McKenna 2022), and the atypical convexity and integrability of the limiting variational problems.

  PublisherDOI

• Exponential growth of random determinants beyond invariance

  Probability and Mathematical Physics · 2022-12-31 · 3 citations

  preprintOpen access

  We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with covariance profiles, Wigner matrices and covariance matrices with subexponential tails, Erdős-Rényi and $d$-regular graphs for any polynomial sparsity parameter, and non-mean-field random matrix models, such as random band matrices for any polynomial bandwidth. The proof builds on recent tools, including the theory of the Matrix Dyson Equation as developed in [Ajanki, Erdős, Krüger 2019]. We use these asymptotics as an important input to identify the complexity of classes of Gaussian random landscapes in our companion papers [Ben Arous, Bourgade, McKenna 2021; McKenna 2021].

  PublisherDOI

• Optimal Local Law and Central Limit Theorem for $$\beta $$-Ensembles

  Communications in Mathematical Physics · 2022 · 35 citations

  1st authorCorresponding
  • Mathematics
  • Statistical physics
  • Mathematical physics
  PublisherOA PDFDOI

• Liouville quantum gravity from random matrix dynamics

  arXiv (Cornell University) · 2022-06-07 · 3 citations

  preprintOpen access1st authorCorresponding

  We establish the first connection between $2d$ Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if $(U_t)$ is a Brownian motion on the unitary group at equilibrium, then the measures $$ |\det(U_t - e^{i θ})|^γ dt dθ$$ converge in the limit of large dimension to the $2d$ LQG measure, a properly normalized exponential of the $2d$ Gaussian free field. Gaussian free field type fluctuations associated with these dynamics were first established by Spohn (1998) and convergence to the LQG measure in $2d$ settings was conjectured since the work of Webb (2014), who proved the convergence of related one dimensional measures by using inputs from Riemann-Hilbert theory. The convergence follows from the first multi-time extension of the result by Widom (1973) on Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols. To prove these, we develop a general surgery argument and combine determinantal point processes estimates with stochastic analysis on Lie group, providing in passing a probabilistic proof of Webb's $1d$ result. We believe the techniques will be more broadly applicable to matrix dynamics out of equilibrium, joint moments of determinants for classes of correlated random matrices, and the characteristic polynomial of non-Hermitian random matrices.

  PublisherOA PDFDOI

• Landscape complexity beyond invariance and the elastic manifold

  arXiv (Cornell University) · 2021-05-11 · 4 citations

  preprintOpen access

  This paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self-interactions in a random medium. We establish the simple-vs.-glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as the Larkin mass, confirming formulas of Fyodorov and Le Doussal. One essential, dynamical, step of the proof also applies to a general signal-to-noise model of soft spins in an anisotropic well, for which we prove a negative-second-moment threshold distinguishing positive from zero complexity. A universal near-critical behavior appears within this phase portrait, namely quadratic near-critical vanishing of the complexity of total critical points, and cubic near-critical vanishing of the complexity of local minima. These two models serve as a paradigm of complexity calculations for Gaussian landscapes exhibiting few distributional symmetries, i.e. beyond the invariant setting. The two main inputs for the proof are determinant asymptotics for non-invariant random matrices from our companion paper [Ben Arous, Bourgade, McKenna 2021], and the atypical convexity and integrability of the limiting variational problems.

  PublisherOA PDFDOI

Recent grants

• Universality of Random Matrices Statistics

  NSF · $148k · 2012–2014

• Analytic Methods for the Random Matrix Universality Class

  NSF · $225k · 2015–2018

• Spectral Properties of Random Matrices

  NSF · $300k · 2018–2022

Frequent coauthors

• Gérard Ben Arous

  New York University

  24 shared

• Horng‐Tzer Yau

  Harvard University

  19 shared

• Ashkan Nikeghbali

  18 shared

• Alain Rouault

  Laboratoire de Mathématiques

  14 shared

• Jun Yin

  12 shared

• Louis‐Pierre Arguin

  9 shared

• Marc Yor

  9 shared

• László Erdős

  8 shared

Education

• B.S., Mathematics and Physics

  École Polytechnique

  2006

• M.S., Computer Science

  Telecom Paris

  2007

• M.S., Probability

  Université Paris 6

  2007

• Ph.D., Mathematics

  Université Paris 6

  2009