Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation - Cornalba, Federico; Fischer, Julian L; Ingmanns, Jonas; Raithel, Claudia (The Annals of Probability)
The Dean–Kawasaki equation—one of the most fundamental SPDEs of fluctuating hydrodynamics—has been proposed as a model for density fluctuations in weakly interacting particle systems. In its original form, it is highly singular and fails to be renormalizable, even by approaches such as regularity structures and paracontrolled distributions, hindering mathematical approaches to its rigorous justification. It has been understood recently that it is natural to introduce a suitable regularization, for example, by applying a formal spatial discretization or by truncating high-frequency noise: This yields well-posed equations that should still precisely approximate the law of the particle density fluctuations. In the present work, we prove that a regularization in the form of a formal discretization of the Dean–Kawasaki equation indeed accurately describes density fluctuations in systems of weakly interacting diffusing particles: We show that, in suitable weak metrics, the law of fluctuations as predicted by the discretized Dean–Kawasaki SPDE approximates the law of fluctuations of the original particle system, up to an error that is of arbitrarily high order in the inverse particle number and a discretization error. In particular, the Dean– Kawasaki equation provides a means for efficient and accurate simulations of density fluctuations in weakly interacting particle systems.
Keywords: Weakly interacting particle systems; fluctuating hydrodynamics; Dean-Kawasaki equation; stochastic PDEs; numerical approximation

On the spectral edge of non-Hermitian random matrices - Campbell, Andrew J; Cipolloni, Giorgio; Erdös, László; Ji, Hong Chang (The Annals of Probability)
For general non-Hermitian large random matrices X and deterministic deformation matrices A, we prove that the local eigenvalue statistics of A+X close to the typical edge points of its spectrum are universal. Furthermore, we show that, under natural assumptions, on A the spectrum of A+X does not have outliers at a distance larger than the natural fluctuation scale of the eigenvalues. As a consequence, the number of eigenvalues in each component of Spec(A+X) is deterministic.

Tracy-Widom limit for free sum of random matrices - Ji, Hong Chang; Park, Jaewhi (The Annals of Probability)
We consider fluctuations of the largest eigenvalues of the random matrix model A + UBU∗ where A and B are N × N deterministic Hermitian (or symmetric) matrices and U is a Haar-distributed unitary (or orthogonal) matrix. We prove that the largest eigenvalue weakly converges to the GUE (or GOE) Tracy–Widom distribution, under mild assumptions on A and B to guarantee that the density of states of the model decays as square root around the upper edge. Our proof is based on the comparison of the Green function along the Dyson Brownian motion starting from the matrix A + UBU∗ and ending at time N−1/3+o(1). As a byproduct of our proof, we also prove an optimal local law for the Dyson Brownian motion up to the constant time scale.

On the rightmost eigenvalue of non-Hermitian random matrices - Cipolloni, Giorgio; Erdös, László; Schröder, Dominik J; Xu, Yuanyuan (The Annals of Probability)
We establish a precise three-term asymptotic expansion, with an optimal estimate of the error term, for the rightmost eigenvalue of an n×n random matrix with independent identically distributed complex entries as n tends to infinity. All terms in the expansion are universal.
Keywords: Statistics; Probability and Uncertainty; Statistics and Probability

The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold - Dello Schiavo, Lorenzo (Annals of Probability)
We construct a recurrent diffusion process with values in the space of probability measures over an arbitrary closed Riemannian manifold of dimension d≥2. The process is associated with the Dirichlet form defined by integration of the Wasserstein gradient w.r.t. the Dirichlet–Ferguson measure, and is the counterpart on multidimensional base spaces to the modified massive Arratia flow over the unit interval described in V. Konarovskyi and M.-K. von Renesse (Comm. Pure Appl. Math. 72 (2019) 764–800). Together with two different constructions of the process, we discuss its ergodicity, invariant sets, finite-dimensional approximations, and Varadhan short-time asymptotics.

Normal fluctuation in quantum ergodicity for Wigner matrices - Cipolloni, Giorgio; Erdös, László; Schröder, Dominik J (Annals of Probability)
We consider the quadratic form of a general high-rank deterministic matrix on the eigenvectors of an N×N Wigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large N limit. The proof is a combination of the energy method for the Dyson Brownian motion inspired by Marcinek and Yau (2021) and our recent multiresolvent local laws (Comm. Math. Phys. 388 (2021) 1005–1048).

On words of non-Hermitian random matrices - Dubach, Guillaume; Peled, Yuval (The Annals of Probability)
We consider words Gi1⋯Gim involving i.i.d. complex Ginibre matrices and study tracial expressions of their eigenvalues and singular values. We show that the limit distribution of the squared singular values of every word of length m is a Fuss–Catalan distribution with parameter m+1. This generalizes previous results concerning powers of a complex Ginibre matrix and products of independent Ginibre matrices. In addition, we find other combinatorial parameters of the word that determine the second-order limits of the spectral statistics. For instance, the so-called coperiod of a word characterizes the fluctuations of the eigenvalues. We extend these results to words of general non-Hermitian matrices with i.i.d. entries under moment-matching assumptions, band matrices, and sparse matrices. These results rely on the moments method and genus expansion, relating Gaussian matrix integrals to the counting of compact orientable surfaces of a given genus. This allows us to derive a central limit theorem for the trace of any word of complex Ginibre matrices and their conjugate transposes, where all parameters are defined topologically.
Keywords: Statistics; Probability and Uncertainty; Statistics and Probability

Correlated random matrices: Band rigidity and edge universality - Alt, Johannes; Erdös, László; Krüger, Torben H; Schröder, Dominik J (Annals of Probability)
We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of the self-consistent density of states. In particular, we establish a strong form of band rigidity which excludes mismatches between location and label of eigenvalues close to internal edges in these general models.

Boundary regularity of stochastic PDEs - Gerencser, Mate (Annals of Probability)
The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly—and in a sense, arbitrarily—bad: as shown by Krylov[ SIAM J. Math. Anal.34(2003) 1167–1182], for any α>0 one can find a simple 1-dimensional constant coefficient linear equation whose solution at the boundary is not α-Hölder continuous.We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on C1 domains are proved to be α-Hölder continuous up to the boundary with some α>0.

Local single ring theorem on optimal scale - Bao, Zhigang; Erdös, László; Schnelli, Kevin (Annals of Probability)
Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix X:=UΣV∗ converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in ℂ. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N−1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).

Sample path properties of the stochastic flows - Koralov, Leonid; Kaloshin, Vadim; Dolgopyat, Dmitry (The Annals of Probability)