Dyson Brownian Motion Research Articles

Circular Rosenzweig-Porter random matrix ensemble

The Rosenzweig-Porter random matrix ensemble serves as a qualitative phenomenological model for the level statistics and fractality of eigenstates across the many-body localization transition in static systems. We propose a unitary (circular) analogue of this ensemble, which similarly captures the phenomenology of many-body localization in periodically driven (Floquet) systems. We define this ensemble as the outcome of a Dyson Brownian motion process. We show numerical evidence that this ensemble shares some key statistical properties with the Rosenzweig-Porter ensemble for both the eigenvalues and the eigenstates.

Kinetic Dyson Brownian motion

We study the spectrum of the kinetic Brownian motion in the space of d×d Hermitian matrices, d≥2. We show that the eigenvalues stay distinct for all times, and that the process Λ of eigenvalues is a kinetic diffusion (i.e. the pair (Λ,Λ˙) of Λ and its derivative is Markovian) if and only if d=2. In the large scale and large time limit, we show that Λ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.

Stability of large complex systems with heterogeneous relaxation dynamics

We study the probability of stability of a large complex system of size N within the framework of a generalized May model, which assumes a linear dynamics of each population size n i (with respect to its equilibrium value): . The a i > 0’s are the intrinsic decay rates, J ij is a real symmetric (N × N) Gaussian random matrix and measures the strength of pairwise interaction between different species. Our goal is to study how inhomogeneities in the intrinsic damping rates a i affect the stability of this dynamical system. As the interaction strength T increases, the system undergoes a phase transition from a stable phase to an unstable phase at a critical value T = T c. We reinterpret the probability of stability in terms of the hitting time of the level b = 0 of an associated Dyson Brownian motion (DBM), starting at the initial position a i and evolving in ‘time’ T. In the large N → ∞ limit, using this DBM picture, we are able to completely characterize T c for arbitrary density μ(a) of the a i ’s. For a specific flat configuration , we obtain an explicit parametric solution for the limiting (as N → ∞) spectral density for arbitrary T and σ. For finite but large N, we also compute the large deviation properties of the probability of stability on the stable side T < T c using a Coulomb gas representation.

Central Limit Theorem for Linear Eigenvalue Statistics of Non‐Hermitian Random Matrices

AbstractWe consider large non‐Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Extreme gaps between eigenvalues of Wigner matrices

This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the entries. The proof relies on the Erdős–Schlein–Yau dynamic approach. We exhibit a new observable that satisfies a stochastic advection equation and reduces local relaxation of the Dyson Brownian motion to a maximum principle. This observable also provides a simple and unified proof of gap universality in the bulk and the edge, which is quantitative. To illustrate this, we give the first explicit rate of convergence to the Tracy–Widom distribution for generalized Wigner matrices.

Bulk properties of the Airy line ensemble

The Airy line ensemble is a central object in random matrix theory and last passage percolation defined by a determinantal formula. The goal of this paper is to provide a set of tools which allow for precise probabilistic analysis of the Airy line ensemble. The two main theorems are a representation in terms of independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines. Along the way, we give tail bounds and moduli of continuity for nonintersecting Brownian ensembles, and a quick proof of tightness for Dyson's Brownian motion converging to the Airy line ensemble.

N-sided radial Schramm–Loewner evolution

We use the interpretation of the Schramm–Loewner evolution as a limit of path measures tilted by a loop term in order to motivate the definition of n-radial SLE going to a particular point. In order to justify the definition we prove that the measure obtained by an appropriately normalized loop term on n-tuples of paths has a limit. The limit measure can be described as n paths moving by the Loewner equation with a driving term of Dyson Brownian motion. While the limit process has been considered before, this paper shows why it naturally arises as a limit of configurational measures obtained from loop measures.

Potential kernels for radial Dunkl Laplacians

AbstractWe derive two-sided bounds for the Newton and Poisson kernels of the W-invariant Dunkl Laplacian in the geometric complex case when the multiplicity $k(\alpha )=1$ i.e., for flat complex symmetric spaces. For the invariant Dunkl–Poisson kernel $P^{W}(x,y)$ , the estimates are $$ \begin{align*} P^{W}(x,y)\asymp \frac{P^{\mathbf{R}^{d}}(x,y)}{\prod_{\alpha&gt; 0 \ }|x-\sigma_{\alpha} y|^{2k(\alpha)}}, \end{align*} $$ where the $\alpha $ ’s are the positive roots of a root system acting in $\mathbf {R}^{d}$ , the $\sigma _{\alpha }$ ’s are the corresponding symmetries and $P^{\mathbf {R}^{d}}$ is the classical Poisson kernel in ${\mathbf {R}^{d}}$ . Analogous bounds are proven for the Newton kernel when $d\ge 3$ .The same estimates are derived in the rank one direct product case $\mathbb {Z}_{2}^{N}$ and conjectured for general W-invariant Dunkl processes.As an application, we get a two-sided bound for the Poisson and Newton kernels of the classical Dyson Brownian motion and of the Brownian motions in any Weyl chamber.

How Dynamical Quantum Memories Forget

Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the ``mixed'' phase, a maximally mixed initial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems—i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears—purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.

Fermionic eigenvector moment flow

We exhibit new functions of the eigenvectors of the Dyson Brownian motion which follow an equation similar to the Bourgade-Yau eigenvector moment flow. These observables can be seen as a Fermionic counterpart to the original (Bosonic) ones. By analyzing both Fermionic and Bosonic observables, we obtain new correlations between eigenvectors. The fluctuations $\sum_{\alpha\in I}u_k(\alpha)^2-{\vert I\vert}/{N}$ decorrelate for distinct eigenvectors as the dimension $N$ grows and an optimal estimate on the partial inner product $\sum_{\alpha\in I}u_k(\alpha)u_\ell(\alpha)$ between two eigenvectors is given. These static results obtained by integrable dynamics are stated for generalized Wigner matrices and should apply to wide classes of mean field models.

Fluctuation around the circular law for random matrices with real entries

We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.

Applications of mesoscopic CLTs in random matrix theory

We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of “homogenization” for Dyson Brownian motion, this yields the universality of quantities which depend on the behavior of single eigenvalues of Wigner matrices and $\beta$-ensembles. Among the results we obtain are the Gaussian fluctuations of single eigenvalues for Wigner matrices (without an assumption of 4 matching moments) and classical $\beta$-ensembles ($\beta=1,2,4$), Gaussian fluctuations of the eigenvalue counting function, and an asymptotic expansion up to order $o(N^{-1})$ for the expected value of eigenvalues in the bulk of the spectrum. The latter result solves a conjecture of Tao and Vu.

Universality of local spectral statistics of products of random matrices

We derive exact analytical expressions for correlation functions of singular values of the product of M Ginibre matrices of size N in the double scaling limit M,N→∞. The singular value statistics is described by a determinantal point process with a kernel that interpolates between Gaussian unitary ensemble statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit N→∞, the interpolation parameter is given by the limiting quotient a=N/M. One of our goals is to find an explicit form of the kernel at the hard edge, in the bulk, and at the soft edge for any a. We find that in addition to the standard scaling regimes, there is a transitional regime which interpolates between the hard edge and the bulk. We conjecture that these results are universal, and that they apply to a broad class of products of random matrices from the Gaussian basin of attraction, including correlated matrices. We corroborate this conjecture by numerical simulations. Additionally, we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of Dyson Brownian motion with the initial condition given by equidistant positions, with the crucial difference that this equivalence holds only locally. Finally, we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum.

Phase Transitions for Products of Characteristic Polynomials under Dyson Brownian Motion

We study the averaged products of characteristic polynomials for the Gaussian and Laguerre $\beta$-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are characterised by determining the explicit functional forms of the scaled limits of the averaged products of characteristic polynomials, which are given as certain multidimensional integrals, with dimension equal to the number of products.

Eigenvalue processes of Elliptic Ginibre Ensemble and their overlaps

We consider the non-hermitian matrix-valued process of Elliptic Ginibre Ensemble. This model includes Dyson's Brownian motion model and the time evolution model of Ginibre ensemble by using hermiticity parameter. We show the complex eigenvalue processes satisfy the stochastic differential equations which are very similar to Dyson's model and give an explicit form of overlap correlations. As a corollary, in the case of 2-by-2 matrix, we also mention the relation between the diagonal overlap, which is the speed of eigenvalues, and the distance of the two eigenvalues.

On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion

We study the generalized Dyson Brownian motion (GDBM) of an interacting N-particle system with logarithmic Coulomb interaction and general potential V. Under reasonable condition on V, we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures of GDBM is tight on $$\mathcal {C}([0,T],\mathscr {P}(\mathbb {R}))$$ and all the large N limits satisfy a nonlinear McKean–Vlasov equation. Inspired by previous works due to Biane and Speicher, Carrillo, McCann and Villani, and Blower, we prove that the McKean–Vlasov equation is indeed the gradient flow of the Voiculescu free entropy on the Wasserstein space of probability measures over $$\mathbb {R}$$ . Using the optimal transportation theory, we prove that if $$V''\ge K$$ for some constant $$K\in \mathbb {R}$$ , the McKean–Vlasov equation has a unique weak solution in the space of probability measures $$\mathscr {P}(\mathbb {R})$$ . This establishes the Law of Large Numbers and the propagation of chaos for the empirical measures of GDBM with non-quadratic external potentials which are not necessarily convex. Finally, we prove the longtime convergence of the McKean–Vlasov equation for $$C^2$$ -convex potentials V.

Dyson Brownian motion for general $$\beta $$ and potential at the edge

In this paper, we compare the solutions of the Dyson Brownian motion for general $$\beta $$ and potential V and the associated McKean–Vlasov equation near the edge. Under suitable conditions on the initial data and the potential V, we obtain optimal rigidity estimates of particle locations near the edge after a short time $$t={{\,\mathrm{o}\,}}(1)$$ . Our argument uses the method of characteristics along with a careful estimate involving an equation of the edge. With the rigidity estimates as an input, we prove a central limit theorem for mesoscopic statistics near the edge, which, as far as we know, has been done for the first time in this paper. Additionally, combining our results with Landon and Yau (Edge statistics of Dyson Brownian motion. arXiv:1712.03881 , 2017), we give a proof of the local ergodicity of the Dyson Brownian motion for general $$\beta $$ and potential at the edge, i.e., we show the distribution of extreme particles converges to the Tracy–Widom $$\beta $$ distribution in a short time.

Large time behavior, bi-Hamiltonian structure, and kinetic formulation for a complex Burgers equation

We prove the existence and uniqueness of positive analytical solutions with positive initial data to the mean field equation (the Dyson equation) of the Dyson Brownian motion through the complex Burgers equation with a force term on the upper half complex plane. These solutions converge to a steady state given by Wigner’s semicircle law. A unique global weak solution with nonnegative initial data to the Dyson equation is obtained, and some explicit solutions are given by Wigner’s semicircle laws. We also construct a bi-Hamiltonian structure for the system of real and imaginary components of the complex Burgers equation (coupled Burgers system). We establish a kinetic formulation for the coupled Burgers system and prove the existence and uniqueness of entropy solutions. The coupled Burgers system in Lagrangian variable naturally leads to two interacting particle systems, the Fermi–Pasta–Ulam–Tsingou model with nearest-neighbor interactions, and the Calogero–Moser model. These two particle systems yield the same Lagrangian dynamics in the continuum limit.

Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions

This paper proves an equality in law between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times in a discrete random environment. A consequence describes the distribution of the all-time supremum of Dyson Brownian motion with drift. A finite temperature version relates the point-to-line partition functions of two directed polymers, with an inverse-gamma and a Brownian environment, and generalises Dufresne’s identity. Our proof introduces an interacting system of Brownian motions with an invariant measure given by a field of point-to-line log partition functions for the log-gamma polymer.

Semi-implicit Euler–Maruyama approximation for noncolliding particle systems

We introduce a semi-implicit Euler–Maruyama approximation which preserves the noncolliding property for some class of noncolliding particle systems such as Dyson–Brownian motions, Dyson–Ornstein–Uhlenbeck processes and Brownian particle systems with nearest neighbor repulsion, and study its rates of convergence in both $L^{p}$-norm and pathwise sense.