arXiv subjects

Dang-Zheng Liu

Publications and source records attributed to Dang-Zheng Liu.

At least 19 recordsLinked to original sources

arXiv2026

Subcritical limits of a 1D KPZ growth surface with finite moments

We consider the one-dimensional random growth surface introduced by Adhikari and Chatterjee [AC24], where the height function is defined recursively via the heights at two neighboring sites and an independent noise term at each space-time point. In the subcritical regime, assuming uniformly bounded eighth moments of the noise variables and appropriate regularity of the driving function, we establish convergence in law of the suitably rescaled height function to the solution of the additive stochastic heat equation. Our approach employs a direct recursive renormalization scheme. A central component is a graph representation of the renormalization terms, which enables the derivation of the high-moment bounds essential to the renormalization procedure.

math.PR↗

arXiv2026

Edge Universality for Inhomogeneous Random Matrices II: Markov Chain Comparison and Critical Statistics

The first paper in this series introduced a \emph{short-to-long mixing} condition that captures mean-field GOE/GUE edge universality in the supercritical sparsity regime, for symmetric/Hermitian random matrices with independent entries and a Markov variance profile. This condition reduces the universality problem to the mixing properties of the underlying Markov chains. In this paper, we develop new \emph{short-to-long comparison} conditions that extend the analysis to the subcritical and critical sparsity regimes. Specifically, we prove that two inhomogeneous random matrices exhibit the same universal edge statistics whenever their variance-profile Markov chains are comparable, regardless of the fine details of the matrix entries. To illustrate the power of our Markov chain comparison theorem, we derive the spectral edge statistics for several prototypical models: random band matrices, the Wegner orbital model, and Hankel-profile random matrices. These comparisons uncover a rich landscape of both universal and non-universal phenomena -- shaped by geometric structure, spike patterns, and domains of stable attraction -- features that lie fundamentally beyond the reach of classical random matrix theory.

math.PR↗

arXiv2026

Outliers for deformed inhomogeneous random matrices

Inhomogeneous random matrices with non-trivial variance profiles determined by symmetric stochastic matrices and with independent sub-Gaussian entries up to Hermitian symmetry, encompass a wide range of important models, including sparse Wigner matrices and random band matrices. In these models, the maximum entry variance-a natural proxy for sparsity-serves both as a key structural feature and a primary analytical obstacle. In this paper, we consider low-rank additive perturbations of such matrices and establish a sharp BBP phase transition for extreme eigenvalues at the level of the law of large numbers. Furthermore, in the Gaussian setting, we derive the fluctuations of spectral outliers under suitable conditions on the variance profile and perturbation. These fluctuations exhibit strong non-universality, depending on the eigenvectors, sparsity levels, and the underlying geometric structure. Our proof strategies rely on ribbon graph expansions, upper bounds for diagram functions, large-moment estimates, and the enumeration of typical diagrams.

math.PR↗

arXiv2025

Reinforced Loop Soup via Wilson's Algorithm

The goal of this note is twofold: first, we explain the relation between the isomorphism theorems in the context of vertex reinforced jump process discovered in [BHS19, BHS21] and the standard Markovian isomorphism theorems for Markovian jump processes; second, we introduce the vertex reinforced counterpart of the standard Poissonian loop soup developed by Le Jan [LJ10]. To this end, we propose an algorithm that can be viewed as a variant of Wilson's algorithm with reinforcement. We establish the isomorphism theorems for the erased loops and the random walk from this algorithm, and in particular provide a concrete construction of the reinforced loop soup via a random process with a reinforcement mechanism.

math.PR↗

arXiv2025

Edge Universality for Inhomogeneous Random Matrices

We consider symmetric and Hermitian random matrices whose entries are independent and symmetric random variables with an arbitrary variance pattern. Under a novel Short-to-Long Mixing condition, which is sharp in the sense that it precludes a corrected shift at the spectral edge, we establish GOE/GUE edge universality for such inhomogeneous random matrices. This condition effectively reduces the universality problem to verifying the mixing properties of a random walk governed by the variance profile matrix. Our universality results are applicable to a remarkably broad class of random matrix ensembles that may be highly inhomogeneous, sparse, or far beyond the mean-field setting of classical random matrix theory. Notable examples include: 1. Inhomogeneous Wishart-type random matrices; 2. Random band matrices whose entries are independent random variables with general variance profile, particularly with an optimal bandwidth in dimensions $d \le 2$; 3. Sparse random matrices with structured variance profiles; 4. Generalized Wigner matrices under significantly weaker sparsity constraints and heavy-tailed entry distributions; 5. Wegner orbital models under sharp mixing assumptions; 6. Random 2-lifts of random $d$-regular graphs where $d\geq N^{2/3+ε}$ for any $ε>0$.

math.PR↗

arXiv2025

Local Statistics of Singular Values for Products of Truncated Unitary Matrices

This paper investigates local spectral statistics of singular values for many products of independent large rectangular matrices, sampled from the ensemble of truncated unitary matrices with the invariant Haar measure. Our main contribution establishes a universal three-phase transition in these statistics, demonstrating an interpolation beween GUE statistics and classical Gaussian behavior. While such transition was previously known for products of complex Gaussian matrices\cite{ABK19}\cite{LWW23}, the current work provides the complete characterization in the truncated unitary matrix setting.

math.PR↗

arXiv2025

Edge statistics for random band matrices

We consider Hermitian and symmetric random band matrices on the $d$-dimensional lattice $(\mathbb{Z}/L\mathbb{Z})^d$ with bandwidth $W$, focusing on local eigenvalue statistics at the spectral edge in the limit $W\to\infty$. Our analysis reveals a critical dimension $d_c=6$ and identifies the critical bandwidth scaling as $W_c=L^{(1-d/6)_+}$. In the Hermitian case, we establish the Anderson transition for all dimensions $d<4$, and GUE edge universality when $d\geq 4$ under the condition $W\geq L^{1/3+ε}$ for any $ε>0$. In the symmetric case, we also establish parallel but more subtle transition phenomena after tadpole diagram renormalization. These findings extend Sodin's pioneering work [Ann. Math. 172, 2010], which was limited to the one-dimensional case and did not address the critical phenomena.

math.PR↗

arXiv2025

Critical edge statistics for deformed GinUEs

For the complex Ginibre ensemble subjected to an additive perturbation by a deterministic normal matrix $X_0$, we establish that under specific spectral conditions on $X_0$, only two distinct types of local spectral statistics emerge at the spectral edge: GinUE statistics and critical statistics, which respectively correspond to regular and quadratically vanishing spectral points. The critical statistics, as a non-Hermitian analogue of Pearcey statistics in random matrix theory, describes a novel point process on the complex plane. This identifies the third (and likely final) universal statistics in non-Hermitian random matrix theory, after the established GinUE bulk and edge universality classes, and represents the primary achievement of this paper.

math.PR↗

arXiv2024

Local limit theorems for random walks on a large discrete torus

Inspired by the study of edge statistics of random band matrices, we investigate random walks on large $d$-dimensional periodic lattices, whose transition matrices are determined by discretized density functions. Under certain moment assumption on the density, we prove local limit theorems for random walks in three different regimes according to the bandwidth parameter, random walk length and torus size.

math.PR↗

arXiv2024

Repeated erfc statistics for deformed GinUEs

For an additive perturbation of the complex Ginibre ensemble under a deterministic matrix $X_0$, under certain assumption on $X_0$, we observe that there are only two kinds of local statistical patterns at the spectral edge: GinUE statistics and critical statistics, which corresponds to regular or quadratic vanishing spectral points. As a continuation of our previous study on critical statistics "Critical edge statistics for deformed GinUEs"(arXiv: 2311.13227), in this paper we establish the local statistics of GinUE type at the regular spectral edge, which is characterized by a repeated erfc integral found in "Phase transition of eigenvalues in deformed Ginibre ensembles"(arXiv: 2204.13171v2).

math.PR↗

arXiv2022

Lyapunov exponent, universality and phase transition for products of random matrices

Products of $M$ i.i.d. random matrices of size $N \times N$ are related to classical limit theorems in probability theory ($N=1$ and large $M$), to Lyapunov exponents in dynamical systems (finite $N$ and large $M$), and to universality in random matrix theory (finite $M$ and large $N$). Under the two different limits of $M \to \infty$ and $N \to \infty$, the local singular value statistics display Gaussian and random matrix theory universality, respectively. However, it is unclear what happens if both $M$ and $N$ go to infinity. This problem, proposed by Akemann, Burda, Kieburg \cite{Akemann-Burda-Kieburg14} and Deift \cite{Deift17}, lies at the heart of understanding both kinds of universal limits. In the case of complex Gaussian random matrices, we prove that there exists a crossover phenomenon as the relative ratio of $M$ and $N$ changes from $0$ to $\infty$: sine and Airy kernels from the Gaussian Unitary Ensemble (GUE) when $M/N \to 0$, Gaussian fluctuation when $M/N \to \infty$, and new critical phenomena when $M/N \to γ\in (0,\infty)$. Accordingly, we further prove that the largest singular value undergoes a phase transition between the Gaussian and GUE Tracy-Widom distributions.

math.PR↗

arXiv2022

Phase transition of eigenvalues in deformed Ginibre ensembles

Consider a random matrix of size $N$ as an additive deformation of the complex Ginibre ensemble under a deterministic matrix $X_0$ with a finite rank, independent of $N$. When some eigenvalues of $X_0$ separate from the unit disk, outlier eigenvalues may appear asymptotically in the same locations, and their fluctuations exhibit surprising phenomena that highly depend on the Jordan canonical form of $X_0$. These findings are largely due to Benaych-Georges and Rochet \cite{BR}, Bordenave and Capitaine \cite{BC16}, and Tao \cite{Ta13}. When all eigenvalues of $X_0$ lie inside the unit disk, we prove that local eigenvalue statistics at the spectral edge form a new class of determinantal point processes, for which correlation kernels are characterized in terms of the repeated erfc integrals. This thus completes a non-Hermitian analogue of the BBP phase transition in Random Matrix Theory. Similar results hold for the deformed real quaternion Ginibre ensemble.

math.PR↗

arXiv2020

Phase transitions for products of characteristic polynomials under Dyson Brownian motion

We study the averaged products of characteristic polynomials for the Gaussian and Laguerre $β$-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.

math.PR↗

arXiv2020

On $H^{2|2}$ Isomorphism theorems and reinforced loop soup

We show that supersymmetric (susy) hyperbolic isomorphism theorems that relate Vertex Reinforced Jump Processes and $H^{2|2}$ field, introduced in [2] and [3], are annealed version of isomorphism theorems relating Markov processes and Gaussian free field, with the help of a Bayes formula that relates susy hyperbolic field to susy free field. On the other hand, we also prove a BFS-Dynkin's isomorphism theorem for reinforced loop soup. Moreover, we provide yet another proof of BFS-Dynkin's isomorphism for VRJP a la Feynman-Kac.

math.PR↗

arXiv2019

Phase transitions for infinite products of large non-Hermitian random matrices

Products of $M$ i.i.d. non-Hermitian random matrices of size $N \times N$ relate Gaussian fluctuation of Lyapunov and stability exponents in dynamical systems (finite $N$ and large $M$) to local eigenvalue universality in random matrix theory (finite $M$ and large $N$). The remaining task is to study local eigenvalue statistics as $M$ and $N$ tend to infinity simultaneously, which lies at the heart of understanding two kinds of universal patterns. For products of i.i.d. complex Ginibre matrices, truncated unitary matrices and spherical ensembles, as $M+N\to \infty$ we prove that local statistics undergoes a transition when the relative ratio $M/N$ changes from $0$ to $\infty$: Ginibre statistics when $M/N \to 0$, normality when $M/N\to \infty$, and new critical phenomena when $M/N\to γ\in (0, \infty)$.

math.PR↗

arXiv2019

Singular Value Statistics for the Spiked Elliptic Ginibre Ensemble

The complex elliptic Ginibre ensemble with coupling $τ$ is a complex Gaussian matrix interpolating between the Gaussian Unitary Ensemble (GUE) and the Ginibre ensemble. It has been known for some time that its eigenvalues form a determinantal point process in the complex plane. A recent result of Kanazawa and Kieburg (arXiv:1804.03985) shows that the singular values form a Pfaffian point process. In this paper we turn to consider an extended elliptic Ginibre ensemble, which connects the GUE and the spiked Wishart matrix, and prove that the singular values still build a Pfaffian point process with correlation kernels expressed by contour integral representations. As $τ$ tends to 1 at a certain critical rate, we prove that the limiting distribution of the largest singular value is described as a new Fredholm Pfaffian series, which connects two distributions $F_{\mathrm{GUE}}$ and $F^{2}_{\mathrm{GUE}}$ where $F_{\mathrm{GUE}}$ is the GUE Tracy-Widom distribution. For fixed $τ$, we prove the Baik-Ben Arous-Péché transition of the largest singular value and the sine kernel in the bulk. We also observe a crossover phenomenon at the origin when $τ$ tends to 1 at another critical rate.

math.PR↗

arXiv2018

Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and Péché, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kernels are found in the limit of infinite matrix dimension at the origin of the spectrum. They depend on finite rank perturbations of the correlation and coupling matrices and are shown to be integrable. The first kernel (I) is found for two independent matrices from the second, and two weakly coupled matrices from the third ensemble. It generalises the Meijer $G$-kernel for two independent and uncorrelated matrices. The third kernel (III) is obtained for the generalised Wishart ensemble and for two strongly coupled matrices. It further generalises the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices, provides an interpolation between the kernels (I) and (III), generalising previous findings of part of the authors.

math-ph↗

arXiv2018

Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral

We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib-Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix transformation which maps the space of polynomial ensembles onto itself. This matrix transformation is closely related to the so-called hyperbolic Harish-Chandra-Itzykson-Zuber integral.

math-ph↗