arXiv subjects

Wangjun Yuan

Publications and source records attributed to Wangjun Yuan.

At least 19 recordsLinked to original sources

arXiv2026

On spectrum of sample correlation matrices from large fold tensor vectors

In this paper, we investigate the limiting spectral distribution of the sample correlation matrix, whose sample vectors are $k$-fold tensor products of $n$-dimensional vectors with i.i.d. entries. We focus on the limiting regime $n,k \to \infty$ with $k = o(n)$, and we show that the limiting spectral distribution is the Mar\v{c}enko-Pastur law. As a consequence, we show that the limiting spectral distribution of the Whishart matrix from the $k$-fold tensor product of independent uniformly distributed unit vectors in $\mathbb C^n$ is the Mar\v{c}enko-Pastur law.

math.PR↗

arXiv2026

Operator Norm Bounds for Multi-leg Matrix Tensors and Applications to Random Matrix Theory

We investigate the extremal values of partial traces of matrix tensors under operator norm constraints. To evaluate these multi-linear quantities, we develop a comprehensive graphical formalism that encodes multi-leg partial traces, partial permutations, and their moments using colored directed graphs. With this graphical framework, we establish optimal, sharp bounds for the partial trace $(\mathrm{Tr}_{\sigma_1} \otimes \ldots \otimes \mathrm{Tr}_{\sigma_k})(A_1, \ldots, A_m)$ over matrices bounded by $\|A_i\| \le 1$. Specifically, we prove that this maximum evaluates exactly to $N^{M(\sigma_1,\ldots,\sigma_k)}$, where $N$ is the dimension and $M$ represents the maximal number of directed cycles in the associated graph across all possible internal vertex pairings. We further derive explicit operator norm estimates for matrices generated by partial traces of partial permutations. Finally, we apply these combinatorial bounds to multi-matrix random matrix theory. By examining models involving Ginibre ensembles, we extend concepts of asymptotic freeness to matrix coefficient algebras, establishing operator norm estimates that rigorously separate the asymptotic behavior of non-crossing and crossing pairings.

math.OA↗

arXiv2024

Strong convergence for tensor GUE random matrices

Haagerup and Thorbj{\o}rnsen proved that iid GUEs converge strongly to free semicircular elements as the dimension grows to infinity. Motivated by considerations from quantum physics -- in particular, understanding nearest neighbor interactions in quantum spin systems -- we consider iid GUE acting on multipartite state spaces, with a mixing component on some sites and identity on the remaining sites. We show that under proper assumptions on the dimension of the sites, strong asymptotic freeness still holds. Our proof relies on an interpolation technology recently introduced by Bandeira, Boedihardjo and van Handel.

math.PR↗

arXiv2024

Multiple collisions of eigenvalues and singular values of matrix Gaussian field

Let $X^\beta$ be a real symmetric or complex Hermitian matrix whose entries are independent Gaussian random fields. We provide the sufficient and necessary conditions such that multiple collisions of eigenvalue processes of $A^\beta + T_\beta X^\beta T_\beta^*$ occur with positive probability. In addition, for a real or complex rectangular matrix $W^\beta$ with independent Gaussian random field entries, we obtain the sufficient and necessary conditions under which the probability of multiple collisions of non-trivial singular value processes of $B^\beta + T_\beta W^\beta \tilde T_\beta$ is positive. In both cases, the size of the set of collision times is characterized via Hausdorff dimension.

math.PR↗

arXiv2024

On spectrum of sample covariance matrices from large tensor vectors

In this paper, we investigate the limiting empirical spectral distribution (LSD) of sums of independent rank-one $k$-fold tensor products of $n$-dimensional vectors as $k,n \to \infty$. Assuming that the base vectors are complex random variables with unit modular, we show that the LSD is the Marčenko-Pastur law. Comparing with the existing results, our limiting setting allows $k$ to grow much faster than $n$. Consequently, we obtain the necessary and sufficient conditions for Marčenko-Pastur law to serve as the LSD of our matrix model. Our approach is based on the moment method.

math.PR↗

arXiv2023

On a class of stochastic fractional heat equations

For the fractional heat equation $\frac{\partial}{\partial t} u(t,x) = -(-Δ)^{\fracα{2}}u(t,x)+ u(t,x)\dot W(t,x)$ where the covariance function of the Gaussian noise $\dot W$ is defined by the heat kernel, we establish Feynman-Kac formulae for both Stratonovich and Skorohod solutions, along with their respective moments. In particular, we prove that $d<2+α$ is a sufficient and necessary condition for the equation to have a unique square-integrable mild Skorohod solution. One motivation lies in the occurrence of this equation in the study of a random walk in random environment which is generated by a field of independent random walks starting from a Poisson field.

math.PR↗

arXiv2023

Stochastic partial differential equations associated with Feller processes

For the stochastic partial differential equation $\frac{\partial u}{\partial t}=\mathcal L u +u\dot W$ where $\dot W$ is Gaussian noise colored in time and $\mathcal L$ is the infinitesimal generator of a Feller process $X$, we obtain Feynman-Kac type of representations for the Stratonovich and Skorohod solutions as well as for their moments. The regularity of the law and the H\"older continuity of the solutions are also studied.

math.PR↗

arXiv2023

Gaussian fluctuations for the wave equation under rough random perturbations

In this article, we consider the stochastic wave equation in spatial dimension $d=1$, with linear term $σ(u)=u$ multiplying the noise. This equation is driven by a Gaussian noise which is white in time and fractional in space with Hurst index $H \in (\frac{1}{4},\frac{1}{2})$. First, we prove that the solution is strictly stationary and ergodic in the spatial variable. Then, we show that with proper normalization and centering, the spatial average of the solution converges to the standard normal distribution, and we estimate the rate of this convergence in the total variation distance. We also prove the corresponding functional convergence result.

math.PR↗

arXiv2023

Hyperbolic Anderson model with time-independent rough noise: Gaussian fluctuations

In this article, we study the hyperbolic Anderson model in dimension 1, driven by a time-independent rough noise, i.e. the noise associated with the fractional Brownian motion of Hurst index $H \in (1/4,1/2)$. We prove that, with appropriate normalization and centering, the spatial integral of the solution converges in distribution to the standard normal distribution, and we estimate the speed of this convergence in the total variation distance. We also prove the corresponding functional limit result. Our method is based on a version of the second-order Gaussian Poincaré inequality developed recently in [27], and relies on delicate moment estimates for the increments of the first and second Malliavin derivatives of the solution. These estimates are obtained using a connection with the wave equation with delta initial velocity, a method which is different than the one used in [27] for the parabolic Anderson model.

math.PR↗

arXiv2022

Central limit theorems for heat equation with time-independent noise: the regular and rough cases

In this article, we investigate the asymptotic behaviour of the spatial integral of the solution to the parabolic Anderson model with time independent noise in dimension $d\geq 1$, as the domain of the integral becomes large. We consider 3 cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index $H \in (\frac{1}{4},\frac{1}{2})$ in dimension $d=1$. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.

math.PR↗

arXiv2022

Hitting probabilities of Gaussian random fields and collision of eigenvalues of random matrices

Let $X= \{X(t), t \in \mathbb R^N\}$ be a centered Gaussian random field with values in $\mathbb R^d$ satisfying certain conditions and let $F \subset \mathbb R^d$ be a Borel set. In our main theorem, we provide a sufficient condition for $F$ to be polar for $X$, i.e. $\mathbb P \big( X(t) \in F \hbox{ for some } t \in \mathbb R^N \big) = 0$, which improves significantly the main result in Dalang et al [7], where the case of $F$ being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we solve a problem on the existence of collisions of the eigenvalues of random matrices with Gaussian random field entries that was left open in Jaramillo and Nualart [14] and Song et al [21].

math.PR↗

arXiv2022

Spatial integral of the solution to hyperbolic Anderson model with time-independent noise

In this article, we study the asymptotic behavior of the spatial integral of the solution to the hyperbolic Anderson model in dimension $d\leq 2$, as the domain of the integral gets large (for fixed time $t$). This equation is driven by a spatially homogeneous Gaussian noise, whose covariance function is either integrable, or is given by the Riesz kernel. The novelty is that the noise does not depend on time, which means that Itô's martingale theory for stochastic integration cannot be used. Using a combination of Malliavin calculus with Stein's method, we show that with proper normalization and centering, the spatial integral of the solution converges to a standard normal distribution, by estimating the speed of this convergence in the total variation distance. We also prove the corresponding functional limit theorem for the spatial integral process.

math.PR↗

arXiv2021

On spectral distribution of sample covariance matrices from large dimensional and large $k$-fold tensor products

We study the eigenvalue distributions for sums of independent rank-one $k$-fold tensor products of large $n$-dimensional vectors. Previous results in the literature assume that $k=o(n)$ and show that the eigenvalue distributions converge to the celebrated Marčenko-Pastur law under appropriate moment conditions on the base vectors. In this paper, motivated by quantum information theory, we study the regime where $k$ grows faster, namely $k=O(n)$. We show that the moment sequences of the eigenvalue distributions have a limit, which is different from the Marčenko-Pastur law. As a byproduct, we show that the Marčenko-Pastur law limit holds if and only if $k=o(n)$ for this tensor model. The approach is based on the method of moments.

math.PR↗

arXiv2021

Recent advances on eigenvalues of matrix-valued stochastic processes

Since the introduction of Dyson's Brownian motion in early 1960's, there have been a lot of developments in the investigation of stochastic processes on the space of Hermitian matrices. Their properties, especially, the properties of their eigenvalues have been studied in great details. In particular, the limiting behaviors of the eigenvalues are found when the dimension of the matrix space tends to infinity, which connects with random matrix theory. This survey reviews a selection of results on the eigenvalues of stochastic processes from the literature of the past three decades. For most recent variations of such processes, such as matrix-valued processes driven by fractional Brownian motion or Brownian sheet, the eigenvalues of them are also discussed in this survey. In the end, some open problems in the area are also proposed.

math.PR↗

arXiv2021

On eigenvalue distributions of large auto-covariance matrices

In this article, we establish a limiting distribution for eigenvalues of a class of auto-covariance matrices. The same distribution has been found in the literature for a regularized version of these auto-covariance matrices. The original non-regularized auto-covariance matrices are non invertible which introduce supplementary diffculties for the study of their eigenvalues through Girko's Hermitization scheme. The key result in this paper is a new polynomial lower bound for the least singular value of the resolvent matrices associated to a rank-defective quadratic function of a random matrix with independent and identically distributed entries. Another improvement in the paper is that the lag of the auto-covariance matrices can grow to infinity with the matrix dimension.

math.PR↗

arXiv2021

On eigenvalues of the Brownian sheet matrix

We derive a system of stochastic partial differential equations satisfied by the eigenvalues of the symmetric matrix whose entries are the Brownian sheets. We prove that the sequence $\left\{L_{d}(s,t), (s,t)\in[0,S]\times [0,T]\right\}_{d\in\mathbb N}$ of empirical spectral measures of the rescaled matrices is tight on $C([0,S]\times [0,T], \mathcal P(\mathbb R))$ and hence is convergent as $d$ goes to infinity by Wigner's semicircle law. We also obtain PDEs which are satisfied by the high-dimensional limiting measure.

math.PR↗

arXiv2020

Eigenvalue distributions of high-dimensional matrix processes driven by fractional Brownian motion

In this article, we study high-dimensional behavior of empirical spectral distributions $\{L_N(t), t\in[0,T]\}$ for a class of $N\times N$ symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H \in(1/2,1)$. For Wigner-type matrices, we obtain almost sure relative compactness of $\{L_N(t), t\in[0,T]\}_{N\in\mathbb N}$ in $C([0,T], \mathbf P(\mathbb R))$ following the approach in \cite{Anderson2010}; for Wishart-type matrices, we obtain tightness of $\{L_N(t), t\in[0,T]\}_{N\in\mathbb N}$ on $C([0,T], \mathbf P(\mathbb R))$ by tightness criterions provided in Appendix \ref{subset:tightness argument}. The limit of $\{L_N(t), t\in[0,T]\}$ as $N\to \infty$ is also characterised.

math.PR↗

arXiv2020

On collision of multiple eigenvalues for matrix-valued Gaussian processes

For real symmetric and complex Hermitian Gaussian processes whose values are $d\times d$ matrices, we characterize the conditions under which the probability that at least $k$ eigenvalues collide is positive for $2\le k\le d$, and we obtain the Hausdorff dimension of the set of collision times.

math.PR↗