Beijing Institute of Mathematical Sciences and Applications

"Random matrices arise naturally in various contexts ranging from theoretical physics to computer science. In many of these problems, it is important to know the spectral characteristics of a random matrix of a large but fixed size. We will discuss recent progress in this area illustrating it by problems coming from combinatorics and computer science:

With the recent developments on nonlinear SPDE's, where smoothing of rough noises is needed, one is naturally led to study interacting particle systems whose macroscopic evolution is described by these equations and which possess an in-built smoothing. In this article, our main results are to derive regularized versions of the ill-posed one dimensional SPDE $$\partial_t \rho = \frac{1}{2}\Delta \Phi(\rho) - 2 \nabla \big(W'\Phi(\rho)\big),$$ where the spatial white noise $W'$ is replaced by a regularization $W'_\varepsilon$, as quenched and annealed hydrodynamic limits of zero-range interacting particle systems in $\varepsilon$-regularized Sinai-type random environments.

I will talk two recent works on hypercontractity. The first one is about a natural extension in matricial Schatten-von Neumann setting. The second one is on three point inequalities. The talk is based on works with Shilei Fan, Yong Han, Zipeng Wang.

We construct a compound Poisson process conditioned on its random summation that represents the sizes of the connected components in the sparse Erdős-Rényi random graph G(n,c/n). This new representation depicts a connection between the phase transition in the sparse random graph and the condensation transition in the zero-range model. Under this framework, we can derive moderate deviation principles for the maximun component, total number of connected components and empirical measure of the sizes in the non-critical regimes. Large deviation results are discussed.

This talk first provides a brief introduction of our recent contribution to self-excited systems, including functional limit theorem, scaling limit theorem and mean-field limit. In the second part, we introduce serveral applications in various fields.

For any integer $d\geq 2$, let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices. Consider the random matrix products $G_n := g_n \ldots g_1$. For any starting point $x\in \mathbb{R}^d_+$ with $|x| = 1$ and $y \geq 0$, we define the exit time $\tau_{x, y} = \inf \{ k \geq 1: y + \log |G_k x| < 0 \}$. In this talk, we investigate the conditioned local probability $\mathbb{P} (y + \log |G_n x| \in [0, \Delta] + z, \tau_{x, y} > n)$ under various assumptions on $y$ and $z$. For the case where $y = o(\sqrt{n})$, we establish precise upper and lower bounds for $z$ within a compact interval, and provide exact asymptotic results as $z \to \infty$. Furthermore, we explore the case where $y \asymp \sqrt{n}$ and derive corresponding asymptotic expressions for different values of $z$.

"Branching capacity is a set function introduced in [Zhu 2016], recording hitting probability of a set from afar by a branching random walk.

Locally isotropic Gaussian random fields were first introduced by Kolmogorov in 1941. Such models were used to describe various phenomena in statistical physics. In particular, they were introduced to model a single particle in a random potential by Engel, Mezard and Parisi in 1990s. Using Parisi's award winning replica trick, Fyodorov and Le Doussal predicted the high dimensional limit of the Hessian spectrum at the global minimum of these models, and discovered phase transitions according to different levels of replica symmetry breaking. In this talk, I will present a solution to their conjecture in the so called replica symmetric regime. Our method is based on landscape complexity, or counting the number of critical points of the Hamiltonian. This talk is based on joint works with Antonio Auffinger (Northwestern University), Hao Xu (University of Macau) and Haoran Yang (Peking University).

Abstract: In recent years, a technique has been developed to compute the conformal radii of random domains defined by SLE curves, which is based on the coupling between SLE and Liouville quantum gravity (LQG). Compared to prior methods that compute SLE related quantities via its coupling with LQG, the crucial new input is the exact solvability of structure constants in Liouville conformal field theory. It appears that various percolation exponents can be expressed in terms of conformal radii that can be computed this way. This includes known exponents such as the one-arm and polychromatic two-arm exponents, as well as the backbone exponents, which is unknown previously. In this talk we will review this method using the derivation of the backbone exponent as an example, based on a joint work with Nolin, Qian, and Zhuang

We will report on recent progress regarding the universality of the extreme eigenvalues of a large random matrix with i.i.d. entries. Beyond the radius of the celebrated circular law, we will establish a precise three-term asymptotic expansion for the largest eigenvalue (in modulus) with an optimal error term. Based on this result, we will further show that the properly normalized largest eigenvalue converges to a Gumbel distribution as the dimension goes to infinity. Similar results also apply to the rightmost eigenvalue of the matrix. These results are based on several joint works with Giorgio Cipolloni, Laszlo Erdos, and Dominik Schroder.

The skew Brownian motion (Ito-McKean, 1965, Walsh, 1978) is a diffusion which behaves as a Brownian motion away from 0, but has a ‘local drift’every time it crosses 0. It can be constructed by assigning signs to Brownian excursions away from 0, each excursion being positive with probability $p$ and negative with probability $1-p$. It can equivalently (Harrison- Shepp, 1981) be seen as the strong solution of the SDE $dX_t=dB_t + q dL_t(X)$ where $L_t(X)$ denotes the local time of the diffusion at 0. The skew Brownian flow as studied by Burdzy- Chen (2001) and Burdzy-Kaspi(2004) is the flow of solutions driven by the same Brownian motion but starting from any point of the real line. We propose an alternative, somewhat more geometric, construction of this flow. We will show how intriguing properties of the skew Brownian flow (coalescence, bifurcation points…) appear naturally in this setting. Based on a joint work with Chengshi Wang and Yaolin Yu.

"Random matrices arise naturally in various contexts ranging from theoretical physics to computer science. In many of these problems, it is important to know the spectral characteristics of a random matrix of a large but fixed size. We will discuss recent progress in this area illustrating it by problems coming from combinatorics and computer science: