# Jian Ding - SHAO

## Contact Details

NameJian Ding |
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AffiliationSHAO |
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Location |
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## Pubs By Year |
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## External Links |
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## Pub CategoriesMathematics - Probability (42) Mathematical Physics (4) Mathematics - Mathematical Physics (4) Mathematics - Combinatorics (3) Mathematics - Metric Geometry (2) Computer Science - Learning (2) Mathematics - Functional Analysis (1) High Energy Astrophysical Phenomena (1) Computer Science - Data Structures and Algorithms (1) Computer Science - Computer Science and Game Theory (1) Statistics - Theory (1) Mathematics - Optimization and Control (1) Mathematics - Representation Theory (1) Nonlinear Sciences - Cellular Automata and Lattice Gases (1) Computer Science - Information Theory (1) Mathematics - Information Theory (1) Mathematics - Statistics (1) Nonlinear Sciences - Adaptation and Self-Organizing Systems (1) Physics - Physics and Society (1) Computer Science - Discrete Mathematics (1) |

## Publications Authored By Jian Ding

Place an obstacle with probability $1-\mathsf{p}$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $\mathsf{p}$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite cluster of non-obstacles; we show that for environments with probability tending to 1 as $n\to \infty$, conditioning on survival after $n$ steps the simple random walk is localized in a region of size poly-logarithmic in $n$ with probability tending to 1 as $n\to \infty$. Read More

The intersecting pedestrian flow on the 2D lattice with random update rule is studied. Each pedestrian has three moving directions without the back step. Under periodic boundary conditions, an intermediate phase has been found at which some pedestrians could move along the border of jamming stripes. Read More

Let $G$ be a connected complex simple Lie group, and let $\widehat{G}^{\mathrm{d}}$ be the set of all non-trivial irreducible unitary representations with non-vanishing Dirac cohomology. We show that $\widehat{G}^{\mathrm{d}}$ consists of two parts: finitely many scattered representations (the scattered part), and finitely many strings of representations (the string part). Moreover, the string part of $\widehat{G}^{\mathrm{d}}$ are all in the good range and they come from the scattered part of $\widehat{L}^{\mathrm{d}}$ via cohomological induction, where $L$ runs over the Levi factors of $\theta$-stable parabolic subgroups of $G$. Read More

We study the Liouville heat kernel (in the $L^2$ phase) associated with a class of logarithmically correlated Gaussian fields on the two dimensional torus. We show that for each $\varepsilon>0$ there exists such a field, whose covariance is a bounded perturbation of that of the two dimensional Gaussian free field, and such that the associated Liouville heat kernel satisfies the short time estimates, $$ \exp \left( - t^{ - \frac 1 { 1 + \frac 1 2 \gamma^2 } - \varepsilon } \right) \le p_t^\gamma (x, y) \le \exp \left( - t^{- \frac 1 { 1 + \frac 1 2 \gamma^2 } + \varepsilon } \right) , $$ for $\gamma<1/2$. In particular, these are different from predictions, due to Watabiki, concerning the Liouville heat kernel for the two dimensional Gaussian free field. Read More

We consider the stabilization of an unstable discrete-time linear system that is observed over a channel corrupted by continuous multiplicative noise. Our main result shows that if the system growth is large enough, then the system cannot be stabilized in a second-moment sense. This is done by showing that the probability that the state magnitude remains bounded must go to zero with time. Read More

For a one-dimensional simple random walk $(S_t)$, for each time $t$ we say a site $x$ is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work is inspired by T\'oth (2001), and disproves a conjecture of Erd\"{o}s and R\'ev\'esz (1984) and of T\'oth (2001). Read More

Given any $\gamma>0$ and for $\eta=\{\eta_v\}_{v\in \mathbb Z^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on $\mathbb Z^2$ pinned at the origin, we consider the random walk on $\mathbb Z^2$ among random conductances where the conductance of edge $(u, v)$ is given by $\mathrm{e}^{\gamma(\eta_u + \eta_v)}$. We show that, for almost every $\eta$, this random walk is recurrent and that, with probability tending to 1 as $T\to \infty$, the return probability at time $2T$ decays as $T^{-1+o(1)}$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius $N$ scales as $N^{\psi(\gamma)+o(1)}$ with $\psi(\gamma)>2$ for all $\gamma>0$. Read More

Given a planar continuous Gaussian free field $h$ in a domain $D$ with Dirichlet boundary condition and any $\delta>0$, we let $\{h_\delta(v): v\in D\}$ be a real-valued smooth Gaussian field where $h_\delta(v)$ is the average of $h$ in a circle of radius $\delta$ with center $z$. For $\gamma > 0$, we study the Liouville first passage percolation (in scale $\delta$), i.e. Read More

Let $\{\eta(v): v\in V_N\}$ be a discrete Gaussian free field in a two-dimensional box $V_N$ of side length $N$ with Dirichlet boundary conditions. We study the Liouville first passage percolation, i.e. Read More

Let $R=\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}+\cdots+u^{k}\mathbb{F}_{2^{m}}$ , where $\mathbb{F}_{2^{m}}$ is a finite field with $2^{m}$ elements, $m$ is a positive integer, $u$ is an indeterminate with $u^{k+1}=0.$ In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over $R$. A new Gray map over $R$ is defined and a sufficient and necessary condition for the existence of dual-containing cyclic codes over $R$ is given. Read More

Let $\{\eta_{N, v}: v\in V_N\}$ be a discrete Gaussian free field in a two-dimensional box $V_N$ of side length $N$ with Dirichlet boundary conditions. We study the Liouville first passage percolation, i.e. Read More

We initiate the study on chemical distances of percolation clusters for level sets of two-dimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a hard wall. Read More

Let $\{Y_{\mathfrak{B}}(v):v\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study the Liouville first-passage percolation, in which each vertex is given a weight of $e^{\gamma Y_{\mathfrak{B}}(v)}$ for some $\gamma>0$. We show that for sufficiently small but fixed $\gamma>0$, for any sequence of scales $\{S_k\}$ there exists a subsequence along which the appropriately scaled Liouville FPP metric converges in the Gromov--Hausdorff sense to a random metric on the unit square in $\mathbf{R}^{2}$. Read More

In this paper, the impact of escaping in couples on the evacuation dynamics has been investigated via experiments and modeling. Two sets of experiments have been implemented, in which pedestrians are asked to escape either in individual or in couples. The experiments show that escaping in couples can decrease the average evacuation time. Read More

We consider the branching random walk $\{\mathcal R^N_z: z\in V_N\}$ with Gaussian increments indexed over a two-dimensional box $V_N$ of side length $N$, and we study the first passage percolation where each vertex is assigned weight $e^{\gamma \mathcal R^N_z}$ for $\gamma>0$. We show that for $\gamma>0$ sufficiently small but fixed, the expected FPP distance between the left and right boundaries is at most $O(N^{1 - \gamma^2/10})$. Read More

This paper is concerned with a binary detection problem over a non-secure network. To satisfy the communication rate constraint and against possible cyber attacks, which are modeled as deceptive signals injected to the network, a likelihood ratio based (LRB) scheduler is designed in the sensor side to smartly select sensor measurements for transmission. By exploring the scheduler, some sensor measurements are successfully retrieved from the attacked data at the decision center. Read More

We consider first passage percolation(FPP) where the vertex weight is given by the exponential of two-dimensional log-correlated Gaussian fields. Our work is motivated by understanding the discrete analog for the random metric associated with \emph{Liouville quantum gravity}(LQG), which roughly corresponds to the exponential of a two-dimensional Gaussian free field (GFF). The particular focus of the present paper is an aspect of universality for such FPP among the family of log-correlated Gaussian fields. Read More

We study the weight and length of the minimum mean-weight cycle in the stochastic mean-field distance model, i.e., in the complete graph on $n$ vertices with edges weighted by independent exponential random variables. Read More

We show that the centered maximum of a sequence of log-correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a sequence of random variables, reminiscent of the derivative martingale in the theory of Branching Random Walk and Gaussian Chaos. We also discuss applications of the main convergence theorem and discuss examples that show that for logarithmically correlated fields, some additional structural assumptions of the type we make are needed for convergence of the centered maximum. Read More

For a complete graph of size $n$, assign each edge an i.i.d. Read More

We study the problem of detecting the presence of an underlying high-dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erd\H{o}s-R\'enyi random graph $G(n,p)$. Under the alternative, the graph is generated from the $G(n,p,d)$ model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere $\mathbb{S}^{d-1}$, and two vertices are connected if the corresponding latent vectors are close enough. Read More

We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_*(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha<\alpha_*$, and unsatisfiable for $\alpha>\alpha_*$. We show that the threshold $\alpha_*(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. Read More

We study a new class of online learning problems where each of the online algorithm's actions is assigned an adversarial value, and the loss of the algorithm at each step is a known and deterministic function of the values assigned to its recent actions. This class includes problems where the algorithm's loss is the minimum over the recent adversarial values, the maximum over the recent values, or a linear combination of the recent values. We analyze the minimax regret of this class of problems when the algorithm receives bandit feedback, and prove that when the minimum or maximum functions are used, the minimax regret is $\tilde \Omega(T^{2/3})$ (so called hard online learning problems), and when a linear function is used, the minimax regret is $\tilde O(\sqrt{T})$ (so called easy learning problems). Read More

Let $\eta^*_n$ denote the maximum, at time $n$, of a nonlattice one-dimensional branching random walk $\eta_n$ possessing (enough) exponential moments. In a seminal paper, Aidekon demonstrated convergence of $\eta^*_n$ in law, after recentering, and gave a representation of the limit. We give here a shorter proof of this convergence by employing reasoning motivated by Bramson, Ding and Zeitouni. Read More

Given a finite, connected graph $G$, the lamplighter chain on $G$ is the lazy random walk $X^\diamond$ on the associated lamplighter graph $G^\diamond={\mathbf Z}_2 \wr G$. The mixing time of the lamplighter chain on the torus ${\mathbf Z}_n^d$ is known to have a cutoff at a time asymptotic to the cover time of ${\mathbf Z}_n^d$ if $d=2$, and to half the cover time if $d \ge 3$. We show that the mixing time of the lamplighter chain on $G_n(a)={\mathbf Z}_n^2 \times {\mathbf Z}_{a \log n}$ has a cutoff at $\psi(a)$ times the cover time of $G_n(a)$ as $n \to \infty$, where $\psi$ is an explicit weakly decreasing map from $(0,\infty)$ onto $[1/2,1)$. Read More

We initiate the study of mixing times of Markov chain under monotone censoring. Suppose we have some Markov Chain $M$ on a state space $\Omega$ with stationary distribution $\pi$ and a monotone set $A \subset \Omega$. We consider the chain $M'$ which is the same as the chain $M$ started at some $x \in A$ except that moves of $M$ of the form $x \to y$ where $x \in A$ and $y \notin A$ are {\em censored} and replaced by the move $x \to x$. Read More

We prove two theorems concerning extreme values of general Gaussian fields. Our first theorem concerns with the concept of multiple peaks. A theorem of Chatterjee states that when a centered Gaussian field admits the so-called superconcentration property, it typically attains values near its maximum on multiple near-orthogonal sites, known as multiple peaks. Read More

We consider the random regular $k$-NAE-SAT problem with $n$ variables each
appearing in exactly $d$ clauses. For all $k$ exceeding an absolute constant
$k_0$, we establish explicitly the satisfiability threshold $d_*=d_*(k)$. We
prove that for $d

In this short note, we present a theorem concerning certain "additive structure" for the level sets of non-degenerate Gaussian fields, which yields the multiple valley phenomenon for extremal fields with exponentially many valleys. Read More

We determine the asymptotics of the independence number of the random $d$-regular graph for all $d \ge d_0$. It is highly concentrated, with constant-order fluctuations around $n\alpha_* - c_*\log n$ for explicit constants $\alpha_*(d)$ and $c_*(d)$. Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs. Read More

We study the adversarial multi-armed bandit problem in a setting where the player incurs a unit cost each time he switches actions. We prove that the player's $T$-round minimax regret in this setting is $\widetilde{\Theta}(T^{2/3})$, thereby closing a fundamental gap in our understanding of learning with bandit feedback. In the corresponding full-information version of the problem, the minimax regret is known to grow at a much slower rate of $\Theta(\sqrt{T})$. Read More

In this note, we demonstrate an instance of bounded-degree graphs of size $n$, for which the total variation mixing time for the random walk is decreased by a factor of $\log n/ \log\log n$ if we multiply the edge-conductances by bounded factors in a certain way. Read More

We study the long range percolation model on $\mathbb{Z}$ where sites $i$ and $j$ are connected with probability $\beta |i-j|^{-s}$. Graph distances are now well understood for all exponents $s$ except in the case $s=2$ where the model exhibits non-trivial self-similar scaling. Establishing a conjecture of Benjamini and Berger \cite{BenBer:01}, we prove that the typical distance from site 0 to $n$ grows as a power law $n^{\theta(\beta)}$ up to a multiplicative constant for some exponent $0<\theta(\beta)<1$ as does the diameter of the graph on a box of length $n$. Read More

We consider the two-dimensional Gaussian Free Field on a box of side length $N$, with Dirichlet boundary data, and prove the convergence of the law of the recentered maximum of the field. Read More

For two metric spaces X and Y, say that X {threshold-embeds} into Y if there exist a number K > 0 and a family of Lipschitz maps $f_{\tau} : X \to Y : \tau > 0 \}$ such that for every $x,y \in X$, \[ d_X(x,y) \geq \tau => d_Y(f_{\tau}(x),f_{\tau}(y)) \geq \|\varphi_{\tau}\|_{\Lip} \tau/K \] where $\|f_{\tau}\|_{\Lip}$ denotes the Lipschitz constant of $f_{\tau}$. We show that if a metric space X threshold-embeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. Read More

We consider in this paper the collection of near maxima of the discrete, two dimensional Gaussian free field in a box with Dirichlet boundary conditions. We provide a rough description of the geometry of the set of near maxima, estimates on the gap between the two largest maxima, and an estimate for the right tail up to a multiplicative constant on the law of the centered maximum. Read More

**Category:**Mathematics - Probability

Let $S_n^{(2)}$ denote the iterated partial sums. That is, $S_n^{(2)}=S_1+S_2+ .. Read More

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with $q\geq 3$ states and show that it undergoes a critical slowdown at an inverse-temperature $\beta_s(q)$ strictly lower than the critical $\beta_c(q)$ for uniqueness of the thermodynamic limit. The dynamical critical $\beta_s(q)$ is the spinodal point marking the onset of metastability. We prove that when $\beta<\beta_s(q)$ the mixing time is asymptotically $C(\beta, q) n \log n$ and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order $n$. Read More

In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ as it emerges from the critical window, i.e. for $p = (1+\epsilon)/n$ where $\epsilon^3 n \to\infty$ and $\epsilon=o(1)$, in terms of a tractable contiguous model. Read More

For a complete graph of size $n$, assign each edge an i.i.d. Read More

We study the cover time $\tau_{\mathrm{cov}}$ by (continuous-time) random walk on the 2D box of side length $n$ with wired boundary or on the 2D torus, and show that in both cases with probability approaching 1 as $n$ increases, $\sqrt{\tau_{\mathrm{cov}}}=\sqrt{2n^2}[\sqrt{2/\pi} \log n + O(\log\log n)]$. This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progress towards a conjecture of Bramson and Zeitouni (2009). Read More

We study the tail behavior for the maximum of discrete Gaussian free field on a 2D box with Dirichlet boundary condition after centering by its expectation. We show that it exhibits an exponential decay for the right tail and a double exponential decay for the left tail. In particular, our result implies that the variance of the maximum is of order 1, improving an $o(\log n)$ bound by Chatterjee (2008) and confirming a folklore conjecture. Read More

We compute the second order correction for the cover time of the binary tree of depth $n$ by (continuous-time) random walk, and show that with probability approaching 1 as $n$ increases, $\sqrt{\tau_{\mathrm{cov}}}=\sqrt{|E|}[\sqrt{2\log 2}\cdot n - {\log n}/{\sqrt{2\log 2}} + O((\log\logn)^8]$, thus showing that the second order correction differs from the corresponding one for the maximum of the Gaussian free field on the tree. Read More

In this paper we show that on bounded degree graphs and general trees, the cover time of the simple random walk is asymptotically equal to the product of the number of edges and the square of the expected supremum of the Gaussian free field on the graph, assuming that the maximal hitting time is significantly smaller than the cover time. Previously, this was only proved for regular trees and the 2D lattice. Furthermore, for general trees, we derive exponential concentration for the cover time, which implies that the standard deviation of the cover time is bounded by the geometric mean of the cover time and the maximal hitting time. Read More

We propose a new class of game-theoretic models for network formation in which strategies are not directly related to edge choices, but instead correspond more generally to the exertion of social effort. The observed social network is thus a byproduct of an expressive strategic interaction, which can more naturally explain the emergence of complex social structures. Within this framework, we present a natural network formation game in which agent utilities are locally defined and that, despite its simplicity, produces a rich class of equilibria that exhibit structural properties commonly observed in social networks - such as triadic closure - that have proved elusive in most existing models. Read More

We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph $G$ is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on $G$, scaled by the number of edges in $G$. This allows us to resolve a number of open questions. Read More

The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation. We refine this upper bound, and show that the resulting bound is sharp and explicitly computable in random graphs. Read More

**Affiliations:**

^{1}SHAO,

^{2}SHAO,

^{3}Rice University

**Category:**High Energy Astrophysical Phenomena

Both analytical and numerical works show that magnetic reconnection must occur in hot accretion flows. This process will effectively heat and accelerate electrons. In this paper we use the numerical hybrid simulation of magnetic reconnection plus test-electron method to investigate the electron acceleration and heating due to magnetic reconnection in hot accretion flows. Read More

Consider Glauber dynamics for the Ising model on a graph of $n$ vertices. Hayes and Sinclair showed that the mixing time for this dynamics is at least $n\log n/f(\Delta)$, where $\Delta$ is the maximum degree and $f(\Delta) = \Theta(\Delta \log^2 \Delta)$. Their result applies to more general spin systems, and in that generality, they showed that some dependence on $\Delta$ is necessary. Read More

Let $\mathcal{C}_1$ be the largest component of the Erd\H{o}s--R\'{e}nyi random graph $\mathcal{G}(n,p)$. The mixing time of random walk on $\mathcal {C}_1$ in the strictly supercritical regime, $p=c/n$ with fixed $c>1$, was shown to have order $\log^2n$ by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, $p=(1+\varepsilon)/n$ where $\lambda=\varepsilon^3n$ is bounded, Nachmias and Peres proved that the mixing time on $\mathcal{C}_1$ is of order $n$. Read More