arXiv subjects
Publications and source records attributed to Jian Ding.
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.
For a complete graph of size $n$, assign each edge an i.i.d. exponential variable with mean $n$. For $λ>0$, consider the length of the longest path whose average weight is at most $λ$. It was shown by Aldous (1998) that the length is of order $\log n$ for $λ< 1/\mathrm{e}$ and of order $n$ for $λ> 1/\mathrm{e}$. Aldous (2003) posed the question on detailed behavior at and near criticality $1/\mathrm{e}$. In particular, Aldous asked whether there exist scaling exponents $μ, ν$ such that for $λ$ within $1/\mathrm{e}$ of order $n^{-μ}$, the length for the longest path of average weight at most $λ$ has order $n^ν$. We answer this question by showing that the critical behavior is far richer: For $λ$ around $1/\mathrm{e}$ within a window of $α(\log n)^{-2}$ with a small absolute constant $α>0$, the longest path is of order $(\log n)^3$. Furthermore, for $λ\geq 1/\mathrm{e} + β(\log n)^{-2}$ with $β$ a large absolute constant, the longest path is at least of length a polynomial in $n$. An interesting consequence of our result is the existence of a second transition point in $1/\mathrm{e} + [α(\log n)^{-2}, β(\log n)^{-2}]$. In addition, we demonstrate a smooth transition from subcritical to critical regime. Our results were not known before even in a heuristic sense.
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. An important ingredient for our proof is a result of Bramson and Zeitouni (2010), who proved the tightness of the centered maximum together with an evaluation of the expectation up to an additive constant.
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 $β_s(q)$ strictly lower than the critical $β_c(q)$ for uniqueness of the thermodynamic limit. The dynamical critical $β_s(q)$ is the spinodal point marking the onset of metastability. We prove that when $β<β_s(q)$ the mixing time is asymptotically $C(β, q) n \log n$ and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order $n$. At $β=β_s(q)$ the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order $n^{4/3}$. For $β>β_s(q)$ the mixing time is exponentially large in $n$. Furthermore, as $β\uparrow β_s$ with $n$, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of $O(n^{-2/3})$ around $β_s$. These results form the first complete analysis of mixing around the critical dynamical temperature --- including the critical power law --- for a model with a first order phase transition.
We study the cover time $τ_{\mathrm{cov}}$ 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, $\sqrt{τ_{\mathrm{cov}}}=\sqrt{2n^2}[\sqrt{2/π} \log n + O(\log\log n)]$. This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progress towards a conjecture of Bramson and Zeitouni (2009).
Let $\mathcal{C}_1$ be the largest component of the Erdős--Rényi 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 $λ=\varepsilon^3n$ is bounded, Nachmias and Peres proved that the mixing time on $\mathcal{C}_1$ is of order $n$. However, it was unclear how to interpolate between these results, and estimate the mixing time as the giant component emerges from the critical window. Indeed, even the asymptotics of the diameter of $\mathcal{C}_1$ in this regime were only recently obtained by Riordan and Wormald, as well as the present authors and Kim. In this paper, we show that for $p=(1+\varepsilon)/n$ with $λ=\varepsilon^3n\to\infty$ and $λ=o(n)$, the mixing time on $\mathcal{C}_1$ is with high probability of order $(n/λ)\log^2λ$. In addition, we show that this is the order of the largest mixing time over all components, both in the slightly supercritical and in the slightly subcritical regime [i.e., $p=(1-\varepsilon)/n$ with $λ$ as above].
Let $S_n^{(2)}$ denote the iterated partial sums. That is, $S_n^{(2)}=S_1+S_2+ ... +S_n$, where $S_i=X_1+X_2+ ... s+X_i$. Assuming $X_1, X_2,....,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{(2)}:=\PP(\max_{1\le i \le n}S_i^{(2)}< 0) \le c\sqrt{\frac{\EE|S_{n+1}|}{(n+1)\EE|X_1|}},$$ with $c \le 6 \sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $\min(-X_1,0)$ is bounded or when it has only finite third moment and in addition $X_1$ is squared integrable. Furthermore, $p_n^{(2)}\asymp n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_i$. In contrast, we show that for any $0 < γ< 1/4$ there exist integrable, zero-mean random variables for which the rate of decay of $p_n^{(2)}$ is $n^{-γ}$.
In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erdős-Rényi random graph $G(n,p)$ as it emerges from the critical window, i.e. for $p = (1+ε)/n$ where $ε^3 n \to\infty$ and $ε=o(1)$, in terms of a tractable contiguous model. Here we provide the analogous description for the supercritical giant component, i.e., the largest component of $G(n,p)$ for $p = λ/n$ where $λ>1$ is fixed. The contiguous model is roughly as follows: Take a random degree sequence and sample a random multigraph with these degrees to arrive at the kernel; Replace the edges by paths whose lengths are i.i.d. geometric variables to arrive at the 2-core; Attach i.i.d. Poisson Galton-Watson trees to the vertices for the final giant component. As in the case of the emerging giant, we obtain this result via a sequence of contiguity arguments at the heart of which are Kim's Poisson-cloning method and the Pittel-Wormald local limit theorems.
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. We give a deterministic polynomial-time algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996), showing that for any graph, the blanket and cover times are within an O(1) factor. The best previous approximation factor for both these problems was $O((\log \log n)^2)$ for $n$-vertex graphs, due to Kahn, Kim, Lovasz, and Vu (2000).
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, $\sqrt{τ_{\mathrm{cov}}}=\sqrt{|E|}[\sqrt{2\log 2}\cdot n - {\log n}/{\sqrt{2\log 2}} + O((\log\logn)^8]$, thus showing that the second order correction differs from the corresponding one for the maximum of the Gaussian free field on the tree.
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. Cooper and Frieze showed that the cover time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large d. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman, is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4.
We study the diameter of $C_1$, the largest component of the Erdős-Rényi random graph $G(n,p)$ in the emerging supercritical phase, i.e., for $p = \frac{1+ε}n$ where $ε^3 n \to \infty$ and $ε=o(1)$. This parameter was extensively studied for fixed $ε> 0$, yet results for $ε=o(1)$ outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter, however these did not cover the entire supercritical regime (namely, when $ε^3 n\to\infty$ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of $1000/7$. We show that throughout the emerging supercritical phase, i.e. for any $ε=o(1)$ with $ε^3 n \to \infty$, the diameter of $C_1$ is with high probability asymptotic to $D(ε,n)=(3/ε)\log(ε^3 n)$. This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of $C_1$ is w.h.p. asymptotic to $(2/3)D(ε,n)$, and the maximal distance in $C_1$ between any pair of kernel vertices is w.h.p. asymptotic to $(5/9)D(ε,n)$.
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. Specifically, we consider a game in which players organize networking events at a cost that grows with the number of attendees. An event's cost is assumed by the organizer but the benefit accrues equally to all attendees: a link is formed between any two players who see each other at more than a certain number r of events per time period. The graph of connections so obtained is the social network of the model. We analyze the Nash equilibria of this game when each player derives a benefit a>0 from all her neighbors in the network and when the costs are linear, i.e., when the cost of an event with L invitees is b+cL, with b>0 and c>0. For a/cr > 1 and b sufficiently small, all Nash equilibria have the complete graph as their social network; for a/cr < 1 the Nash equilibria correspond to a rich class of social networks, all of which have substantial clustering in the sense that the clustering coefficient is bounded below by the inverse of the average degree. Additionally, for any degree sequence with finite mean, and not too many vertices of degree one or two, we can construct a Nash equilibrium producing a social network with the given degree sequence.
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. We consider fiducial values of density, temperature, and magnetic parameter $β_e$ (defined as the ratio of the electron pressure to the magnetic pressure) of the accretion flow as $n_{0} \sim 10^{6} {\rm cm^{-3}}$, $T_{e}^0\sim 2\times 10^9 {\rm K}$, and $β_e=1$. We find that electrons are heated to a higher temperature $T_{e}=5\times 10^9$K, and a fraction $η\sim 8%$ of electrons are accelerated into a broken power-law distribution, $dN(γ)\propto γ^{-p}$, with $p\approx 1.5$ and 4 below and above $\sim 1$ MeV, respectively. We also investigate the effect of varying $β$ and $n_0$. We find that when $β_e$ is smaller or $n_0$ is larger, i.e, the magnetic field is stronger, $T_e$, $η$, and $p$ all become larger.
We provide a complete description of the giant component of the Erdős-Rényi random graph $G(n,p)$ as soon as it emerges from the scaling window, i.e., for $p = (1+ε)/n$ where $ε^3 n \to \infty$ and $ε=o(1)$. Our description is particularly simple for $ε= o(n^{-1/4})$, where the giant component $C_1$ is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for $C_1$). Let $Z$ be normal with mean $\frac23 ε^3 n$ and variance $ε^3 n$, and let $K$ be a random 3-regular graph on $2\lfloor Z\rfloor$ vertices. Replace each edge of $K$ by a path, where the path lengths are i.i.d. geometric with mean $1/ε$. Finally, attach an independent Poisson($1-ε$)-Galton-Watson tree to each vertex. A similar picture is obtained for larger $ε=o(1)$, in which case the random 3-regular graph is replaced by a random graph with $N_k$ vertices of degree $k$ for $k\geq 3$, where $N_k$ has mean and variance of order $ε^k n$. This description enables us to determine fundamental characteristics of the supercritical random graph. Namely, we can infer the asymptotics of the diameter of the giant component for any rate of decay of $ε$, as well as the mixing time of the random walk on $C_1$.
In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature $β_c$, the inverse-gap is bounded for $β< β_c$, polynomial in the surface area for $β= β_c$ and exponential in it for $β> β_c$. This has been proved for $\Z^2$ except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for $β< β_c$ and exponential for $β> β_c$ were established, where $β_c$ is the critical spin-glass parameter, and the tree-height $h$ plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the $b$-ary tree, by showing that it is indeed polynomial in $h$ at criticality. The degree of our polynomial bound does not depend on $b$, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for $β> β_c$, the inverse-gap and mixing-time are both $\exp[Θ((β-β_c) h)]$.
We study Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss Model. It is well known that at high temperature ($β< 1$) the mixing time is $Θ(n\log n)$, whereas at low temperature ($β> 1$) it is $\exp(Θ(n))$. Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed $β> 1$, the mixing-time of this model is $Θ(n\log n)$, analogous to the high-temperature regime of the original dynamics. Furthermore, they showed \emph{cutoff} for the original dynamics for fixed $β<1$. The question whether the censored dynamics also exhibits cutoff remained unsettled. In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Currie-Weiss model. Namely, we found a scaling window of order $1/\sqrt{n}$ around the critical temperature $β_c=1$, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging. In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if $β= 1 + δ$ for some $δ> 0$ with $δ^2 n \to \infty$, then the mixing-time has order $(n / δ)\log(δ^2 n)$. The cutoff constant is $(1/2+[2(ζ^2 β/ δ- 1)]^{-1})$, where $ζ$ is the unique positive root of $g(x)=\tanh(βx)-x$, and the cutoff window has order $n / δ$.
The cutoff phenomenon describes a case where a Markov chain exhibits a sharp transition in its convergence to stationarity. In 1996, Diaconis surveyed this phenomenon, and asked how one could recognize its occurrence in families of finite ergodic Markov chains. In 2004, the third author noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing-time and spectral-gap tends to infinity, and conjectured that in many settings, this condition should also be sufficient. Diaconis and Saloff-Coste (2006) verified this conjecture for continuous-time birth-and-death chains, started at an endpoint, with convergence measured in separation. It is natural to ask whether the conjecture holds for these chains in the more widely used total-variation distance. In this work, we confirm the above conjecture for all continuous-time or lazy discrete-time birth-and-death chains, with convergence measured via total-variation distance. Namely, if the product of the mixing-time and spectral-gap tends to infinity, the chains exhibit cutoff at the maximal hitting time of the stationary distribution median, with a window of at most the geometric mean between the relaxation-time and mixing-time. In addition, we show that for any lazy (or continuous-time) birth-and-death chain with stationary distribution $π$, the separation $1 - p^t(x,y)/π(y)$ is maximized when $x,y$ are the endpoints. Together with the above results, this implies that total-variation cutoff is equivalent to separation cutoff in any family of such chains.