Jian Ding

Anatomy of a young giant component in the random graph

We provide a complete description of the giant component of the Erdős-Renyi random graph documentclass{article} usepackage{mathrsfs} usepackage{amsmath} pagestyle{empty} begin{document} begin{align*}{mathcal{G}}(n,p)end{align*} end{document} **image** as soon as it emerges from the scaling window, i.e., for p = (1+e)/n where e3n →∞ and e = o(1). Our description is particularly simple for e = o(n-1/4), where the giant component documentclass{article} usepackage{mathrsfs} usepackage{amsmath} pagestyle{empty} begin{document} begin{align*}{mathcal{C}_1}end{align*} end{document} **image** 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 documentclass{article} usepackage{mathrsfs} usepackage{amsmath} pagestyle{empty} begin{document} begin{align*}{mathcal{C}_1}end{align*} end{document} **image** ). Let Z be normal with mean documentclass{article} usepackage{mathrsfs} usepackage{amsmath} pagestyle{empty} begin{document} begin{align*}frac{2}{3} varepsilon^3 nend{align*} end{document} **image** and variance e3n, and let documentclass{article} usepackage{mathrsfs} usepackage{amsmath} pagestyle{empty} begin{document} begin{align*}mathcal{K}end{align*} end{document} **image** be a random 3-regular graph on documentclass{article} usepackage{mathrsfs} usepackage{amsmath} pagestyle{empty} begin{document} begin{align*}2leftlfloor Zrightrfloorend{align*} end{document} **image** vertices. Replace each edge of documentclass{article} usepackage{mathrsfs} usepackage{amsmath} pagestyle{empty} begin{document} begin{align*}mathcal{K}end{align*} end{document} **image** by a path, where the path lengths are i.i.d. geometric with mean 1/e. Finally, attach an independent Poisson( 1-e )-Galton-Watson tree to each vertex. A similar picture is obtained for larger e = o(1), in which case the random 3-regular graph is replaced by a random graph with Nk vertices of degree k for k ≥ 3, where Nk has mean and variance of order ekn. 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 e, as well as the mixing time of the random walk on documentclass{article} usepackage{mathrsfs} usepackage{amsmath} pagestyle{empty} begin{document} begin{align*}{mathcal{C}_1}end{align*} end{document} **image** .

The mixing time evolution of glauber dynamics for the mean-field ising model

We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time

Persistence of iterated partial sums

{frac{1}{2(1-beta)}nlog n}

Mixing time of critical ising model on trees is polynomial in the height

with a window of order n, whereas the mixing-time at the critical temperature β =xa01 is Θ(n3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n3/2) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β =xa0β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point βcxa0=xa01. In particular, we find a scaling window of order

Bandits with switching costs: T 2/3 regret

{1/sqrt{n}}

Mixing time of near-critical random graphs

around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 <xa0δ < 1 so that δ2n → ∞ with n, the mixing-time has order (n/δ) log(δ2n), and exhibits cutoff with constant