A continuous variant of the inverse Littlewood-Offord problem for quadratic forms

Motivated by the inverse Littlewood-Offord problem for linear forms, we study the concentration of quadratic forms. We show that if this form concentrates on a small ball with high probability, then the coefficients can be approximated by a sum of additive and algebraic structures.Comment: 17 pages. This is the first part of http://arxiv.org/abs/1101.307

Dynamical systems with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles

We study the long-term qualitative behavior of randomly perturbed dynamical systems. More specifically, we look at limit cycles of stochastic differential equations (SDE) with Markovian switching, in which the process switches at random times among different systems of SDEs, when the switching is fast and the diffusion (white noise) term is small. The system is modeled by $$dX^{\epsilon,\delta}(t)=f(X^{\epsilon,\delta}(t), \alpha^\epsilon(t))dt+\sqrt{\delta}\sigma(X^{\epsilon,\delta}(t), \alpha^\epsilon(t))dW(t) , \ X^\epsilon(0)=x,$$ where $\alpha^\epsilon(t)$ is a finite state space Markov chain with irreducible generator $Q=(q_{ij})$. The relative changing rates of the switching and the diffusion are highlighted by the two small parameters $\epsilon$ and $\delta$. We associate to the system the averaged ODE $$d\bar X(t)=\bar f(\bar X(t))dt, \ X(0)=x,$$ where $\bar f(\cdot)=\sum_{i=1}^{m_0}f(\cdot, i)\nu_i$ and $(\nu_1,\dots,\nu_{m_0})$ is the unique invariant probability measure of the Markov chain with generator $Q$. Suppose that for each pair $(\epsilon,\delta)$ of parameters, the process has an invariant probability measure $\mu^{\epsilon,\delta}$, and that the averaged ODE has a limit cycle in which there is an averaged occupation measure $\mu^0$ for the averaged equation. We are able to prove that if $\bar f$ has finitely many unstable or hyperbolic fixed points, then $\mu^{\epsilon,\delta}$ converges weakly to $\mu^0$ as $\epsilon\to 0$ and $\delta \to 0$. Our results generalize to the setting of state-dependent switching $$\mathbb{P}\{\alpha^\epsilon(t+\Delta)=j~|~\alpha^\epsilon=i, X^{\epsilon,\delta}(s),\alpha^\epsilon(s), s\leq t\}=q_{ij}(X^{\epsilon,\delta}(t))\Delta+o(\Delta),~~ i\neq j$$ as long as the generator $Q(\cdot)=(q_{ij}(\cdot))$ is bounded, Lipschitz, and irreducible for all $x\in\mathbb{R}^d$. We conclude our analysis by studying a predator-prey model.Comment: 40 page

Random doubly stochastic matrices: The circular law

Let $X$ be a matrix sampled uniformly from the set of doubly stochastic matrices of size $n\times n$. We show that the empirical spectral distribution of the normalized matrix $\sqrt{n}(X-{\mathbf {E}}X)$ converges almost surely to the circular law. This confirms a conjecture of Chatterjee, Diaconis and Sly.Comment: Published in at http://dx.doi.org/10.1214/13-AOP877 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices

Let $M_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that $M_n$ is non-singular with probability $1-O(n^{-C})$ for any positive constant $C$. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.Comment: Some minor corrections in Section 10 of v

An equal area law for holographic entanglement entropy of the AdS-RN black hole

The Anti-de Sitter-Reissner-Nordstrom (AdS-RN) black hole in the canonical ensemble undergoes a phase transition similar to the liquid-gas phase transition, i.e. the isocharges on the entropy-temperature plane develop an unstable branch when the charge is smaller than a critical value. It was later discovered that the isocharges on the entanglement entropy-temperature plane also exhibit the same van der Waals-like structure, for spherical entangling regions. In this paper, we present numerical results which sharpen this similarity between entanglement entropy and black hole entropy, by showing that both of these entropies obey Maxwell's equal area law to an accuracy of around 1 %. Moreover, we checked this for a wide range of size of the spherical entangling region, and the equal area law holds independently of the size. We also checked the equal area law for AdS-RN in 4 and 5 dimensions, so the conclusion is not specific to a particular dimension. Finally, we repeated the same procedure for a similar, van der Waals-like transition of the dyonic black hole in AdS in a mixed ensemble (fixed electric potential and fixed magnetic charge), and showed that the equal area law is not valid in this case. Thus the equal area law for entanglement entropy seems to be specific to the AdS-RN background.Comment: 17 pages, multiple figures. v4: matches published versio

Normal vector of a random hyperplane

Let v_1,...,v_{n-1} be n-1 independent vectors in R^n (or C^n). We study x, the unit normal vector of the hyperplane spanned by the v_i. Our main finding is that x resembles a random vector chosen uniformly from the unit sphere, under some randomness assumption on the v_i. Our result has applications in random matrix theory. Consider an n by n random matrix with iid entries. We first prove an exponential bound on the upper tail for the least singular value, improving the earlier linear bound by Rudelson and Vershynin. Next, we derive optimal delocalization for the eigenvectors corresponding to eigenvalues of small modulus.Comment: 24 pages, 2 figures. A portion of this work substantially improves a previous preprint of the first autho