Showing papers by "Van Vu published in 2014"

A simple SVD algorithm for finding hidden partitions

Abstract: Finding a hidden partition in a random environment is a general and important problem, which contains as subproblems many famous questions, such as finding a hidden clique, finding a hidden coloring, finding a hidden bipartition etc. In this paper, we provide a simple SVD algorithm for this purpose, answering a question of McSherry. This algorithm is very easy to implement and works for sparse graphs with optimal density.

Random matrices: Law of the determinant

Abstract: Let $A_{n}$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_{n}|$ satisfies a central limit theorem. More precisely, \begin{eqnarray*}&&\sup_{x\in{\mathbf {R}}}\biggl|{\mathbf{P} }\biggl(\frac{\log(|\det A_{n}|)-({1}/{2})\log(n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf{P} }\bigl(\mathbf{N} (0,1)\le x\bigr)\biggr|\\&&\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}

Empirical Test on the Liquidity-Adjusted Capital Asset Pricing Model

Abstract: In this study, we examine the effects of systematic liquidity risk on stock returns in the Australian market We find that liquidity risk, in the form of (i) the co-movement between individual stock liquidity and market liquidity, (ii) the co-movement between stock returns and market liquidity, and (iii) the co-movement between stock liquidity and market returns, is priced individually and jointly in Australian equities The results are robust to the use of alternative liquidity proxies and after controlling for other factors that are known to affect stock returns The analysis across different market conditions shows that the net liquidity risk is approximately eight times higher in bearish markets than in bullish markets Our overall results support the importance of liquidity risk in the generation of stock returns, particularly during market downturns

Universality of local eigenvalue statistics in random matrices with external source

Abstract: Consider a random matrix of the form , where Mn is a Wigner matrix and Dn is a real deterministic diagonal matrix (Dn is commonly referred to as an external source in the mathematical physics literature). We study the universality of the local eigenvalue statistics of Wn for a general class of Wigner matrices Mn and diagonal matrices Dn. Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrates the remarkable phenomenon that local laws are more resilient than global ones. The universality of the correlation functions follows from a four moment theorem, which we prove using a variant of the approach used earlier by Tao and Vu.

Random matrices have simple spectrum

Abstract: Let $M_n = (\xi_{ij})_{1 \leq i,j \leq n}$ be a real symmetric random matrix in which the upper-triangular entries $\xi_{ij}, i

Real roots of random polynomials: expectation and repulsion

Abstract: Let $P_{n}(x)= \sum_{i=0}^n \xi_i x^i$ be a Kac random polynomial where the coefficients $\xi_i$ are iid copies of a given random variable $\xi$. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a double root. As an application, we consider the problem of estimating the number of real roots of $P_n$, which has a long history and in particular was the main subject of a celebrated series of papers by Littlewood and Offord from the 1940s. We show, for a large and natural family of atom variables $\xi$, that the expected number of real roots of $P_n(x)$ is exactly $\frac{2}{\pi} \log n +C +o(1)$, where $C$ is an absolute constant depending on the atom variable $\xi$. Prior to this paper, such a result was known only for the case when $\xi$ is Gaussian.

Products of independent elliptic random matrices

Abstract: For fixed $m > 1$, we study the product of $m$ independent $N \times N$ elliptic random matrices as $N$ tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $1$, to the $m$-th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. Our result also generalizes earlier results of G\"otze-Tikhomirov and O'Rourke-Soshnikov concerning the product of independent iid random matrices.

On the number of real roots of random polynomials

Abstract: Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the expectation of the number of real roots is $\frac{2}{\pi} \log n + o(\log n)$. In this paper, we determine the true nature of the error term by showing that the expectation equals $\frac{2}{\pi}\log n + O(1)$. Prior to this paper, such estimate has been known only in the gaussian case, thanks to works of Edelman and Kostlan.

Random walks with different directions: Drunkards beware !

Abstract: As an extension of Polya's classical result on random walks on the square grids ($\Z^d$), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability after $n$ steps is at most $n^{-d/2 - d/(d-2) +o(1)}$, which is sharp. The real surprise is in dimensions 2 and 3. In dimension 2, where the traditional grid walk is recurrent, our upper bound is $n^{-\omega (1)}$, which is much worse than higher dimensions. In dimension 3, we prove an upper bound of order $n^{-4 +o(1)}$. We discover a new conjecture concerning incidences between spheres and points in $\R^3$, which, if holds, would improve the bound to $n^{-9/2 +o(1)}$, which is consistent % with the $d \ge 4$ case. to the $d \ge 4$ case. This conjecture resembles Szemer\'edi-Trotter type results and is of independent interest.