Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and sample covariance matrices \cite{Baik:2005}. We consider the case when the $n\times n$ external source matrix has two distinct real eigenvalues: $a$ with multiplicity $r$ and zero with multiplicity $n-r$. The source is small in the sense that $r$ is finite or $r=\mathcal O(n^\gamma)$, for $0< \gamma<1$. For a Gaussian potential, P\'ech\'e \cite{Peche:2006} showed that for $|a|$ sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for $|a|$ sufficiently large (the supercritical regime) $r$ eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of $r\times r$ Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.Comment: 41 pages, 4 figure

Random walks and random fixed-point free involutions

A bijection is given between fixed point free involutions of $\{1,2,...,2N\}$ with maximum decreasing subsequence size $2p$ and two classes of vicious (non-intersecting) random walker configurations confined to the half line lattice points $l \ge 1$. In one class of walker configurations the maximum displacement of the right most walker is $p$. Because the scaled distribution of the maximum decreasing subsequence size is known to be in the soft edge GOE (random real symmetric matrices) universality class, the same holds true for the scaled distribution of the maximum displacement of the right most walker.Comment: 10 page

Genetic Classification of Populations using Supervised Learning

There are many instances in genetics in which we wish to determine whether two candidate populations are distinguishable on the basis of their genetic structure. Examples include populations which are geographically separated, case--control studies and quality control (when participants in a study have been genotyped at different laboratories). This latter application is of particular importance in the era of large scale genome wide association studies, when collections of individuals genotyped at different locations are being merged to provide increased power. The traditional method for detecting structure within a population is some form of exploratory technique such as principal components analysis. Such methods, which do not utilise our prior knowledge of the membership of the candidate populations. are termed \emph{unsupervised}. Supervised methods, on the other hand are able to utilise this prior knowledge when it is available. In this paper we demonstrate that in such cases modern supervised approaches are a more appropriate tool for detecting genetic differences between populations. We apply two such methods, (neural networks and support vector machines) to the classification of three populations (two from Scotland and one from Bulgaria). The sensitivity exhibited by both these methods is considerably higher than that attained by principal components analysis and in fact comfortably exceeds a recently conjectured theoretical limit on the sensitivity of unsupervised methods. In particular, our methods can distinguish between the two Scottish populations, where principal components analysis cannot. We suggest, on the basis of our results that a supervised learning approach should be the method of choice when classifying individuals into pre-defined populations, particularly in quality control for large scale genome wide association studies.Comment: Accepted PLOS On

The Statistics of the Points Where Nodal Lines Intersect a Reference Curve

We study the intersection points of a fixed planar curve $\Gamma$ with the nodal set of a translationally invariant and isotropic Gaussian random field \Psi(\bi{r}) and the zeros of its normal derivative across the curve. The intersection points form a discrete random process which is the object of this study. The field probability distribution function is completely specified by the correlation G(|\bi{r}-\bi{r}'|) = . Given an arbitrary G(|\bi{r}-\bi{r}'|), we compute the two point correlation function of the point process on the line, and derive other statistical measures (repulsion, rigidity) which characterize the short and long range correlations of the intersection points. We use these statistical measures to quantitatively characterize the complex patterns displayed by various kinds of nodal networks. We apply these statistics in particular to nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of special interest is the observation that for monochromatic random waves, the number variance of the intersections with long straight segments grows like $L \ln L$, as opposed to the linear growth predicted by the percolation model, which was successfully used to predict other long range nodal properties of that field.Comment: 33 pages, 13 figures, 1 tabl

Vicious Walkers and Hook Young Tableaux

We consider a generalization of the vicious walker model. Using a bijection map between the path configuration of the non-intersecting random walkers and the hook Young diagram, we compute the probability concerning the number of walker's movements. Applying the saddle point method, we reveal that the scaling limit gives the Tracy--Widom distribution, which is same with the limit distribution of the largest eigenvalues of the Gaussian unitary ensemble.Comment: 23 pages, 5 figure

Best-shot versus weakest-link in political lobbying: an application of group all-pay auction

We analyze a group political lobbying all-pay auction with a group specific public good prize, in which one group follows a weakest-link and the other group follows a best-shot impact function. We completely characterize all semi-symmetric equilibria. There are two types of equilibria: (1) each player in the best-shot group puts mass at the upper bound of the support, whereas each player in the other group puts mass at the lower bound of the support; (2) players in the best-shot group put masses at both the lower and the upper bounds, while the other group randomizes without a mass point. An earlier and longer version of this study was circulated under the title “The Group All-pay Auction with Heterogeneous Impact Functions.” We appreciate the comments of an Associate Editor and two anonymous referees, Kyung Hwan Baik, Walter Enders, Matt Van Essen, Paan Jindapon, David Malueg, Paul Pecorino, Seth Streitmatter, Ted Turocy, the participants at the 2015 conference of ‘Contest: Theory and Evidence’ at the University of East Anglia, and the seminar participants at the University of Alabama and Korea University. Iryna Topolyan gratefully acknowledges the support from the Charles Phelps Taft Research Center. Any remaining errors are our own

Characteristic Polynomials of Sample Covariance Matrices: The Non-Square Case

We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are given for real sample covariance matrices

Last passage percolation and traveling fronts

We consider a system of N particles with a stochastic dynamics introduced by Brunet and Derrida. The particles can be interpreted as last passage times in directed percolation on {1,...,N} of mean-field type. The particles remain grouped and move like a traveling wave, subject to discretization and driven by a random noise. As N increases, we obtain estimates for the speed of the front and its profile, for different laws of the driving noise. The Gumbel distribution plays a central role for the particle jumps, and we show that the scaling limit is a L\'evy process in this case. The case of bounded jumps yields a completely different behavior