We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine β , a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane.The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine β is continuous in the gap size and β, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at β = 2. Α tThe Brownian carousel and the winding angle α λ *
We consider a tree that grows randomly in time. Each time a new vertex appears, it chooses exactly one of the existing vertices and attaches to it. The probability that the new vertex chooses vertex x is proportional to w(deg(x)), a weight function of the actual degree of x. The weight function w : N → R + is the parameter of the model.In [4] and [11] the authors derive the asymptotic degree distribution for a model that is equivalent to the special case, when the weight function is linear. The proof therein strongly relies on the linear choice of w.Using well established results from the theory of general branching processes we give the asymptotical degree distribution for a wide range of weight functions. Moreover, we provide the asymptotic distribution of the tree itself as seen from a randomly selected vertex. The latter approach gives greater insight to the limiting structure of the tree.Our proof is robust and we believe that the method may be used to answer several other questions related to the model. It relies on the fact that considering the evolution of the random tree in continuous time, the process may be viewed as a general branching process, this way classical results can be applied.
In the multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of Z is occupied by a particle labeled with some number, and two neighboring particles are interchanged at rate one if their labels are in increasing order. Consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [-1,1]. We study the joint distribution of these speeds: the TASEP speed process. We prove that the TASEP speed process is stationary with respect to the multi-type TASEP dynamics. Consequently, every ergodic stationary measure is given as a projection of the speed process measure. This generalizes previous descriptions restricted to finitely many classes. By combining this result with known stationary measures for TASEPs with finitely many types, we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribution is that two speeds are equal with positive probability and for any given particle there are infinitely many others with the same speed. We also study the partially asymmetric simple exclusion process (ASEP). We prove that the states of the ASEP with the above initial configuration, seen as permutations of Z, are symmetric in distribution. This allows us to extend some of our results, including the stationarity and description of all ergodic stationary measures, also to the ASEP.Comment: Published in at http://dx.doi.org/10.1214/10-AOP561 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org
We consider two models of one-dimensional discrete random Schrödinger operatorsHere ω k are independent random variables with mean 0 and variance 1.We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem.In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the β-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.arXiv:1107.3058v1 [math.PR]
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