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1.
László Erdős Sandrine Péché José A. Ramírez Benjamin Schlein Horng‐Tzer Yau 《纯数学与应用数学通讯》2010,63(7):895-925
We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e?U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C6( \input amssym $\Bbb R$ ) with at most polynomially growing derivatives and ν(x) ≥ Ce?C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc. 相似文献
2.
We prove the Wigner‐Dyson‐Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.© 2016 Wiley Periodicals, Inc. 相似文献
3.
4.
James T. Albrecht Cy P. Chan Alan Edelman 《Foundations of Computational Mathematics》2009,9(4):461-483
This paper proposes that the study of Sturm sequences is invaluable in the numerical computation and theoretical derivation
of eigenvalue distributions of random matrix ensembles. We first explore the use of Sturm sequences to efficiently compute
histograms of eigenvalues for symmetric tridiagonal matrices and apply these ideas to random matrix ensembles such as the
β-Hermite ensemble. Using our techniques, we reduce the time to compute a histogram of the eigenvalues of such a matrix from
O(n
2+m) to O(mn) time where n is the dimension of the matrix and m is the number of bins (with arbitrary bin centers and widths) desired in the histogram (m is usually much smaller than n). Second, we derive analytic formulas in terms of iterated multivariate integrals for the eigenvalue distribution and the
largest eigenvalue distribution for arbitrary symmetric tridiagonal random matrix models. As an example of the utility of
this approach, we give a derivation of both distributions for the β-Hermite random matrix ensemble (for general β). Third, we explore the relationship between the Sturm sequence of a random matrix and its shooting eigenvectors. We show using Sturm sequences that assuming the eigenvector contains no zeros, the number of sign changes in a shooting
eigenvector of parameter λ is equal to the number of eigenvalues greater than λ. Finally, we use the techniques presented in the first section to experimentally demonstrate a O(log n) growth relationship between the variance of histogram bin values and the order of the β-Hermite matrix ensemble.
This research was supported by NSF Grant DMS–0411962. 相似文献
5.
6.
We review and study a one-parameter family of functional transformations, denoted by (S
(β))
β∈ℝ, which, in the case β<0, provides a path realization of bridges associated to the family of diffusion processes enjoying the time-inversion property.
This family includes Brownian motions, Bessel processes with a positive dimension and their conservative h-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary-crossing
times for these diffusions over a given function f to those over the image of f by the mapping S
(β), for some fixed β∈ℝ. We give some new examples of boundary-crossing problems for the Brownian motion and the family of Bessel processes. We
also provide, in the Brownian case, an interpretation of the results obtained by the standard method of images and establish
connections between the exact asymptotics for large time of the densities corresponding to various curves of each family. 相似文献
7.
Richard F. Bass Nathalie Eisenbaum Zhan Shi 《Probability Theory and Related Fields》2000,116(3):391-404
Let X be a symmetric stable process of index α∈ (1,2] and let L
x
t
denote the local time at time t and position x. Let V(t) be such that L
t
V(t)
= sup
x∈
ℝ
L
t
x
. We call V(t) the most visited site of X up to time t. We prove the transience of V, that is, lim
t
→∞ |V(t)| = ∞ almost surely. An estimate is given concerning the rate of escape of V. The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension
of the Ray–Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and
further to the winding problem for planar Brownian motion.
Received: 14 October 1998 / Revised version: 8 June 1999 / Published online: 7 February 2000 相似文献
8.
Dong Sheng WU Yi Min XIAO 《数学学报(英文版)》2007,23(4):613-622
Let B^α = {B^α(t),t E R^N} be an (N,d)-fractional Brownian motion with Hurst index α∈ (0, 1). By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension results for the images of B^α when N 〉 αd. Our results extend those of Kaufman for one-dimensional Brownian motion. 相似文献
9.
Jean-Stéphane Dhersin Jean-François Le Gall 《Probability Theory and Related Fields》1997,108(1):103-129
We prove a Wiener-type criterion for super-Brownian motion and the Brownian snake.If F is a Borel subset of ℝ
d
and x ∈ ℝ
d
, we provide a necessary and sufficientcondition for super-Brownian motion started at δ
x
to immediately hit the set F. Equivalently, this condition is necessary and sufficient for the hitting time of F by theBrownian snake with initial point x to be 0. A key ingredient of the proof isan estimate showing that the hitting probability of F is comparable, up to multiplicative constants,to the relevant capacity of F. This estimate, which is of independent interest, refines previous results due to Perkins and Dynkin. An important role is
played by additivefunctionals of the Brownian snake, which are investigated here via the potentialtheory of symmetric Markov
processes. As a direct application of our probabilisticresults, we obtain a necessary and sufficient condition for the existence
in a domain D of a positivesolution of the equation Δ; u = u
2
which explodes at a given point of ∂ D.
Received: 5 January 1996 / In revised form: 30 October 1996 相似文献
10.
We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure ν ij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution ν ij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector–eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk. 相似文献
11.
Any solution of the functional equation
where B is a Brownian motion, behaves like a reflected Brownian motion, except when it attains a new maximum: we call it an α-perturbed
reflected Brownian motion. Similarly any solution of
behaves like a Brownian motion except when it attains a new maximum or minimum: we call it an α,β-doubly perturbed Brownian
motion. We complete some recent investigations by showing that for all permissible values of the parameters α, α and β respectively,
these equations have pathwise unique solutions, and these are adapted to the filtration of B.
Received: 7 November 1997 / Revised version: 13 July 1998 相似文献
12.
L. Saloff-Coste 《Journal of Geometric Analysis》2004,14(4):715-733
Let M =G/H be an irreducible homogeneous compact manifold of dimension n equipped with its canonical Riemannian metric. Let γ be
the lowest nonzero eigenvalue of the Laplace operator. Let μ be the normalized Haar measure and μ
t be the heat diffusion measure, i.e., the law of Brownian motion started at a fixed origin in M. We show that the total variation
distance between μt and μ is not small for t ≪λ
−1 logn.This is sharp, up to a factor of two, in the case of compact irreducible simply connected symmetric spaces. 相似文献
13.
14.
We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the
bond dilute Ising model on ℤ
d
at inverse temperature β. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any β, with probability one, the spectral gap
of the generator of the dyamics in a box of side L centered at the origin scales like L
−2. Such an estimate is then used to prove a decay to equilibrium for local functions of the form where ε is positive and arbitrarily small and α = ? for d = 1, α=1 for d≥2. In particular our result shows that, contrary to what happes for the Glauber dynamics, there is no dynamical phase transition
when β crosses the critical value β
c
of the pure system.
Received: 10 April 2000 / Revised version: 23 October 2000 / Published online: 5 June 2001 相似文献
15.
Dorin Bucur Ioan R. Ionescu 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(6):1042-1056
We consider an eigenvalue problem associated to the antiplane shearing on a system of collinear faults under a slip-dependent
friction law. Firstly we consider a periodic system of faults in the whole plane. We prove that the first eigenvalues/eigenfunctions
of different physical periodicity are all equal and that the other eigenvalues converge to this first common eigenvalue as
their physical period becomes indefinitely large. Secondly we consider a large scale fault system composed on a small scale
collinear faults periodically disposed. If β0* is the first eigenvalue of the periodic problem in the whole plane, we prove that the first eigenvalue of the microscopic
problem behaves as β0*/∈ when ∈→ 0 regardless the geometry of the domain (here ∈ is the scale quotient). The geophysical implications of this result
is that the macroscopic critical slip Dc scales with Dc∈/∈ (here Dc∈ is the small scale critical slip). 相似文献
16.
Consider a system of particles which move in Rd according to a symmetric α-stable motion, have a lifetime distribution of finite mean, and branch with an offspring law of index 1+β. In case of the critical dimension d=α/β the phenomenon of multi-scale clustering occurs. This is expressed in an fdd scaling limit theorem, where initially we start
with an increasing localized population or with an increasing homogeneous Poissonian population. The limit state is uniform,
but its intensity varies in line with the scaling index according to a continuous-state branching process of index 1+β. Our result generalizes the case α=2 of Brownian particles of Klenke (1998), where p.d.e. methods had been used which are not available in the present setting.
Supported in part by the DFG.
Supported in part by the grants RFBR 02-01-00266 and Russian Scientific School 1758.2003.1. 相似文献
17.
Juan Luis Vázquez 《Israel Journal of Mathematics》1982,43(3):255-272
The semilinear perturbation of Poisson’s equation (E): −Δu+β(u)∋f, where β is a maximal monotone graph inR, has been investigated by Ph. Bénilan, H. Brézis and M. Crandall forf∈L
1(R
N
),N≧1, under the assumptions 0∈β(0) ifN≧3 and 0∈β(0) ∩ Int β(R) ifN=1,2. We discuss in this paper the solvability and well-posedness of (E) in terms of any maximal monotone graph β. In particular,
if β takes only positive values andN≧3 we prove that no solution exists; ifN=2 we give necessary and sufficient conditions on β andf for (E) to be solvable in a natural sense. 相似文献
18.
Summary. We consider the superposition of a speeded up symmetric simple exclusion process with a Glauber dynamics, which leads to
a reaction diffusion equation. Using a method introduced in [Y] based on the study of the time evolution of the H
−1
norm, we prove that the mean density of particles on microscopic boxes of size N
α
, for any 12/13<α<1, converges to the solution of the hydrodynamic equation for times up to exponential order in N, provided the initial state is in the basin of attraction of some stable equilibrium of the reaction–diffusion equation.
From this result we obtain a lower bound for the escape time of a domain in the basin of attraction of the stable equilibrium
point.
Received: 3 March 1995 / In revised form: 2 February 1996 相似文献
19.
Let X be the solution of the stochastic differential equation where B
H
is a fractional Brownian motion with Hurst parameter H. In this paper we compute the Onsager-Machlup functional of X for the supremum norm and H?lder norms of order β with in the case and for H?lder norms of order β with when .
Received: 16 July 2001 / Revised version: 12 March 2002 / Published online: 10 September 2002 相似文献
20.
Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1. For 0 < β < ? we show that a function Φ: (0, T) → ℒ(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H-cylindrical fractional Brownian motion. 相似文献