共查询到20条相似文献,搜索用时 78 毫秒
1.
Nathanaël Enriquez Jacques Franchi Yves Le Jan 《Probability Theory and Related Fields》2001,119(2):213-255
Let ℳ be a geometrically finite hyperbolic surface with infinite volume, having at least one cusp. We obtain the limit law
under the Patterson-Sullivan measure on T
1ℳ of the windings of the geodesics of ℳ around the cusps. This limit law is stable with parameter 2δ− 1, where δ is the Hausdorff
dimension of the limit set of the subgroup Γ of M?bius isometries associated with ℳ. The normalization is t
−1/(2δ−1), for geodesics of length t. Our method relies on a precise comparison between geodesics and diffusion paths, for which we need to approach the Patterson-Sullivan
measure mentioned above by measures that are regular along the stable leaves.
Received: 8 October 1999 / Revised version: 2 June 2000 / Published online: 21 December 2000 相似文献
2.
Filippo Cesi 《Probability Theory and Related Fields》2001,120(4):569-584
We show that the entropy functional exhibits a quasi-factorization property with respect to a pair of weakly dependent σ-algebras.
As an application we give a simple proof that the Dobrushin and Shlosmans complete analyticity condition, for a Gibbs specification
with finite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously
proven using several different techniques. The advantage of our approach is that it relies almost entirely on a general property
of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus
discrete and continuous spins can be treated in the same way.
Received: 7 July 2000 / Revised version: 10 October 2000 / Published online: 5 June 2001 相似文献
3.
Anton Bovier Michael Eckhoff Véronique Gayrard Markus Klein 《Probability Theory and Related Fields》2001,119(1):99-161
We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field
models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem
to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin
theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic
sequence of “admissible transitions”. For these admissible transitions we give upper and lower bounds on the expected transition
times that differ only by a constant factor. The distributions of the rescaled transition times are shown to converge to the
exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model.
Received: 26 November 1998 / Revised version: 21 March 2000 / Published online: 14 December 2000 相似文献
4.
Christof Külske 《Probability Theory and Related Fields》2001,119(1):1-30
Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spin-space and disorder-space
be represented as (suitably generalized) Gibbs measures of an “annealed system”? - We prove that there is always a potential
(depending on both spin and disorder variables) that converges absolutely on a set of full measure w.r.t. the joint measure
(“weak Gibbsianness”). This “positive” result is surprising when contrasted with the results of a previous paper [K6], where
we investigated the measure of the set of discontinuity points of the conditional expectations (investigation of “a.s. Gibbsianness”).
In particular we gave natural “negative” examples where this set is even of measure one (including the random field Ising
model). Further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint
potential in terms of the decay of the disorder average over certain quenched correlations. We apply them to various examples.
From this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero.
Our proof uses a martingale argument that allows to cut (an infinite-volume analogue of) the quenched free energy into local
pieces, along with generalizations of Kozlov's constructions.
Received: 11 November 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000
RID="*"
ID="*" Work supported by the DFG Schwerpunkt `Wechselwirkende stochastische Systeme hoher Komplexit?t' 相似文献
5.
We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept
allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function
technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application,
we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for
the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic
equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class
of models.
Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001 相似文献
6.
Kragujevac (M. L. Kragujevac: On the Laplacian energy of a graph, Czech. Math. J. 56(131) (2006), 1207–1213) gave the definition of Laplacian energy of a graph G and proved LE(G) ⩾ 6n-8; equality holds if and only if G = P
n
. In this paper we consider the relation between the Laplacian energy and the chromatic number of a graph G and give an upper bound for the Laplacian energy on a connected graph. 相似文献
7.
We investigate the limiting fluctuations of the order parameter in the Hopfield model of spin glasses and neural networks
with finitely many patterns at the critical temperature 1/β
c
= 1. At the critical temperature, the measure-valued random variables given by the distribution of the appropriately scaled
order parameter under the Gibbs measure converge weakly towards a random measure which is non-Gaussian in the sense that it
is not given by a Dirac measure concentrated in a Gaussian distribution. This remains true in the case of β = β
N
→β
c
= 1 as N→∞ provided β
N
converges to β
c
= 1 fast enough, i.e., at speed ?(1/). The limiting distribution is explicitly given by its (random) density.
Received: 12 May 1998 / Revised version: 14 October 1998 相似文献
8.
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
9.
A. G. Bashkirov 《Theoretical and Mathematical Physics》2006,149(2):1559-1573
To describe a complex system, we propose using the Renyi entropy depending on the parameter q (0 < q ≤ 1) and passing into
the Gibbs-Shannon entropy at q = 1. The maximum principle for the Renyi entropy yields a Renyi distribution that passes into
the Gibbs canonical distribution at q = 1. The thermodynamic entropy of the complex system is defined as the Renyi entropy
for the Renyi distribution. In contrast to the usual entropy based on the Gibbs-Shannon entropy, the Renyi entropy increases
as the distribution deviates from the Gibbs distribution (the deviation is estimated by the parameter η = 1 − q) and reaches
its maximum at the maximum possible value ηmax. As this occurs, the Renyi distribution becomes a power-law distribution. The parameter η can be regarded as an order parameter.
At η = 0, the derivative of the thermodynamic entropy with respect to η exhibits a jump, which indicates a kind of phase transition
into a more ordered state. The evolution of the system toward further order in this phase state is accompanied by an entropy
gain. This means that in accordance with the second law of thermodynamics, a natural evolution in the direction of self-organization
is preferable.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 299–317, November, 2006. 相似文献
10.
Michel Talagrand 《Probability Theory and Related Fields》2001,119(2):187-212
We give a completely rigorous proof that the replica-symmetric solution holds at high enough temperature for the random K-sat problem. The most notable feature of this problem is that the order parameter of the system is a function and not a number.
Received: 21 April 1998 / Revised version: 24 April 2000 / Published online: 21 December 2000 相似文献
11.
Sompolinski and Zippelius (1981) propose the study of dynamical systems whose invariant measures are the Gibbs measures for
(hard to analyze) statistical physics models of interest. In the course of doing so, physicists often report of an “aging”
phenomenon. For example, aging is expected to happen for the Sherrington-Kirkpatrick model, a disordered mean-field model
with a very complex phase transition in equilibrium at low temperature. We shall study the Langevin dynamics for a simplified
spherical version of this model. The induced rotational symmetry of the spherical model reduces the dynamics in question to
an N-dimensional coupled system of Ornstein-Uhlenbeck processes whose random drift parameters are the eigenvalues of certain random
matrices. We obtain the limiting dynamics for N approaching infinity and by analyzing its long time behavior, explain what is aging (mathematically speaking), what causes
this phenomenon, and what is its relationship with the phase transition of the corresponding equilibrium invariant measures.
Received: 8 July 1999 / Revised version: 2 June 2000 / Published online: 6 April 2001 相似文献
12.
Dinah Rosenberg Eilon Solan Nicolas Vieille 《Probability Theory and Related Fields》2001,119(3):433-451
We study stopping games in the setup of Neveu. We prove the existence of a uniform value (in a sense defined below), by allowing
the players to use randomized strategies. In constrast with previous work, we make no comparison assumption on the payoff
processes. Moreover, we prove that the value is the limit of discounted values, and we construct ε-optimal strategies.
Received: 10 May 1999 / Revised version: 18 May 2000 / Published online: 15 February 2001 相似文献
13.
We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words
of length N from an alphabet of k letters. (In the limit as k→∞ these become the corresponding distributions for permutations on N letters.) We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlevé V equations. We show
further that in the weakly increasing case the generating unction gives the distribution of the smallest eigenvalue in the
k×k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N→∞ limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k×k hermitian matrices of trace zero.
Received: 9 September 1999 / Revised version: 24 May 2000 / Published online: 24 January 2001 相似文献
14.
Michel Talagrand 《Probability Theory and Related Fields》2000,117(3):303-360
We prove that, just below the critical temperature, the mean field p-spins interaction model, for p suitably large, spontaneously decomposes into different states. The asymptotic overlaps between any two different states
are zero. Under a mild (unproven) hypothesis on the weight distribution of these states, we prove that they are pure states.
This situation is called in physics “one level of symmetry breaking”.
Received: 15 January 1998 / Revised version: 10 November 1999 / Published online: 21 June 2000 相似文献
15.
We consider d-dimensional Brownian motion in a scaled Poissonian potential and the principal Dirichlet eigenvalue (ground state energy)
of the corresponding Schr?dinger operator. The scaling is chosen to be of critical order, i.e. it is determined by the typical
size of large holes in the Poissonian cloud. We prove existence of a phase transition in dimensions d≥ 4: There exists a critical scaling constant for the potential. Below this constant the scaled infinite volume limit of the
corresponding principal Dirichlet eigenvalue is linear in the scale. On the other hand, for large values of the scaling constant
this limit is strictly smaller than the linear bound. For d > 4 we prove that this phase transition does not take place on that scale. Further we show that the analogous picture holds
true for the partition sum of the underlying motion process.
Received: 10 December 1999 / Revised version: 14 July 2000/?Published online: 15 February 2001 相似文献
16.
Branching cellular automata (BCA) are introduced as generalisations of fractal percolation by admitting neighbour dependence.
We associate sequences of random sets with BCA's and study their convergence. In case of convergence we derive the Hausdorff
dimension of the limit set and of its boundary. To accomplish the latter we proof that the boundary of a set generated by
a BCA is again a set generated by a BCA.
Received: 7 October 1999 / Revised version: 25 August 2000 / Published online: 26 April 2001 相似文献
17.
We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on
graphs, including the simple examples of the discrete cube and the lattice ZZ
d
. Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities
we prove are well adapted to describe the tail behaviour of various functionals such as the graph distance in this setting.
Received: 6 April 1998 / Revised version: 15 March 1999 / Published on line: 14 February 2000 相似文献
18.
Kenneth S. Alexander 《Probability Theory and Related Fields》2001,120(3):395-444
We introduce the asymmetric random cluster (or ARC) model, which is a graphical representation of the Potts lattice gas,
and establish its basic properties. The ARC model allows a rich variety of comparisons (in the FKG sense) between models with
different parameter values; we give, for example, values (β, h) for which the 0‘s configuration in the Potts lattice gas is dominated by the “+” configuration of the (β, h) Ising model. The Potts model, with possibly an external field applied to one of the spins, is a special case of the Potts
lattice gas, which allows our comparisons to yield rigorous bounds on the critical temperatures of Potts models. For example,
we obtain 0.571 ≤ 1 − exp(−β
c
) ≤ 0.600 for the 9-state Potts model on the hexagonal lattice. Another comparison bounds the movement of the critical line
when a small Potts interaction is added to a lattice gas which otherwise has only interparticle attraction. ARC models can
also be compared to related models such as the partial FK model, obtained by deleting a fraction of the nonsingleton clusters
from a realization of the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds on the effects of small
annealed site dilution on the critical temperature of the Potts model.
Received: 27 August 2000 / Revised version: 31 August 2000 / Published online: 8 May 2001 相似文献
19.
20.
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition
densities are proved.
Received: 28 August 1998 / Revised version: 6 September 1999 / Published online: 14 June 2000 相似文献