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1.
László Erdo˝s 《Probability Theory and Related Fields》2001,121(2):219-236
We establish the exact low-energy asymptotics of the integrated density of states (Lifschitz tail) in a homogeneous magnetic
field and Poissonian impurities with a repulsive single-site potential of Gaussian decay. It has been known that the Gaussian
potential tail discriminates between the so-called “classical” and “quantum” regimes, and precise asymptotics are known in
these cases. For the borderline case, the coexistence of the classical and quantum regimes was conjectured. Here we settle
this last remaining open case to complete the full picture of the magnetic Lifschitz tails.
Received: 28 March 2000 / Revised version: 22 December 2000 / Published online: 24 July 2001 相似文献
2.
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
3.
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
4.
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
5.
We consider problems of the following type. Assign independently to each vertex of the square lattice the value +1, with
probability p, or −1, with probability 1 −p. We ask whether an infinite path π exists, with the property that the partial sums of the ±1s along π are uniformly bounded,
and whether there exists an infinite path π' with the property that the partial sums along π' are equal to zero infinitely
often. The answers to these question depend on the type of path one allows, the value of p and the uniform bound specified. We show that phase transitions occur for these phenomena. Moreover, we make a surprising
connection between the problem of finding a path to infinity (not necessarily self-avoiding, but visiting each vertex at most
finitely many times) with a given bound on the partial sums, and the classical Boolean model with squares around the points
of a Poisson process in the plane. For the recurrence problem, we also show that the probability of finding such a path is
monotone in p, for p≥?.
Received: 10 January 2000 / Revised version: 14 August 2000 / Published online: 9 March 2001 相似文献
6.
József Lörinczi Christian Maes Koen Vande Velde 《Probability Theory and Related Fields》1998,112(1):121-147
We study local transformations of Gibbs measures. We establish sufficient conditions for the quasilocality of the images
and obtain results on the existence and continuity properties of their relative energies. General results are illustrated
by simple examples.
Received: 11 November 1997 / Revised version: 20 February 1998 相似文献
7.
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
8.
We consider the low temperature Ising model in a uniform magnetic field h > 0 with minus boundary conditions and conditioned on having no internal contours. This simple contour model defines a non-Gibbsian
spin state. For large enough magnetic fields (h >: h
c
) this state is concentrated on the single spin configuration of all spins up. For smaller values (h≤h
c
), the spin state is non-trivial. At the critical point h
c
≠ 0 the magnetization jumps discontinuously. Freezing provides also an example of a translation invariant weakly Gibbsian
state which is not almost Gibbsian.
Received: 10 November 1998 相似文献
9.
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
10.
M. E. Palistrant 《Theoretical and Mathematical Physics》2011,168(3):1290-1302
We investigate phase transitions in quasi-two-dimensional systems with an anisotropic energy spectrum and a deviation from
the half-filling of the energy band (μ ≠ 0). We demonstrate the possibility of the transition of an insulator into a half-metallic state when the nesting condition
is violated because the parameter μ ≠ 0 and of taking the umklapp processes into account. We obtain the basic equations for
the parameters of the superconducting (Δ) and magnetic (M) orders and determine the conditions for the emergence of superconductivity on the background of a spin-density-wave
state and also for the coexistence of superconductivity and magnetism. We show that the transition of a magnetic system into
a superconducting state as the parameter μ increases can be a first-order phase transition at low temperatures. We also obtain
an expression for the heat capacity jump C
S
-C
N
at T = T
c
, which depends on M and μ and differs essentially from the case of the Bardeen-Cooper-Schrieffer theory. We also consider
the transformations related to the density of electron states of the relevant anisotropic system, which can undergo essential
changes under pressure or doping. 相似文献
11.
Hirofumi Osada 《Probability Theory and Related Fields》1998,112(1):53-90
We prove the positivity of the self-diffusion matrix of interacting Brownian particles with hard core when the dimension of
the space is greater than or equal to 2. Here the self-diffusion matrix is a coefficient matrix of the diffusive limit of
a tagged particle. We will do this for all activities, z>0, of Gibbs measures; in particular, for large z– the case of high density particles. A typical example of such a particle system is an infinite amount of hard core Brownian
balls.
Received: 22 September 1997 / Revised version: 15 January 1998 相似文献
12.
Summary. We study the stationary measures of an infinite Hamiltonian system of interacting particles in ℝ
3
subject to a stochastic local perturbation conserving energy and momentum. We prove that the translation invariant measures
that are stationary for the deterministic Hamiltonian dynamics, reversible for the stochastic dynamics, and with finite entropy
density, are convex combination of “Gibbs” states. This result implies hydrodynamic behavior for the systems under consideration.
Received: 17 December 1994/In revised form: 12 April 1996 相似文献
13.
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
14.
《偏微分方程通讯》2013,38(7-8):1187-1213
Abstract We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent 0 < q < 1. The random Anderson-type potential is constructed with a nonnegative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly supported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179--209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is Hölder continuous as a function of the nonzero magnetic field strength. 相似文献
15.
Consider the standard continuous percolation in ℝ4, and choose the parameters so that the induced percolation on a fixed two dimensional linear subspace is critical. Although
two dimensional critical percolation dies, we show that there are exceptional two dimensional linear subspaces, in which percolation
occurs.
Received: 1 April 1997 / Revised version: 20 January 1998 相似文献
16.
Chih-Chung Chang Claudio Landim Stefano Olla 《Probability Theory and Related Fields》2001,119(3):381-409
We consider an asymmetric exclusion process in dimension d≥ 3 under diffusive rescaling starting from the Bernoulli product measure with density 0 < α < 1. We prove that the density
fluctuation field Y
N
t
converges to a generalized Ornstein–Uhlenbeck process, which is formally the solution of the stochastic differential equatin
dY
t
= ?Y
t
dt + dB
∇
t
, where ? is a second order differential operator and B
∇
t
is a mean zero Gaussian field with known covariances.
Received: 31 May 1999 / Revised version: 15 June 2000 / Published online: 24 January 2001 相似文献
17.
Approximation of a laminated microstructure for a rotationally invariant,double well energy density 总被引:3,自引:0,他引:3
Mitchell Luskin 《Numerische Mathematik》1996,75(2):205-221
Summary. We give error estimates for the approximation of a laminated microstructure which minimizes the energy for a rotationally invariant, double well energy density . We present error estimates for the convergence of the deformation in the convergence of directional derivatives of the deformation in the “twin planes,” the weak convergence of the deformation
gradient, the convergence of the microstructure (or Young measure) of the deformation gradients, and the convergence of nonlinear
integrals of the deformation gradient.
Received July 25, 1995 / Revised version received November 20, 1995 相似文献
18.
F. den Hollander F.R. Nardi E. Olivieri E. Scoppola 《Probability Theory and Related Fields》2003,125(2):153-194
The goal of this paper is to describe metastability and nucleation for a local version of the three-dimensional lattice gas
with Kawasaki dynamics at low temperature and low density.
Let $\Lambda\subseteq{\mathbb Z}^3$ be a large finite box. Particles perform simple exclusion on $\Lambda$, but when they
occupy neighboring sites they feel a binding energy $-U<0$ that slows down their dissociation. Along each bond touching the
boundary of $\Lambda$ from the outside, particles are created with rate $\rho=e^{-\Delta\beta}$ and are annihilated with rate
1, where $\beta$ is the inverse temperature and $\D>0$ is an activity parameter. Thus, the boundary of $\Lambda$ plays the
role of an infinite gas reservoir with density $\rho$.
We consider the regime where $\Delta\in (U,3U)$ and the initial configuration is such that $\Lambda$ is empty. For large $\beta$,
the system wants to fill $\Lambda$ but is slow in doing so. We investigate how the transition from empty to full takes place
under the dynamics. In particular, we identify the size and shape of the critical droplet\/ and the time of its creation in the limit as $\beta\to\infty$.
Received: 23 February 2002 / Revised version: 24 June 2002 / Published online: 24 October 2002
Mathematics Subject Classification (2000): 60K35, 82B43, 82C43, 82C80
Key words or phrases: Lattice gas – Kawasaki dynamics – Metastability – Critical droplet – Large deviations – Discrete isoperimetric inequalities 相似文献
19.
Mario V. Wüthrich 《Probability Theory and Related Fields》1998,112(3):299-319
We consider d-dimensional Brownian motion in a truncated Poissonian potential (d≥ 2). If Brownian motion starts at the origin and ends in the closed ball with center y and radius 1, then the transverse fluctuation of the path is expected to be of order |y|ξ, whereas the distance fluctuation is of order |y|χ. Physics literature tells us that ξ and χ should satisfy a scaling identity 2ξ− 1 = χ. We give here rigorous results for
this conjecture.
Received: 31 December 1997 / Revised version: 14 April 1998 相似文献
20.
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