共查询到20条相似文献,搜索用时 31 毫秒
1.
Summary. We study the 2D Ising model in a rectangular box Λ
L
of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑
t∈ΛL
σ(t) when L→∞ for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m
* is strictly positive. We study in particular boundary effects due to an arbitrary real-valued boundary magnetic field. Using
the self-duality of the model a large part of the analysis consists in deriving properties of the covariance function <σ(0)σ(t)>, as |t|→∞, at dual values of the parameters of the model. To do this analysis we establish new results about the high-temperature
representation of the model. These results are valid for dimensions D≥2 and up to the critical temperature. They give a complete non-perturbative exposition of the high-temperature representation.
We then study the Gibbs measure conditioned by {|∑
t∈ΛL
σ(t) −m|Λ
L
||≤|Λ
L
|L
−
c
}, with 0<c<1/4 and −m
*<m<m
*. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric
type.
Received: 17 October 1996 / In revised form: 7 March 1997 相似文献
2.
Starting from the simple symmetric random walk {Sn}n, we introduce a new process whose path measure is weighted by a factor exp with α,h≥0, {Wn}n a typical realization of an IID process and N a positive integer. We are looking for results in the large N limit. This factor favors Sn>0 if Wn+h>0 and Sn<0 if Wn+h<0. The process can be interpreted as a model for a random heterogeneous polymer in the proximity of an interface separating
two selective solvents. It has been shown [6] that this model undergoes a (de)localization transition: more precisely there
exists a continuous increasing function λ↦hc(λ) such that if h<hc(λ) then the model is localized while it is delocalized if h≥hc(λ). However, localization and delocalization were not given in terms of path properties, but in a free energy sense. Later
on it has been shown that free energy localization does indeed correspond to a (strong) form of path localization [3]. On
the other hand, only weak results on the delocalized regime have been known so far.
We present a method, based on concentration bounds on suitably restricted partition functions, that yields much stronger results on the path behavior in the interior of the delocalized region, that
is for h>hc(λ). In particular we prove that, in a suitable sense, one cannot expect more than O( log N) visits of the walk to the lower half plane. The previously known bound was o(N). Stronger O(1)–type results are obtained deep inside the delocalized region.
The same approach is also helpful for a different type of question: we prove in fact that the limit as α tends to zero of
hc(λ)/λ exists and it is independent of the law of ω1, at least when the random variable ω1 is bounded or it is Gaussian. This is achieved by interpolating between this class of variables and the particular case of
ω1 taking values ±1 with probability 1/2, treated in [6]. 相似文献
3.
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
4.
Hyun Jae Yoo 《Mathematische Zeitschrift》2006,252(1):27-48
We consider fermion (or determinantal) random point fields on Euclidean space ℝd. Given a bounded, translation invariant, and positive definite integral operator J on L2(ℝd), we introduce a determinantal interaction for a system of particles moving on ℝd as follows: the n points located at x1,· · ·,xn ∈ ℝd have the potential energy given by where j(x−y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d≥2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I+J)−1 is a Gibbs measure admitted to the specification. 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.
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
7.
Alberto Gandolfi 《Probability Theory and Related Fields》1999,114(4):419-430
This paper studies a particular line in the parameter space of the FK random interaction random cluster model for spin glasses
following Katsura ([K]) and Mazza ([M]). We show that, after averaging over the random couplings, the occupied FK bonds have
exactly a Bernoulli distribution. Comparison with explicit calculations on trees confirms the marginal role of FK percolation
in determining phase transitions.
Received: 1 October 1997 / Revised version: 18 May 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.
Remco van der Hofstad Frank den Hollander Gordon Slade 《Probability Theory and Related Fields》1998,111(2):253-286
Summary. We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding
walk on ℤ
d
where loops of length m are penalised by a factor e
−β/m p
(0<β≪1) when: (1) d>4, p≥0; (2) d≤4, . In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d>4, p=0. In addition, we prove a local central limit theorem, with the exception of the case d>4, p=0.
Received: 29 October 1997 / In revised form: 15 January 1998 相似文献
10.
Christian Mazza 《Probability Theory and Related Fields》1999,113(2):171-190
We consider the ±J spin glass on a finite graph G=(V,E), with i.i.d. couplings. Our approach considers the Z
2 local gauge invariance of the system. We see the gauge group as a graph theoretic linear code ? over GF(2). The gauge is fixed by choosing a convenient linear supplement of ?. Assuming some relation between the disorder parameter
p and the inverse temperature of the thermal bath β
pb
, we study percolation in the random interaction random cluster model, and extend the results to dilute spin glasses.
Received: 5 May 1997 / Revised version: 9 April 1998 相似文献
11.
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
12.
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
13.
László Erdős 《Probability Theory and Related Fields》1998,112(3):321-371
We obtain the Lifschitz tail, i.e. the exact low energy asymptotics of the integrated density of states (IDS) of the two-dimensional
magnetic Schr?dinger operator with a uniform magnetic field and random Poissonian impurities. The single site potential is
repulsive and it has a finite but nonzero range. We show that the IDS is a continuous function of the energy at the bottom
of the spectrum. This result complements the earlier (nonrigorous) calculations by Brézin, Gross and Itzykson which predict
that the IDS is discontinuous at the bottom of the spectrum for zero range (Dirac delta) impurities at low density. We also
elucidate the reason behind this apparent controversy. Our methods involve magnetic localization techniques (both in space
and energy) in addition to a modified version of the “enlargement of obstacles” method developed by A.-S. Sznitman.
Received: 20 July 1997 / Revised version: 20 April 1998 相似文献
14.
Y. Kifer 《Probability Theory and Related Fields》1995,103(2):223-248
Summary I introduce random multidimensional subshifts of finite type which generalize models of spin-glasses and establish the “almost
sure” large deviations bounds for Gibbs measures there. The paper is sequel to [EKW] where the corresponding results were
obtained for deterministic multidimensional subshifts of finite type.
Partially supported by US-Israel BSF 相似文献
15.
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
16.
Let M(N) be a sequence of integers with M→∞ as N→∞ and M=o(N). For bounded i.i.d. r.v. ξ
i
k
and bounded i.i.d. r.v. σ
i
, we study the large deviation of the family of (ordered) scalar products X
k
=N
−1∑
i
=1
N
σ
i
ξ
i
k
,k≤M, under the distribution conditioned on the ξ
i
k
's. To get a full large deviation principle, it is necessary to specify also the total norm(∑
k
≤
M
(X
k
)2)1/2, which turns to be associated with some extra Gaussian distribution. Our results apply to disordered, mean-field systems,
including generalized Hopfield models in the regime of a sublinear number of patterns. We build also a class of examples where
this norm is the crucial order parameter.
Received: 6 April 1999 / Revised version: 29 May 2000 /?Published online: 24 July 2001 相似文献
17.
M.S. Bernabei 《Probability Theory and Related Fields》2001,119(3):410-432
The Central Limit Theorem for a model of discrete-time random walks on the lattice ℤν in a fluctuating random environment was proved for almost-all realizations of the space-time nvironment, for all ν > 1 in
[BMP1] and for all ν≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for
ν≥ 3 is finite. In the present paper we consider the cases ν = 1,2 and prove the Central Limit Theorem as T→∞ for the random correction to the first two cumulants. The rescaling factor for theaverage is for ν = 1 and (ln T), for ν=2; for the covariance it is , ν = 1,2.
Received: 25 November 1999 / Revised version: 7 June 2000 / Published online: 15 February 2001 相似文献
18.
Martin Grothaus Yuri G. Kondratiev Michael Röckner 《Probability Theory and Related Fields》2007,137(1-2):121-160
We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle
systems on ℝ
d
,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ
d
with finite volume (Lebesgue measure) V=|Λ|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate
relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are
identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves
are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a
property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques
we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one
invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides
as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary
condition. 相似文献
19.
Summary. Standard large deviation estimates or the use of the Hubbard–Stratonovich transformation reduce the analysis of the distribution
of the overlap parameters essentially to that of an explicitly known random function Φ
N,β
on ℝ
M
. In this article we present a rather careful study of the structure of the minima of this random function related to the
retrieval of the stored patterns. We denote by m
*
(β ) the modulus of the spontaneous magnetization in the Curie–Weiss model and by α the ratio between the number of the stored patterns and the system size. We show that there exist strictly positive numbers
0 < γ
a
< γ
c
such that (1) If √α≦γ
a
(m
*
(β ))
2
, then the absolute minima of Φ are located within small balls around the points ± m
*
e
μ
, where e
μ
denotes the μ-th unit vector while (2) if √α≦γ
c
(m
*
(β ))
2
at least a local minimum surrounded by extensive energy barriers exists near these points. The random location of these minima
is given within precise bounds. These are used to prove sharp estimates on the support of the Gibbs measures.
Received: 5 August 1995 / In revised form: 22 May 1996 相似文献
20.
Alain Toubol 《Probability Theory and Related Fields》1998,110(4):497-534
Comets and Neveu have initiated in [5] a method to prove convergence of the partition function of disordered systems to a
log-normal random variable in the high temperature regime by means of stochastic calculus. We generalize their approach to
a multidimensional Sherrington-Kirkpatrick model with an application to the Heisenberg model of uniform spins on a sphere
of ℝ
d
, see [9]. The main tool that we use is a truncation of the partition function outside a small neighbourhood of the typical
energy path.
Received: 30 October 1996 / In revised form: 13 October 1997 相似文献