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1.
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
2.
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
3.
We consider a mechanical model in the plane, consisting of a vertical rod, subject to a constant horizontal force f and to elastic collisions with the particles of a free gas which is “horizontally” in equilibrium at some inverse temperature
β. In a previous paper we proved that, in the appropriate space and time scaling, the motion of the rod is described as a drift
term plus a diffusion term. In this paper we prove that the drift d(f) and the diffusivity σ
2
(f) are continuous functions of f, and moreover that the Einstein relation holds, i.e.,
lim
f → 0
d(f)f = β2 σ
2
(0) .
Received: 26 January 1996 / In revised form: 2 October 1996 相似文献
4.
Martin T. Barlow Robin Pemantle Edwin A. Perkins 《Probability Theory and Related Fields》1997,107(1):1-60
Summary. We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to α
−n
, where α < 1 is a positive real parameter. The heights of these clusters are shown to increase linearly with their total size; this complements
known results that show the height increases only logarithmically when α≧ 1. Results are obtained using stochastic monotonicity and regeneration results which may be of independent interest. Our motivation
comes from two other ways in which the model may be viewed: as a problem in first-passage percolation, and as a version of
diffusion-limited aggregation (DLA), adjusted so that “fingering” occurs.
Received: 16 August 1994 / In revised form: 18 March 1996 相似文献
5.
The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated
as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution
at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation
of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence
the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it
can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled
systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which
enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic
fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities”
argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes
into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP
scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the
“acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach
of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization
of the Godunov scheme. 相似文献
6.
7.
Andreas Greven Achim Klenke Anton Wakolbinger 《Probability Theory and Related Fields》2001,120(1):85-117
We study the longtime behaviour of interacting systems in a randomly fluctuating (space–time) medium and focus on models
from population genetics. There are two prototypes of spatial models in population genetics: spatial branching processes and
interacting Fisher–Wright diffusions. Quite a bit is known on spatial branching processes where the local branching rate is
proportional to a random environment (catalytic medium).
Here we introduce a model of interacting Fisher–Wright diffusions where the local resampling rate (or genetic drift) is proportional
to a catalytic medium. For a particular choice of the medium, we investigate the longtime behaviour in the case of nearest
neighbour migration on the d-dimensional lattice.
While in classical homogeneous systems the longtime behaviour exhibits a dichotomy along the transience/recurrence properties
of the migration, now a more complicated behaviour arises. It turns out that resampling models in catalytic media show phenomena
that are new even compared with branching in catalytic medium.
Received: 15 November 1999 / Revised version: 16 June 2000 / Published online: 6 April 2001 相似文献
8.
In this paper, we consider a new class of random dynamical systems that contains, in particular, neural networks and complicated
circuits. For these systems, we consider the viability problem: we suppose that the system survives only the system state
is in a prescribed domain Π of the phase space. The approach developed here is based on some fundamental ideas proposed by
A. Kolmogorov, R. Thom, M. Gromov, L. Valiant, L. Van Valen, and others. Under some conditions it is shown that almost all
systems from this class with fixed parameters are unstable in the following sense: the probability P
t
to leave Π within the time interval [0, t] tends to 1 as t → ∞. However, it is allowed to change these parameters sometimes (“evolutionary” case), then it may happen that P
t
< 1 − δ < 1 for all t (“stable evolution”). Furthermore, we study the properties of such a stable evolution assuming that the system parameters
are encoded by a dicsrete code. This allows us to apply complexity theory, coding, algorithms, etc. Evolution is a Markov
process of modification of this code. Under some conditions we show that the stable evolution of unstable systems possesses
the following general fundamental property: the relative Kolmogorov complexity of the code cannot be bounded by a constant
as t → ∞. For circuit models, we define complexity characteristics of these circuits. We find that these complexities also have
a tendency to increase during stable evolution. We give concrete examples of stable evolution. Bibliography: 80 titles.
To the memory of A. N. Livshitz
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 31–69. 相似文献
9.
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
10.
By a “reproducing” method forH =L
2(ℝ
n
) we mean the use of two countable families {e
α : α ∈A}, {f
α : α ∈A}, inH, so that the first “analyzes” a function h ∈H by forming the inner products {<h,e
α >: α ∈A} and the second “reconstructs” h from this information:h = Σα∈A <h,e
α >:f
α.
A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature
in common: they are generated by a single or a finite collection of functions by applying to the generators two countable
families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor
systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety
of wavelets) involve translations and dilations.
A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article
we establish a result that “unifies” all of these characterizations by means of a relatively simple system of equalities.
Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach
that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need
not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on ℝ
n
. Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations
for different kinds of dilation matrices. 相似文献
11.
Alain-Sol Sznitman 《Probability Theory and Related Fields》1999,115(3):287-323
We consider a d-dimensional random walk in random environment for which transition probabilities at each site are either neutral or present
an effective drift “pointing to the right”. We obtain large deviation estimates on the probability that the walk moves in
a too slow ballistic fashion, both under the annealed and quenched measures. These estimates underline the key role of large
neutral pockets of the medium in the occurrence of slowdowns of the walk.
Received: 12 March 1998 / Revised version: 19 February 1999 相似文献
12.
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
13.
Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of
their ergodic invariant measures. The question arises whether a.s.the process eventually stays close to one of these ergodic
states, or if it changes between the attainable ergodic states infinitely often (“recurrence”). Under the assumption that
there exists a convergence–determining class of distributions that is (strongly) preserved under the dynamics, we show that
the system is in fact recurrent in the above sense.
We apply our method to several interacting particle systems, obtaining new or improved recurrence results. In addition, we
answer a question raised by Ed Perkins concerning the change of the locally predominant type in a model of mutually catalytic
branching.
Received: 22 January 1999 / Revised version: 24 May 1999 相似文献
14.
Rainer Buckdahn Marc Quincampoix Aurel Răşcanu 《Probability Theory and Related Fields》2000,116(4):485-504
In the present paper, we study conditions under which the solutions of a backward stochastic differential equation remains
in a given set of constraints. This property is the so-called “viability property”. In a separate section, this condition
is translated to a class of partial differential equations.
Received: 23 April 1998 / Published online: 14 February 2000 相似文献
15.
In this paper, we prove the validity of the Chern conjecture in affine geometry [18], namely that an affine maximal graph
of a smooth, locally uniformly convex function on two dimensional Euclidean space, R
2, must be a paraboloid. More generally, we shall consider the n-dimensional case, R
n
, showing that the corresponding result holds in higher dimensions provided that a uniform, “strict convexity” condition holds.
We also extend the notion of “affine maximal” to non-smooth convex graphs and produce a counterexample showing that the Bernstein
result does not hold in this generality for dimension n≥10.
Oblatum 16-IV-1999 & 4-XI-1999?Published online: 21 February 2000 相似文献
16.
The main result in this paper states that if a one-parameter Gaussian process has C
2k
paths and satisfies a non-degeneracy condition, then the distribution of its maximum on a compact interval is of class C
k
. The methods leading to this theorem permit also to give bounds on the successive derivatives of the distribution of the
maximum and to study their asymptotic behaviour as the level tends to infinity.
Received: 14 May 1999 / Revised version: 18 October 1999 / Published online: 14 December 2000 相似文献
17.
In this paper we consider the issue of sliding motion in Filippov systems on the intersection of two or more surfaces. To
this end, we propose an extension of the Filippov sliding vector field on manifolds of co-dimension p, with p ≥ 2. Our model passes through the use of a multivalued sign function reformulation. To justify our proposal, we will restrict
to cases where the sliding manifold is attractive. For the case of co-dimension p = 2, we will distinguish between two types of attractive sliding manifold: “node-like” and “spiral-like”. The case of node-like
attractive manifold will be further extended to the case of p ≥ 3. Finally, we compare our model to other existing methodologies on some examples. 相似文献
18.
In this paper we present a martingale related to the exit measures of super Brownian motion. By changing measure with this
martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown
to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application
is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain.
The results are similar in flavor to the “immortal particle” picture of conditioned super Brownian motion but more general,
as the change of measure is given by a martingale which need not arise from a single harmonic function.
Received: 27 August 1998 / Revised version: 8 January 1999 相似文献
19.
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