共查询到20条相似文献,搜索用时 11 毫秒
1.
2.
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
3.
Nobuo Yoshida 《Probability Theory and Related Fields》1999,115(1):1-40
We consider a ferromagnetic spin system with unbounded interactions on the d-dimensional integer lattice (d > 1). Under mild assumptions on the one-body interactions (so that arbitrarily deep double wells are allowed), we prove that
if the coupling constants are small enough, then the finite volume Gibbs states satisfy the log-Sobolev inequality uniformly
in the volume and the boundary condition.
Received: 11 November 1997 / Revised version: 17 July 1998 相似文献
4.
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
5.
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
6.
Stochastic Ising and voter models on ℤ
d
are natural examples of Markov processes with compact state spaces. When the initial state is chosen uniformly at random,
can it happen that the distribution at time t has multiple (subsequence) limits as t→∞? Yes for the d = 1 Voter Model with Random Rates (VMRR) – which is the same as a d = 1 rate-disordered stochastic Ising model at zero temperature – if the disorder distribution is heavy-tailed. No (at least in a weak sense) for the VMRR when the tail is light or d≥ 2. These results are based on an analysis of the “localization” properties of Random Walks with Random Rates.
Received: 10 August 1998 相似文献
7.
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
8.
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
9.
We demonstrate the or mapping properties of several rough operators. In all cases these estimates are sharp in the sense that the Lorentz exponent
2 cannot be replaced by any lower number.
Received December 10, 1999 / Published online April 12, 2001 相似文献
10.
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
11.
Let f be an entire transcendental function with rational coefficients in its power series about the origin. Further, let f satisfy a functional equation f(qz)= (z−c)f(z)+Q(z) with and some particular c∈ℚ. Then the linear independence of 1,f(α), f(−α) over ℚ for non-zero α∈ℚ is proved, and a linear independence measure for these numbers is given. Clearly, for Q= 0 the function f can be written as an infinite product.
Received: 19 September 2000 / Revised version: 14 March 2001 相似文献
12.
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
13.
Kenneth S. Alexander 《Probability Theory and Related Fields》1998,110(4):441-471
Summary. For lattice models on ℤ
d
, weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance
from that region. For a wide class of models on ℤ2, including all finite range models, we show that weak mixing is a consequence of Gibbs uniqueness, exponential decay of an
appropriate form of connectivity, and a natural coupling property. In particular, on ℤ2, the Fortuin-Kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially,
and the q-state Potts model above the critical temperature is weak mixing whenever correlations decay exponentially, a hypothesis satisfied
if q is sufficiently large. Ratio weak mixing is the property that uniformly over events A and B occurring on subsets Λ and Γ, respectively, of the lattice, |P(A∩B)/P(A)P(B)−1| decreases exponentially in the distance between Λ and Γ. We show that under mild hypotheses, for example finite range,
weak mixing implies ratio weak mixing.
Received: 27 August 1996 / In revised form: 15 August 1997 相似文献
14.
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
15.
Radha Kessar 《manuscripta mathematica》2001,104(2):145-162
T. Jost proved that Donovan's conjecture holds for the unipotent blocks of the finite general linear groups over a fixed
field. In this paper, we show that the Morita equivalences exhibited by Jost are in fact equivalences between the source algebras
of the corresponding blocks, and thus that Puig's conjecture holds for the unipotent blocks of finite general groups over
a fixed field.
Received: 3 October 2000 相似文献
16.
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
17.
This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n,?) with a Siegel modular form of degree n for the full modular group Γ
n
=Sp(2n,ℤ), generalizing the well-known procedure for n=1. This will show that the so-called “standard” and ldquo;spinor”L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type
F
<4, is then used to establish meromorphic continuation for the spinor L-function when n=3.
Received: 28 March 2000 / Revised version: 25 October 2000 相似文献
18.
Knut Smoczyk 《Mathematische Zeitschrift》2002,240(4):849-883
We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. .
Received: 11 April 2001 / Published online: 29 April 2002 相似文献
19.
Annalisa Baldi Bruno Franchi 《Calculus of Variations and Partial Differential Equations》2003,16(3):283-298
Abstract. In this paper we study the notion of perimeter associated with doubling metric measures or strongly weights. We prove that the metric perimeter in the sense of L. Ambrosio and M. Miranda jr. coincides with the metric Minkowski
content and can be obtained also as a -limit of Modica-Mortola type degenerate integral functionals.
Received: 27 August 2001 / Accepted: 29 November 2001 / Published online: 10 June 2002
Investigation supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy. The authors
are very grateful to Luigi Ambrosio and Francesco Serra Cassano for making their preprints available to them, for listening
with patience and for many unvaluable suggestions. 相似文献
20.
We prove that a for a mapping f of finite distortion , the -Hausdorff measure of any point preimage is zero provided is integrable, with , and the multiplicity function of f is essentially bounded. As a consequence for we obtain that the mapping is then open and discrete.
Received: 18 June 2001 / Revised version: 31 January 2002 / Published online: 27 June 2002 相似文献