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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.
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
3.
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
4.
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
5.
Summary. Let X,X
1,X
2,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝ
d
. Assume that E
X=0 and that X is not concentrated in a proper subspace of ℝ
d
. Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[S
N
] of the normalized sums S
N
=N
−1/2(X
1+⋯+X
N
) and show that
provided that d≥9 and the fourth moment of X exists. The bound ?(N
−1) is optimal and improves, e.g., the well-known bound ?(N
−
d
/(
d
+1)) due to Esseen (1945). The result extends to the case of random vectors taking values in a Hilbert space. Furthermore, we
provide explicit bounds for Δ
N
and for the concentration function of the random variable ℚ[S
N
].
Received: 9 January 1997 / In revised form: 15 May 1997 相似文献
6.
Xiaoping Xu 《manuscripta mathematica》1999,100(4):489-518
In this paper, we introduce two new families of infinite-dimensional simple Lie algebras and a new family of infinite-dimensional
simple Lie superalgebras. These algebras can be viewed as generalizations of the Block algebras.
Received: 3 March 1999 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.
Summary. This is a continuation of our previous work [6] on the investigation of intermittency for the parabolic equation (∂/∂t)u=Hu on ℝ+×ℤ
d
associated with the Anderson Hamiltonian H=κΔ+ξ(·) for i.i.d. random potentials ξ(·). For the Cauchy problem with nonnegative
homogeneous initial condition we study the second order asymptotics of the statistical moments <u(t,0)
p
> and the almost sure growth of u(t,0) as t→∞. We point out the crucial role of double exponential tails of ξ(0) for the formation of high intermittent peaks of the
solution u(t,·) with asymptotically finite size. The challenging motivation is to achieve a better understanding of the geometric structure
of such high exceedances which in one or another sense provide the essential contribution to the solution.
Received: 10 December 1996 / In revised form: 30 September 1997 相似文献
9.
We consider oriented bond or site percolation on ℤ
d
+. In the case of bond percolation we denote by P
p
the probability measure on configurations of open and closed bonds which makes all bonds of ℤ
d
+ independent, and for which P
p
{e is open} = 1 −P
p
e {is closed} = p for each fixed edge e of ℤ
d
+. We take X(e) = 1 (0) if e is open (respectively, closed). We say that ρ-percolation occurs for some given 0 < ρ≤ 1, if there exists an oriented infinite path v
0 = 0, v
1, v
2, …, starting at the origin, such that lim inf
n
→∞ (1/n) ∑
i=1
n
X(e
i
) ≥ρ, where e
i
is the edge {v
i−1
, v
i
}. [MZ92] showed that there exists a critical probability p
c
= p
c
(ρ, d) = p
c
(ρ, d, bond) such that there is a.s. no ρ-percolation for p < p
c
and that P
p
{ρ-percolation occurs} > 0 for p > p
c
. Here we find lim
d
→∞
d
1/ρ
p
c
(ρd, bond) = D
1
, say. We also find the limit for the analogous quantity for site percolation, that is D
2 = lim
d
→∞
d
1/ρ
p
c
(ρ, d, site). It turns out that for ρ < 1, D
1
< D
2
, and neither of these limits equals the analogous limit for the regular d-ary trees.
Received: 7 January 1999 / Published online: 14 June 2000 相似文献
10.
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
11.
Michel Talagrand 《Probability Theory and Related Fields》1998,110(2):177-275
Summary. We perform a thorough investigation of the main aspects of the Hopfield model with many patterns. Advances are made toward
the validity of the “replica symmetric” solution. Strong evidence of the validity of this solution is given over the entire
domain where this validity is conjectured; complete proof is given in a subregion that contains strictly the ergodic region.
Received: 22 May 1996 / In revised form: 20 May 1997 相似文献
12.
Dmitry Ioffe 《Probability Theory and Related Fields》1995,102(3):313-330
Summary We prove an upper large deviation bound for the block spin magnetization in the 2D Ising model in the phase coexistence region. The precise rate (given by the Wulff construction) is shown to hold true for all > c. Combined with the lower bounds derived in [I] those results yield an exact second order large deviation theory up to the critical temperature. 相似文献
13.
Alexander D. Wentzell 《Probability Theory and Related Fields》1999,113(2):255-271
. For a certain class of families of stochastic processes ηε(t), 0≤t≤T, constructed starting from sums of independent random variables, limit theorems for expectations of functionals F(ηε[0,T]) are proved of the form
where w
0 is a Wiener process starting from 0, with variance σ2 per unit time, A
i
are linear differential operators acting on functionals, and m=1 or 2. Some intricate differentiability conditions are imposed on the functional.
Received: 12 September 1995 / Revised version: 6 April 1998 相似文献
14.
John Verzani 《Probability Theory and Related Fields》1997,107(4):517-526
Summary. For the Brownian path-valued process of Le Gall (or Brownian snake) in , the times at which the process is a cone path are considered as a function of the size of the cone and the terminal position
of the path. The results show that the paths for the path-valued process have local properties unlike those of a standard
Brownian motion.
Received: 29 January 1996 / In revised form: 21 June 1996 相似文献
15.
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
16.
Summary. In standard first-passage percolation on (with ), the time-minimizing paths from a point to a plane at distance are expected to have transverse fluctuations of order . It has been conjectured that with the inequality strict (superdiffusivity) at least for low and with . We prove (versions of) for all and .
Received: 30 August 1995 / In revised form: 28 March 1996 相似文献
17.
Michel Talagrand 《Probability Theory and Related Fields》1998,110(2):109-176
Summary. The Sherrington–Kirkpatrick (SK) model for spin glasses is deceptively simple to state. Yet its rigorous study represents
a considerable challenge. We report here some modest progresses (obtained through elementary methods). Even in the supposedly
simple high temperature region, a number of basic questions remain unsolved.
Received: 7 December 1995 / In revised form: 6 March 1997 相似文献
18.
On the long time behavior of the stochastic heat equation 总被引:2,自引:0,他引:2
We consider the stochastic heat equation in one space dimension and compute – for a particular choice of the initial datum
– the exact long time asymptotic. In the Carmona-Molchanov approach to intermittence in non stationary random media this corresponds
to the identification of the sample Lyapunov exponent. Equivalently, by interpreting the solution as the partition function of a directed polymer in a random environment, we obtain
a weak law of large numbers for the quenched free energy. The result agrees with the one obtained in the physical literature
via the replica method. The proof is based on a representation of the solution in terms of the weakly asymmetric exclusion
process.
Received: 11 November 1997 / Revised version: 31 July 1998 相似文献
19.
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
20.
Yoichi Nishiyama 《Probability Theory and Related Fields》1997,108(4):459-494
Summary. This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ∞(Ψ)-valued continuous-time stochastic processes t⇝X
t
n
=(X
t
n
,ψ|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X
t
n
,ψ is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued
random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented.
We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model
for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior
of log-likelihood ratio random fields of certain continuous semimartingales is derived.
Received: 6 May 1996 / In revised form: 4 February 1997 相似文献