共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
C. Bahadoran 《Probability Theory and Related Fields》1998,110(3):287-331
Summary. We prove hydrodynamical limit for spatially heterogeneous, asymmetric simple exclusion processes on Z
d
. The jump rate of particles depends on the macroscopic position x through some nonnegative, smooth velocity profile α(x). Hydrodynamics are described by the entropy solution to a spatially heterogeneous conservation law of the form
To derive this result, we prove an alternative characterization of entropy solutions involving stationary solutions, and work
with macroscopically stationary states rather than the unknown stationary measures of the process. The method can be extended
to spatially heterogeneous, asymmetric misanthrope processes with slow birth and death.
Received: 11 November 1996/In revised form: 10 October 1997 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.
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
5.
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
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.
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
8.
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
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.
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
11.
For a natural number k, define an oriented site percolation on ℤ2 as follows. Let x
i
, y
j
be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ2
closed if x
i
= y
j
, and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open.
I.e., there is an infinite path P = (i
0, j
0)(i
1, j
1) · · · such that 0 = i
0≤i
1≤· · ·, 0 = j
0≤j
1≤· · ·, and each site (i
n
, j
n
) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive.
Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x
i
) and (y
j
). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods.
Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000 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.
Summary. Consider (independent) first-passage percolation on the edges of ℤ
2
. Denote the passage time of the edge e in ℤ
2
by t(e), and assume that P{t(e) = 0} = 1/2, P{0<t(e)<C
0
} = 0 for some constant C
0
>0 and that E[t
δ
(e)]<∞ for some δ>4. Denote by b
0,n
the passage time from 0 to the halfplane {(x,y): x ≧ n}, and by T(
0
,nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0<C
1
, C
2
<∞ and γ
n
such that C
1
(
log
n)
1/2
≦γ
n
≦ C
2
(
log
n)
1/2
and such that γ
n
−1
[b
0,n
−Eb
0,n
] and (√ 2γ
n
)
−1
[T(
0
,nu) − ET(
0
,nu)] converge in distribution to a standard normal variable (as n →∞, u fixed).
A similar result holds for the site version of first-passage percolation on ℤ
2
, when the common distribution of the passage times {t(v)} of the vertices satisfies P{t(v) = 0} = 1−P{t(v) ≧ C
0
} = p
c
(ℤ
2
,
site
) := critical probability of site percolation on ℤ
2
, and E[t
δ
(u)]<∞ for some δ>4.
Received: 6 February 1996 / In revised form: 17 July 1996 相似文献
14.
Thomas M. Liggett 《Probability Theory and Related Fields》1996,106(4):495-519
Summary. Branching random walks and contact processes on the homogeneous tree in which each site has d+1 neighbors have three possible types of behavior (for d≧ 2): local survival, local extinction with global survival, and global extinction. For branching random walks, we show that
if there is local extinction, then the probability that an individual ever has a descendent at a site n units away from that individual’s location is at most d
− n/2
, while if there is global extinction, this probability is at most d
−n
. Next, we consider the structure of the set of invariant measures with finite intensity for the system, and see how this
structure depends on whether or not there is local and/or global survival. These results suggest some problems and conjectures for contact processes on trees. We prove some and
leave others open. In particular, we prove that for some values of the infection parameter λ, there are nontrivial invariant measures which have a density tending to zero in all directions, and hence are different
from those constructed by Durrett and Schinazi in a recent paper.
Received: 26 April 1996/In revised form: 20 June 1996 相似文献
15.
. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996),
we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p
2>p
1>p
c
, each infinite p
2-cluster contains an infinite p
1-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that
the probability θ
v
(p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p
c
. All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group.
Received: 22 December 1997 / Revised version: 1 July 1998 相似文献
16.
Summary. We study the asymptotic behavior of Brownian motion and its conditioned process in cones using an infinite series representation
of its transition density. A concise probabilistic interpretation of this series in terms of the skew product decomposition
of Brownian motion is derived and used to show properties of the transition density.
Received: 2 April 1996 / In revised form: 21 December 1996 相似文献
17.
Summary. Hyperbolic branching Brownian motion is a branching diffusion process in which individual particles follow independent Brownian
paths in the hyperbolic plane ?
2
, and undergo binary fission(s) at rate λ > 0. It is shown that there is a phase transition in λ: For λ≦ 1/8 the number of particles in any compact region of ?
2
is eventually 0, w.p.1, but for λ > 1/8 the number of particles in any open set grows to ∞ w.p.1. In the subcritical case (λ≦ 1/8) the set Λ of all limit points in ∂?
2
(the boundary circle at ∞) of particle trails is a Cantor set, while in the supercritical case (λ > 1/8) the set Λ has full Lebesgue measure. For λ≦ 1/8 it is shown that w.p.1 the Hausdorff dimension of Λ is δ = (1−√1−8 λ)/2.
Received: 2 November 1995 / In revised form: 22 October 1996 相似文献
18.
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
19.
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