共查询到20条相似文献,搜索用时 15 毫秒
1.
Roberto H. Schonmann 《Probability Theory and Related Fields》1999,113(2):287-300
. A recent theorem by Häggström and Peres concerning independent percolation is extended to all the quasi-transitive graphs. This theorem states that if 0<p 1<p 2≤1 and percolation occurs at level p 1, then every infinite cluster at level p 2 contains some infinite cluster at level p 1. Consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster, i.e., if at level p 1 there is a unique infinite cluster then the same holds at level p 2. These results are further generalized to graphs with a “uniform percolation” property. The threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls. 相似文献
2.
. 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
3.
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
4.
A. F. Ramírez 《Probability Theory and Related Fields》1998,110(3):369-395
Summary. Let η be a diffusion process taking values on the infinite dimensional space T
Z
, where T is the circle, and with components satisfying the equations dη
i
=σ
i
(η) dW
i
+b
i
(η) dt for some coefficients σ
i
and b
i
, i∈Z. Suppose we have an initial distribution μ and a sequence of times t
n
→∞ such that lim
n
→∞μS
tn
=ν exists, where S
t
is the semi-group of the process. We prove that if σ
i
and b
i
are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf
i
,ησ
i
(η)>0, then ν is invariant.
Received: 12 September 1996 / In revised form: 10 November 1997 相似文献
5.
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
6.
Minoru W. Yoshida 《Probability Theory and Related Fields》1996,106(2):265-297
Summary. By the theory of quasi-regular Dirichletforms and the associated special standard processes, the existence of symmetric diffusion
processes taking values in the space of non-negative integer valued Radon measures on and having Gibbs invariant measures associated with some given pair potentials is considered. The existence of such diffusions
can be shown for a wide class of potentials involving some singular ones. Also, as a consequence of an application of stochastic
calculus, a representation for the diffusion by means of a stochastic differential equation is derived.
Received: 5 September 1995 / In revised form: 14 March 1996 相似文献
7.
R. A. Doney 《Probability Theory and Related Fields》1997,107(4):451-465
Summary. If {S
n
,n≧0} is an integer-valued random walk such that S
n
/a
n
converges in distribution to a stable law of index α∈ (0,1) as n→∞, then Gnedenko’s local limit theorem provides a useful estimate for P{S
n
=r} for values of r such that r/a
n
is bounded. The main point of this paper is to show that, under certain circumstances, there is another estimate which is
valid when r/a
n
→ +∞, in other words to establish a large deviation local limit theorem. We also give an asymptotic bound for P{S
n
=r} which is valid under weaker assumptions. This last result is then used in establishing some local versions of generalized
renewal theorems.
Received: 9 August 1995 / In revised form: 29 September 1996 相似文献
8.
9.
10.
Feng-Yu Wang 《Probability Theory and Related Fields》1997,108(1):87-101
Summary. This paper presents some explicit lower bound estimates of logarithmic Sobolev constant for diffusion processes on a compact
Riemannian manifold with negative Ricci curvature. Let Ric≧−K for some K>0 and d, D be respectively the dimension and the diameter of the manifold. If the boundary of the manifold is either empty or convex,
then the logarithmic Sobolev constant for Brownian motion is not less than
max
{(d d+2)
d
1 2(d+1)D
2
exp
[−1−(3d+2)D
2
K], (d−1 d+1)
d
K
exp
[−4D√d K]} .
Next, the gradient estimates of heat semigroups (including the Neumann heat semigroup and the Dirichlet one) are studied by
using coupling method together with a derivative formula modified from [11]. The resulting estimates recover or improve those
given in [7, 21] for harmonic functions.
Received: 19 September 1995 / In revised form 11 April 1996 相似文献
11.
12.
13.
14.
15.
We investigate the limiting fluctuations of the order parameter in the Hopfield model of spin glasses and neural networks
with finitely many patterns at the critical temperature 1/β
c
= 1. At the critical temperature, the measure-valued random variables given by the distribution of the appropriately scaled
order parameter under the Gibbs measure converge weakly towards a random measure which is non-Gaussian in the sense that it
is not given by a Dirac measure concentrated in a Gaussian distribution. This remains true in the case of β = β
N
→β
c
= 1 as N→∞ provided β
N
converges to β
c
= 1 fast enough, i.e., at speed ?(1/). The limiting distribution is explicitly given by its (random) density.
Received: 12 May 1998 / Revised version: 14 October 1998 相似文献
16.
Hiroshi Kaneko 《Probability Theory and Related Fields》2000,117(4):533-550
In this paper, we will give sufficient conditions for the existence of the reflecting diffusion process on a locally compact
space. In constructing reflecting diffusion process, we consider the corresponding Martin–Kuramochi boundary as the reflecting
barrier and introduce the notion of strong (ℰ, u)-Caccioppoli set. Our method covers reflecting diffusion processes with diffusion coefficient degenerating on the boundary.
Received: 23 June 1997 / Revised version: 28 September 1991/ Published online: 14 June 2000 相似文献
17.
Consider a d-dimensional Brownian motion X = (X
1,…,X
d
) and a function F which belongs locally to the Sobolev space W
1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f
k
(X), X
k
] involving the weak first partial derivatives f
k
of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X
k
] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation
results for forward and backward stochastic integrals.
Received: 16 March 1998 / Revised version: 4 April 1999 相似文献
18.
19.
John Urbas 《Mathematische Zeitschrift》2001,236(3):625-641
We derive a monotonicity formula for smooth solutions u of degenerate two dimensional Monge-Ampère equations, and use this to obtain a local H?lder gradient estimate, depending
on for some .
Received August 9, 1999; in final form December 8, 1999/ Published online December 8, 2000 相似文献