共查询到20条相似文献,搜索用时 425 毫秒
1.
John C. Wierman 《Random Structures and Algorithms》2002,20(4):507-518
Rigorous bounds for the bond percolation critical probability are determined for three Archimedean lattices: .7385 < pc((3, 122) bond) < .7449, .6430 < pc((4, 6, 12) bond) < .7376, .6281 < pc((4, 82) bond) < .7201. Consequently, the bond percolation critical probability of the (3, 122) lattice is strictly larger than those of the other ten Archimedean lattices. Thus, the (3, 122) bond percolation critical probability is possibly the largest of any vertex‐transitive graph with bond percolation critical probability that is strictly less than one. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 507–518, 2002 相似文献
2.
Alexander E. Holroyd 《Probability Theory and Related Fields》2003,125(2):195-224
In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour
as p→0 and L→∞ simultaneously of the probability I(L,p) that the entire square is eventually active. We prove that I(L,p)→1 if , and I(L,p)→0 if , where λ=π2/18. We prove the same behaviour, with the same threshold λ, for the probability J(L,p) that a site is active by time L in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but
with threshold λ′=π2/6. The existence of the thresholds λ,λ′ settles a conjecture of Aizenman and Lebowitz [3], while the determination of their values corrects numerical predictions
of Adler, Stauffer and Aharony [2].
Received: 12 May 2002 / Revised version: 12 August 2002 / Published online: 14 November 2002
Research funded in part by NSF Grant DMS-0072398
Mathematics Subject Classification (2000): Primary 60K35; Secondary 82B43
Key words or phrases: Bootstrap percolation – Cellular automaton – Metastability – Finite-size scaling 相似文献
3.
Friendly walkers is a stochastic model obtained from independent one-dimensional simple random walks {S
k
j
}
j≥0
, k=1,2,…,d by introducing ``non-crossing condition': and ``reward for collisions' characterized by parameters . Here, the reward for collisions is described as follows. If, at a given time n, a site in ℤ is occupied by exactly m≥2 walkers, then the site increases the probabilistic weight for the walkers by multiplicative factor exp (β
m
)≥1. We study the localization transition of this model in terms of the positivity of the free energy and describe the location
and the shape of the critical surface in the (d−1)-dimensional space for the parameters .
Received: 13 June 2002 / Revised version: 24 August 2002 Published online: 28 March 2003
Mathematics Subject Classification (2000): 82B41, 82B26, 82D60, 60G50
Key words or phrases: Random walks – Random surfaces – Lattice animals – Phase transitions – Polymers – Random walks 相似文献
4.
We study percolation in the following random environment: let Z be a Poisson process of constant intensity on ℝ2, and form the Voronoi tessellation of ℝ2 with respect to Z. Colour each Voronoi cell black with probability p, independently of the other cells. We show that the critical probability is 1/2. More precisely, if p>1/2 then the union of the black cells contains an infinite component with probability 1, while if p<1/2 then the distribution of the size of the component of black cells containing a given point decays exponentially. These
results are analogous to Kesten's results for bond percolation in ℤ2.
The result corresponding to Harris' Theorem for bond percolation in ℤ2 is known: Zvavitch noted that one of the many proofs of this result can easily be adapted to the random Voronoi setting.
For Kesten's results, none of the existing proofs seems to adapt. The methods used here also give a new and very simple proof
of Kesten's Theorem for ℤ2; we hope they will be applicable in other contexts as well.
Research supported in part by NSF grant ITR 0225610 and DARPA grant F33615-01-C-1900
Research partially undertaken during a visit to the Forschungsinstitut für Mathematik, ETH Zürich, Switzerland 相似文献
5.
Yasunari Higuchi 《Probability Theory and Related Fields》1982,61(1):75-81
Summary We show that the critical probability p
c
is strictly greater than 1/2 for the square lattice site percolation. 相似文献
6.
We consider the harmonic crystal, or massless free field, , , that is the centered Gaussian field with covariance given by the Green function of the simple random walk on ℤ
d
. Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition to be larger than , is an IID field (which is also independent of ϕ), for every x in a large region , with N a positive integer and D a bounded subset of ℝ
d
. We are mostly motivated by results for given typical realizations of σ (quenched set–up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, living in a (d+1)–dimensional space, constrained not to go below an inhomogeneous substrate that acts as a hard wall. We consider various
types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of σ
0
is heavier than Gaussian, while essentially no effect is observed if the tail is sub–Gaussian. In the critical case, that
is the one of approximately Gaussian tail, the interplay of the two sources of randomness, ϕ and σ, leads to an enhanced repulsion effect of additive type. This generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal Large
Deviation type asymptotics as of the probability that ϕ lies above σ in D
N
.
Received: 6 February 2002 / Revised version: 23 May 2002 / Published online: 30 September 2002
Mathematics Subject Classification (2000): 82B24, 60K35, 60G15
Keywords or phrases: Harmonic Crystal – Rough Substrate – Quenched and Annealed Models – Entropic Repulsion – Gaussian fields – Extrema of Random
Fields – Large Deviations – Random Walks 相似文献
7.
In this note we study the geometry of the largest component C1\mathcal {C}_{1} of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. (Random Struct. Algorithms 27:137–184, 2005). There it is shown that this component is of size n
2/3, and here we show that its diameter is n
1/3 and that the simple random walk takes n steps to mix on it. By Borgs et al. (Ann. Probab. 33:1886–1944, 2005), our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus
\mathbbZnd\mathbb{Z}_{n}^{d} (with d large and n→∞) and the Hamming cube {0,1}
n
. 相似文献
8.
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
9.
We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters
inside boxes in Z
d
. We show that for d≥2 and p>p
c
(Z
d
), the mixing time of simple random walk on the largest cluster inside is Θ(n
2
) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance
method. This is the first non-trivial application of this method which yields a tight result.
Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002 相似文献
10.
The Alexander-Orbach conjecture holds in high dimensions 总被引:1,自引:0,他引:1
We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established,
namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral
dimension
ds=\frac43d_{s}=\frac{4}{3}
, that is, p
t
(x,x)=t
−2/3+o(1). This establishes a conjecture of Alexander and Orbach (J. Phys. Lett. (Paris) 43:625–631, 1982). En route we calculate the one-arm exponent with respect to the intrinsic distance. 相似文献
11.
F. den Hollander F.R. Nardi E. Olivieri E. Scoppola 《Probability Theory and Related Fields》2003,125(2):153-194
The goal of this paper is to describe metastability and nucleation for a local version of the three-dimensional lattice gas
with Kawasaki dynamics at low temperature and low density.
Let $\Lambda\subseteq{\mathbb Z}^3$ be a large finite box. Particles perform simple exclusion on $\Lambda$, but when they
occupy neighboring sites they feel a binding energy $-U<0$ that slows down their dissociation. Along each bond touching the
boundary of $\Lambda$ from the outside, particles are created with rate $\rho=e^{-\Delta\beta}$ and are annihilated with rate
1, where $\beta$ is the inverse temperature and $\D>0$ is an activity parameter. Thus, the boundary of $\Lambda$ plays the
role of an infinite gas reservoir with density $\rho$.
We consider the regime where $\Delta\in (U,3U)$ and the initial configuration is such that $\Lambda$ is empty. For large $\beta$,
the system wants to fill $\Lambda$ but is slow in doing so. We investigate how the transition from empty to full takes place
under the dynamics. In particular, we identify the size and shape of the critical droplet\/ and the time of its creation in the limit as $\beta\to\infty$.
Received: 23 February 2002 / Revised version: 24 June 2002 / Published online: 24 October 2002
Mathematics Subject Classification (2000): 60K35, 82B43, 82C43, 82C80
Key words or phrases: Lattice gas – Kawasaki dynamics – Metastability – Critical droplet – Large deviations – Discrete isoperimetric inequalities 相似文献
12.
Campanino Massimo Ioffe Dmitry Velenik Y van 《Probability Theory and Related Fields》2003,125(3):305-349
We derive a precise Ornstein-Zernike asymptotic formula for the decay of the two-point function 〈Σ0Σ
x
〉β in the general context of finite range Ising type models on ℤ
d
. The proof relies in an essential way on the a-priori knowledge of the strict exponential decay of the two-point function
and, by the sharp characterization of phase transition due to Aizenman, Barsky and Fernández, goes through in the whole of
the high temperature region β<β
c
. As a byproduct we obtain that for every β<β
c
, the inverse correlation length ξβ is an analytic and strictly convex function of direction.
Received: 10 January 2002 / Revised version: 19 June 2002 / Published online: 14 November 2002
Partly supported by Italian G. N. A. F. A, EC grant SC1-CT91-0695 and the University of Bologna. Funds for selected research
topics.
Partly supported by the ISRAEL SCIENCE FOUNDATION founded by The Israel Academy of Science and Humanities.
Partly supported by the Swiss National Science Foundation grant #8220-056599.
Mathematics Subject Classification (2000): 60F15, 60K15, 60K35, 82B20, 37C30
Key words or phrases: Ising model – Ornstein-Zernike decay of correlations – Ruelle operator – Renormalization – Local limit theorems 相似文献
13.
Yasunari Higuchi 《Probability Theory and Related Fields》1991,90(2):203-221
Summary We prove the existence of a real valued random field with parameters in thed-dimensional cubic lattice, such that the distribution of the level set of this random field is a Gibbs state for the nearest neighbour ferromagnetic Ising model. Using this, we prove the continuity of the percolation probability with respect to the parameter (,h) in the uniqueness region except on the critical curve
c
={(,h
c
())}, whereh
c() is the critical level of the external field above which percolation takes place.Supported in part by JSPS, BiBoS, Grant in Aid for Cooperative research no. 62303006 and Grant in Aid for Scientific Research no. 63540168 相似文献
14.
It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges
in distribution to the trace of chordal SLE
6. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant
scaling limit (given by Cardy’s formula). The version of convergence to SLE
6 that we prove suffices for the Smirnov–Werner derivation of certain critical percolation crossing exponents and for our analysis
of the critical percolation full scaling limit as a process of continuum nonsimple loops.
Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696. 相似文献
15.
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
16.
Hugo Duminil-Copin Marcelo R. Hilário Gady Kozma Vladas Sidoravicius 《Israel Journal of Mathematics》2018,225(1):479-501
We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on Z. Select vertical columns at random independently with a given positive probability. Keep (respectively remove) vertical edges in the selected columns, with probability p (respectively 1?p). All horizontal edges and vertical edges lying in unselected columns are kept (respectively removed) with probability q (respectively 1 ? q). We show that, if p > pc(Z2) (the critical point for homogeneous Bernoulli bond percolation), then q can be taken strictly smaller than pc(Z2) in such a way that the probability that the origin percolates is still positive. 相似文献
17.
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
18.
The number of infinite clusters in dynamical percolation 总被引:2,自引:2,他引:0
Summary. Dynamical percolation is a Markov process on the space of subgraphs of a given graph, that has the usual percolation measure
as its stationary distribution. In previous work with O. H?ggstr?m, we found conditions for existence of infinite clusters
at exceptional times. Here we show that for ℤ
d
, with p>p
c
, a.s. simultaneously for all times there is a unique infinite cluster, and the density of this cluster is θ(p). For dynamical percolation on a general tree Γ, we show that for p>p
c
, a.s. there are infinitely many infinite clusters at all times. At the critical value p=p
c
, the number of infinite clusters may vary, and exhibits surprisingly rich behaviour. For spherically symmetric trees, we
find the Hausdorff dimension of the set T
k
of times where the number of infinite clusters is k, and obtain sharp capacity criteria for a given time set to intersect T
k
. The proof of this capacity criterion is based on a new kernel truncation technique.
Received: 5 May 1997 / In revised form: 24 November 1997 相似文献
19.
It is proved that the Stokes operator on a bounded domain, an exterior domain, or a perturbed half-space Ω admits a bounded
H
∞
-calculus on L
q
(Ω) if q(1,∞).
Received: 25 January 2002; in final form: 2 October 2002 /
Published online: 16 May 2003 相似文献
20.
In the bootstrap percolation on the n-dimensional hypercube, in the initial position each of the 2n sites is occupied with probability p and empty with probability 1−p, independently of the state of the other sites. Every occupied site remains occupied for ever, while an empty site becomes
occupied if at least two of its neighbours are occupied. If at the end of the process every site is occupied, we say that
the (initial) position spans the hypercube. We shall show that there are constants c1,c2>0 such that for the probability of spanning tends to 1 as n→∞, while for the probability tends to 0. Furthermore, we shall show that for each n the transition has a sharp threshold function.
J. Balogh: work was done while at The University of Memphis, USA
Research supported in part by NSF grant DMS0302804
Research supported in part by NSF grant ITR 0225610 and DARPA grant F33615-01-C-1900 相似文献