1,606 research outputs found
Lace expansion for the Ising model
The lace expansion has been a powerful tool for investigating mean-field
behavior for various stochastic-geometrical models, such as self-avoiding walk
and percolation, above their respective upper-critical dimension. In this
paper, we prove the lace expansion for the Ising model that is valid for any
spin-spin coupling. For the ferromagnetic case, we also prove that the
expansion coefficients obey certain diagrammatic bounds that are similar to the
diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk.
As a result, we obtain Gaussian asymptotics of the critical two-point function
for the nearest-neighbor model with d>>4 and for the spread-out model with d>4
and L>>1, without assuming reflection positivity.Comment: 54 pages, 12 figure
The quenched critical point for self-avoiding walk on random conductors
Following similar analysis to that in Lacoin (PTRF 159, 777-808, 2014), we
can show that the quenched critical point for self-avoiding walk on random
conductors on the d-dimensional integer lattice is almost surely a constant,
which does not depend on the location of the reference point. We provide its
upper and lower bounds that are valid for all dimensions.Comment: 14 page
Critical two-point functions for long-range statistical-mechanical models in high dimensions
We consider long-range self-avoiding walk, percolation and the Ising model on
that are defined by power-law decaying pair potentials of the
form with . The upper-critical dimension
is for self-avoiding walk and the Ising
model, and for percolation. Let and assume
certain heat-kernel bounds on the -step distribution of the underlying
random walk. We prove that, for (and the spread-out
parameter sufficiently large), the critical two-point function
for each model is asymptotically
, where the constant is expressed in
terms of the model-dependent lace-expansion coefficients and exhibits crossover
between . We also provide a class of random walks that
satisfy those heat-kernel bounds.Comment: Published in at http://dx.doi.org/10.1214/13-AOP843 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation
We prove that the Fourier transform of the properly-scaled normalized
two-point function for sufficiently spread-out long-range oriented percolation
with index \alpha>0 converges to e^{-C|k|^{\alpha\wedge2}} for some
C\in(0,\infty) above the upper-critical dimension 2(\alpha\wedge2). This
answers the open question remained in the previous paper [arXiv:math/0703455].
Moreover, we show that the constant C exhibits crossover at \alpha=2, which is
a result of interactions among occupied paths. The proof is based on a new
method of estimating fractional moments for the spatial variable of the
lace-expansion coefficients.Comment: 20 pages, 1 figur
Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation
We consider random walk and self-avoiding walk whose 1-step distribution is
given by , and oriented percolation whose bond-occupation probability is
proportional to . Suppose that decays as with
. For random walk in any dimension and for self-avoiding walk and
critical/subcritical oriented percolation above the common upper-critical
dimension , we prove large-
asymptotics of the gyration radius, which is the average end-to-end distance of
random walk/self-avoiding walk of length or the average spatial size of an
oriented percolation cluster at time . This proves the conjecture for
long-range self-avoiding walk in [Ann. Inst. H. Poincar\'{e} Probab. Statist.
(2010), to appear] and for long-range oriented percolation in [Probab. Theory
Related Fields 142 (2008) 151--188] and [Probab. Theory Related Fields 145
(2009) 435--458].Comment: Published in at http://dx.doi.org/10.1214/10-AOP557 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Gaussian scaling for the critical spread-out contact process above the upper critical dimension
We consider the critical spread-out contact process in \Zd with ,
whose infection range is denoted by . The two-point function
is the probability that x\in\Zd is infected at time by the
infected individual located at the origin o\in\Zd at time 0. We prove
Gaussian behavior for the two-point function with for some finite
for . When , we also perform a local mean-field
limit to obtain Gaussian behaviour for with fixed and when the infection range depends on such that for any
.
The proof is based on the lace expansion and an adaptation of the inductive
approach applied to the discretized contact process. We prove the existence of
several critical exponents and show that they take on mean-field values. The
results in this paper provide crucial ingredients to prove convergence of the
finite-dimensional distributions for the contact process towards the canonical
measure of super-Brownian motion, which we defer to a sequel of this paper.Comment: 50 pages, 5 figure
Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: I. The higher-point functions
We consider the critical spread-out contact process in Z^d with d\ge1, whose
infection range is denoted by L\ge1. In this paper, we investigate the r-point
function \tau_{\vec t}^{(r)}(\vec x) for r\ge3, which is the probability that,
for all i=1,...,r-1, the individual located at x_i\in Z^d is infected at time
t_i by the individual at the origin o\in Z^d at time 0. Together with the
results of the 2-point function in [van der Hofstad and Sakai, Electron. J.
Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs crucially
rely, we prove that the r-point functions converge to the moment measures of
the canonical measure of super-Brownian motion above the upper-critical
dimension 4. We also prove partial results for d\le4 in a local mean-field
setting.Comment: 75 pages, 12 figure
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