992 research outputs found
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
A Monotonicity Result for the Range of a Perturbed Random Walk
We consider a discrete time simple symmetric random walk on Z^d, d>=1, where
the path of the walk is perturbed by inserting deterministic jumps. We show
that for any time n and any deterministic jumps that we insert, the expected
number of sites visited by the perturbed random walk up to time n is always
larger than or equal to that for the unperturbed walk. This intriguing problem
arises from the study of a particle among a Poisson system of moving traps with
sub-diffusive trap motion. In particular, our result implies a variant of the
Pascal principle, which asserts that among all deterministic trajectories the
particle can follow, the constant trajectory maximizes the particle's survival
probability up to any time t>0.Comment: 10 pages, 1 figure. To appear in Journal of Theoretical Probabilit
Spanning forests on the Sierpinski gasket
We present the numbers of spanning forests on the Sierpinski gasket
at stage with dimension equal to two, three and four, and determine the
asymptotic behaviors. The corresponding results on the generalized Sierpinski
gasket with and are obtained. We also derive the
upper bounds of the asymptotic growth constants for both and .Comment: 31 pages, 9 figures, 7 table
Critical two-point function for long-range models with power-law couplings: The marginal case for
Consider the long-range models on of random walk,
self-avoiding walk, percolation and the Ising model, whose
translation-invariant 1-step distribution/coupling coefficient decays as
for some . In the previous work (Ann. Probab., 43,
639--681, 2015), we have shown in a unified fashion for all that,
assuming a bound on the "derivative" of the -step distribution (the
compound-zeta distribution satisfies this assumed bound), the critical
two-point function decays as above the
upper-critical dimension , where for
self-avoiding walk and the Ising model and for percolation.
In this paper, we show in a much simpler way, without assuming a bound on the
derivative of the -step distribution, that for the marginal
case decays as whenever (with a large
spread-out parameter ). This solves the conjecture in the previous work,
extended all the way down to , and confirms a part of predictions in
physics (Brezin, Parisi, Ricci-Tersenghi, J. Stat. Phys., 157, 855--868, 2014).
The proof is based on the lace expansion and new convolution bounds on power
functions with log corrections.Comment: 31 pages, 1 figure, 3 diagrams in equation
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