17 research outputs found
Random walk on the high-dimensional IIC
We study the asymptotic behavior the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by [26]. We do this by obtaining bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander-Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation
High-dimensional incipient infinite clusters revisited
The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second author and Jarai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also obtain estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls
Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time
For critical bond-percolation on high-dimensional torus, this paper proves
sharp lower bounds on the size of the largest cluster, removing a logarithmic
correction in the lower bound in Heydenreich and van der Hofstad (2007). This
improvement finally settles a conjecture by Aizenman (1997) about the role of
boundary conditions in critical high-dimensional percolation, and it is a key
step in deriving further properties of critical percolation on the torus.
Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on
diameter and mixing time of the largest clusters. We further prove that the
volume bounds apply also to any finite number of the largest clusters. The main
conclusion of the paper is that the behavior of critical percolation on the
high-dimensional torus is the same as for critical Erdos-Renyi random graphs.
In this updated version we incorporate an erratum to be published in a
forthcoming issue of Probab. Theory Relat. Fields. This results in a
modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming
issue of Probab. Theory Relat. Field
Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk
We consider self-avoiding walk, percolation and the Ising model with long and
finite range. By means of the lace expansion we prove mean-field behavior for
these models if for self-avoiding walk and the Ising
model, and for percolation, where denotes the
dimension and the power-law decay exponent of the coupling function.
We provide a simplified analysis of the lace expansion based on the
trigonometric approach in Borgs et al. (2007)Comment: 43 pages, many figures. Version v2 with various (minor) changes (in
particular in Sections 1.4 and A.1), and Sect. 4 is shortened. Journal of
Statistical Physics (to appear
Sleep deprivation increases oleoylethanolamide in human cerebrospinal fluid
This study investigated the role of two fatty acid ethanolamides, the endogenous cannabinoid anandamide and its structural analog oleoylethanolamide in sleep deprivation of human volunteers. Serum and cerebrospinal fluid (CSF) samples were obtained from 20 healthy volunteers before and after a night of sleep deprivation with an interval of about 12 months. We found increased levels of oleoylethanolamide in CSF (P = 0.011) but not in serum (P = 0.068) after 24 h of sleep deprivation. Oleoylethanolamide is an endogenous lipid messenger that is released after neural injury and activates peroxisome proliferator-activated receptor-α (PPAR-α) with nanomolar potency. Exogenous PPAR-α agonists, such as hypolipidemic fibrates and oleoylethanolamide, exert both neuroprotective and neurotrophic effects. Thus, our results suggest that oleoylethanolamide release may represent an endogenous neuroprotective signal during sleep deprivation
High-Dimensional Incipient Infinite Clusters Revisited
The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second-named author and JĂĄrai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also get estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls. Keywords: Percolation; Incipient infinite cluster; Lace expansion; Critical behavio