26 research outputs found
Bounds for avalanche critical values of the Bak-Sneppen model
We study the Bak-Sneppen model on locally finite transitive graphs , in
particular on Z^d and on T_Delta, the regular tree with common degree Delta. We
show that the avalanches of the Bak-Sneppen model dominate independent site
percolation, in a sense to be made precise. Since avalanches of the Bak-Sneppen
model are dominated by a simple branching process, this yields upper and lower
bounds for the so-called avalanche critical value . Our main
results imply that 1/(Delta+1) <= \leq p_c^{BS}(T_Delta) \leq 1/(Delta -1)1/(2d+1)\leq p_c^{BS}(Z^d)\leq 1/(2d)+ 1/(2d)^2+O(d^{-3}), as
d\to\infty.Comment: 19 page
Anisotropic percolation in high dimensions: the non-oriented case
We consider inhomogeneous non-oriented Bernoulli bond percolation on ,
where each edge has a parameter, depending on its direction. We prove that,
under certain conditions, if the sum of the parameters is strictly greater than
1/2, we have percolation for sufficiently high dimensions. The main tool is a
dynamical coupling between the models in different dimensions with different
set of parameters
Percolation on nonunimodular transitive graphs
We extend some of the fundamental results about percolation on unimodular
nonamenable graphs to nonunimodular graphs. We show that they cannot have
infinitely many infinite clusters at critical Bernoulli percolation. In the
case of heavy clusters, this result has already been established, but it also
follows from one of our results. We give a general necessary condition for
nonunimodular graphs to have a phase with infinitely many heavy clusters. We
present an invariant spanning tree with on some nonunimodular graph.
Such trees cannot exist for nonamenable unimodular graphs. We show a new way of
constructing nonunimodular graphs that have properties more peculiar than the
ones previously known.Comment: Published at http://dx.doi.org/10.1214/009117906000000494 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
New Lower Bounds on the Self-Avoiding-Walk Connective Constant
We give an elementary new method for obtaining rigorous lower bounds on the
connective constant for self-avoiding walks on the hypercubic lattice .
The method is based on loop erasure and restoration, and does not require exact
enumeration data. Our bounds are best for high , and in fact agree with the
first four terms of the expansion for the connective constant. The bounds
are the best to date for dimensions , but do not produce good results
in two dimensions. For , respectively, our lower bound is within
2.4\%, 0.43\%, 0.12\%, 0.044\% of the value estimated by series extrapolation.Comment: 35 pages, 388480 bytes Postscript, NYU-TH-93/02/0
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
I show that there exist universal constants such that, for
all loopless graphs of maximum degree , the zeros (real or complex)
of the chromatic polynomial lie in the disc . Furthermore,
. This result is a corollary of a more general result
on the zeros of the Potts-model partition function in the
complex antiferromagnetic regime . The proof is based on a
transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of to a polymer gas, followed by verification of the
Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model
partition function. I also show that, for all loopless graphs of
second-largest degree , the zeros of lie in the disc . Along the way, I give a simple proof of a generalized (multivariate)
Brown-Colbourn conjecture on the zeros of the reliability polynomial for the
special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs
of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of
Proposition 4.1, and adds related discussion. To appear in Combinatorics,
Probability & Computin
Expansion for the critical point of site percolation: the first three terms
We expand the critical point for site percolation on the -dimensional
hypercubic lattice in terms of inverse powers of , and we obtain the first
three terms rigorously. This is achieved using the lace expansion.Comment: 22 page
Conformally invariant scaling limits in planar critical percolation
This is an introductory account of the emergence of conformal invariance in
the scaling limit of planar critical percolation. We give an exposition of
Smirnov's theorem (2001) on the conformal invariance of crossing probabilities
in site percolation on the triangular lattice. We also give an introductory
account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of
conformally invariant random curves discovered by Schramm (2000). The article
is organized around the aim of proving the result, due to Smirnov (2001) and to
Camia and Newman (2007), that the percolation exploration path converges in the
scaling limit to chordal SLE(6). No prior knowledge is assumed beyond some
general complex analysis and probability theory.Comment: 55 pages, 10 figure