136 research outputs found
When is 0.999... equal to 1?
A doubly infinite sum, numerically evaluated at between 0.999 and 1.001,
turns out to have a nice value
No directed fractal percolation in zero area
We show that fractal (or "Mandelbrot") percolation in two dimensions produces
a set containing no directed paths, when the set produced has zero area. This
improves a similar result by the first author in the case of constant retention
probabilities to the case of retention probabilities approaching 1
Random walks in random Dirichlet environment are transient in dimension
We consider random walks in random Dirichlet environment (RWDE) which is a
special type of random walks in random environment where the exit probabilities
at each site are i.i.d. Dirichlet random variables. On , RWDE are
parameterized by a -uplet of positive reals. We prove that for all values
of the parameters, RWDE are transient in dimension . We also prove that
the Green function has some finite moments and we characterize the finite
moments. Our result is more general and applies for example to finitely
generated symmetric transient Cayley graphs. In terms of reinforced random
walks it implies that directed edge reinforced random walks are transient for
.Comment: New version published at PTRF with an analytic proof of lemma
High-Precision Entropy Values for Spanning Trees in Lattices
Shrock and Wu have given numerical values for the exponential growth rate of
the number of spanning trees in Euclidean lattices. We give a new technique for
numerical evaluation that gives much more precise values, together with
rigorous bounds on the accuracy. In particular, the new values resolve one of
their questions.Comment: 7 pages. Revision mentions alternative approach. Title changed
slightly. 2nd revision corrects first displayed equatio
Characterization of the critical values of branching random walks on weighted graphs through infinite-type branching processes
We study the branching random walk on weighted graphs; site-breeding and
edge-breeding branching random walks on graphs are seen as particular cases. We
describe the strong critical value in terms of a geometrical parameter of the
graph. We characterize the weak critical value and relate it to another
geometrical parameter. We prove that, at the strong critical value, the process
dies out locally almost surely; while, at the weak critical value, global
survival and global extinction are both possible.Comment: 14 pages, corrected some typos and minor mistake
Transient Random Walks in Random Environment on a Galton-Watson Tree
We consider a transient random walk in random environment on a
Galton--Watson tree. Under fairly general assumptions, we give a sharp and
explicit criterion for the asymptotic speed to be positive. As a consequence,
situations with zero speed are revealed to occur. In such cases, we prove that
is of order of magnitude , with . We also
show that the linearly edge reinforced random walk on a regular tree always has
a positive asymptotic speed, which improves a recent result of Collevecchio
\cite{Col06}
Dynamical percolation on general trees
H\"aggstr\"om, Peres, and Steif (1997) have introduced a dynamical version of
percolation on a graph . When is a tree they derived a necessary and
sufficient condition for percolation to exist at some time . In the case
that is a spherically symmetric tree, H\"aggstr\"om, Peres, and Steif
(1997) derived a necessary and sufficient condition for percolation to exist at
some time in a given target set . The main result of the present paper
is a necessary and sufficient condition for the existence of percolation, at
some time , in the case that the underlying tree is not necessary
spherically symmetric. This answers a question of Yuval Peres (personal
communication). We present also a formula for the Hausdorff dimension of the
set of exceptional times of percolation.Comment: 24 pages; to appear in Probability Theory and Related Field
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