1.
Uniform spanning forests - Benjamini, Itai; Lyons, Russell; Peres, Yuval; Schramm, Oded
We study uniform spanning forest measures on infinite graphs, which
are weak limits of uniform spanning tree measures from finite subgraphs. These
limits can be taken with free (FSF )or wired (WSF ) boundary conditions.
Pemantle proved that the free and wired spanning forests coincide in
$\mathbb{Z}^d$ and that they give a single tree iff $d
2.
Vertex-reinforced random walk on arbitrary graphs - Volkov, Stanislav
Vertex-reinforced random walk (VRRW), defined by Pemantle, is a
random process in a continuously changing environment which is more likely to
visit states it has visited before. We consider VRRW on arbitrary graphs and
show that on almost all of them, VRRW visits only finitely many vertices with a
positive probability. We conjecture that on all graphs of bounded degree, this
happens with probability 1, and provide a proof only for trees of this
type.
¶ We distinguish between several different patterns of localization
and explicitly describe the long-run structure of VRRW, which depends on
whether a graph contains triangles or not.
¶ While the results of this paper generalize...
3.
Power-law corrections to exponential decay of connectivities and
correlations in lattice models - Alexander, Kenneth S.
Consider a translation-invariant bond percolation model on the
integer lattice which has exponential decay of connectivities, that is, the
probability of a connection $0 \leftrightarrow x$ by a path of open bonds
decreases like $\exp\{-m(\theta)|x|\}$ for some positive constant $m(\theta)$
which may depend on the direction $\theta = x/|x|$. In two and three
dimensions, it is shown that if the model has an appropriate mixing property
and satisfies a special case of the FKG property, then there is at most a
power-law correction to the exponential decaythere exist $A$ and $C$
such that $\exp\{-m(\theta)|x|\} \ge P(0 \leftrightarrow x) \ge A|x|^{-C}
\exp\{-m(\theta)|x|\}$ for all nonzero $x$ . In four or...
4.
A correlation inequality for connection events in
percolation - van den Berg, J.; Kahn, J.
It is well-known in percolation theory (and intuitively plausible)
that two events of the form there is an open path from s to
a are positively correlated. We prove the (not intuitively
obvious) fact that this is still true if we condition on an event of the form
there is no open path from s to t.
6.
Asymptotic density in a threshold coalescing and annihilating
random walk - Stephenson, David
We consider an interacting random walk on $\mathbb{Z}^d$ where
particles interact only at times when a particle jumps to a site at which there
are at least $n - 1$ other particles present. If there are $i \ge n - 1$
particles present, then the particle coalesces (is removed from the system)
with probability $c_i$ and annihilates (is removed along with another particle)
with probability $a_i$. We call this process the $n$-threshold randomly
coalescing and annihilating random walk. We show that, for $n \ge 3$, if both
$a_i$ and $a_i + c_i$ are increasing in $i$ and if the dimension $d$ is at
least $2n + 4$, then
$P\{\text{the...
7.
Perturbation of the equilibrium for a totally asymmetric stick
process in one dimension - Seppäläinen, Timo
We study the evolution of a small perturbation of the equilibrium of
a totally asymmetric one-dimensional interacting system. The model we take as
an example is Hammersley's process as seen from a tagged particle, which can be
viewed as a process of interacting positive-valued stick heights on the sites
of $\mathbf{Z}$. It is known that under Euler scaling (space and time scale $n$
) the empirical stick profile obeys the Burgers equation. We refine this result
in two ways. If the process starts close enough to equilibrium, then over times
$n^\nu$ for $1 \le \nu < 3$, and up to errors that vanish in hydrodynamic
scale, the dynamics...
8.
Greedy lattice animals: negative values and unconstrained
maxima - Dembo, Amir; Gandolfi, Alberto; Kesten, Harry
Let $\{X_v, v \in \mathbb{Z}^d\}$ be i.i.d. random variables, and
$S(\xi) = \sum_{v \in \xi} X_v$ be the weight of a lattice animal $\xi$. Let
$N_n = \max\{S(\xi) : |\xi| = n$ \text{and $\xi$ contains the origin}\}$ and
$G_n = \max\{S(\xi) : \xi \subseteq [-n,n]^d\}$ . We show that, regardless of
the negative tail of the distribution of $X_v$ , if $\mathbf{E}( X_v^+)^d
(\log^+ X_v^+))^{d+a} < + \infty$ for some $a>0$, then first, $\lim_n
n^{-1} N_n = N exists, is finite and constant a.e.; and, second, there is a
transition in the asymptotic behavior of $G_n$ depending on the sign of $N$: if
$N > 0$ then $G_n...
9.
Critical large deviations in harmonic crystals with long-range
interactions - Caputo, P.; Deuschel, J.-D.
We continue our study of large deviations of the empirical measures
of a massless Gaussian field on $Z^d$, whose covariance is given by the Green
function of a long-range random walk. In this paper we extend techniques and
results of Bolthausen and Deuschel to the nonlocal case of a random walk
in the domain of attraction of the symmetric $\alpha$-stable law, with $\alpha
\in (0, 2 \wedge d)$. In particular, we show that critical large deviations
occur at the capacity scale $N^{d-\alpha}$, with a rate function given by the
Dirichlet form of the embedded $\alpha$-stable process. We also prove that if
we impose zero boundary conditions, the rate...
10.
Spectral gap for Kac's model of Boltzmann equation - Janvresse, Elise
We consider a random walk on $S^{n-1}$ , the standard sphere of
dimension $n -1$, generated by random rotations on randomly selected coordinate
planes $i,j$ with $1 \le i < j \le n$. This dynamic was used by Marc Kac as
a model for the spatially homogeneous Boltzmann equation. We prove that the
spectral gap on $S^{n-1}$ is $n^{-1}$ up to a constant independent of $n$.
11.
Kolmogorov's test for the Brownian snake - Delmas, Jean-François; Dhersin, Jean-Stéphane
We present a Kolmogorovs test for the Brownian snake. This
result was conjectured by Le Gall in 1998. It has to be compared with
Kolmogorovs test for super Brownian motion by Dhersin and Le Gall.
12.
Superprocesses of stochastic flows - Ma, Zhi-Ming; Xiang, Kai-Nan
We construct a continuous superprocess X on M
(R d) which is the unique weak Feller extension of the
empirical process of consistent k-point motions generated by a family of
differential operators. The process X differs from known
DawsonWatanabe type, FlemingViot type and
OrnsteinUhlenbeck type superprocesses. This new type of superprocess
provides a connection between stochastic flows and measure-valued processes,
and determines a stochastic coalescence which is similar to those of
Smoluchowski. Moreover, the support of X describes how an initial
measure on R d is transported under the flow. As an
example, the process realizes a viewpoint of Darling about the isotropic
stochastic flows under certain conditions.
13.
Eternal additive coalescents and certain bridges with exchangeable
increments - Bertoin, Jean
Aldous and Pitman have studied the asymptotic behavior of the
additive coalescent processes using a nested family random forests derived by
logging certain inhomogeneous continuum random trees. Here we propose a
different approach based on partitions of the unit interval induced by certain
bridges with exchangeable increments. The analysis is made simple by an
interpretation in terms of an aggregating server system.
14.
On the distribution of ranked heights of excursions of a Brownian
bridge - Pitman, Jim; Yor, Marc
The distribution of the sequence of ranked maximum and minimum
values attained during excursions of a standard Brownian bridge $(B^{br}_t, 0
\le t \le 1)$ is described. The height $M^{br +}_j$of the $j$th highest maximum
over a positive excursion of the bridge has the same distribution as $M^{br
+}_1 /j$, where the distribution of $M^{br +}_1 = \sup_{0 x) = e^{2x^{2}}$.
The probability density of the height $M^{br}_j$ of the$j$th highest maximum of
excursions of the reflecting Brownian bridge $(|B^{br}_t|, 0 \le t \le 1)$ is
given by a modification of the known $\theta$-function series for the density
of...
15.
Mean value theorems for stochastic integrals - Krylov, N. V.
The distributions of stochastic integrals are approximated by the
distributions of stochastic integrals of piece-wise constant processes. The
rate of approximation in some negative Sobolev spaces is estimated.
Generalizations are given for problems arising in control theory.
16.
Majorizing measures without measures - Talagrand, Michel
We give a reformulation of majorizing measures that does not involve
measures, but rather special sequences of partitions. This formulation is more
convenient to perform chaining with different distances.
17.
Strongly supermedian kernels and Revuz measures - Beznea, Lucian; Boboc, Nicu
In the frame of Borel right Markov processes, we investigate,
following an analytical point of view, the Revuz correspondence between classes
of potential kernels and their associated measures, improving upon the results
of Revuz, Azéma, Getoor and Sharpe, Fitzsimmons, Fitzsimmons and Getoor
and Dellacherie, Maisonneuve and Meyer. In the probabilistic approach of the
problem, the kernels that occur are the potential operators of different types
of homogeneous random measures. We completely characterize the hypothesis
(B) of Hunt in terms of Revuz measures.
18.
On the existence of a quasistationary measure for a Markov
chain - Lasserre, Jean B.; Pearce, Charles E. M.
We consider a Markov chain on a locally compact metric space with an
absorbing set. Necessary and sufficient conditions are provided for the
existence of a quasistationary probability distribution.
19.
Measuring the magnitude of sums of independent random
variables - Hitczenko, Pawe?; Montgomery-Smith, Stephen
This paper considers how to measure the magnitude of the sum of
independent random variables in several ways. We give a formula for the tail
distribution for sequences that satisfy the so called Lévy property. We
then give a connection between the tail distribution and the pth moment,
and between the pth moment and the rearrangement invariant norms.
20.
On convergence toward an extreme value distribution in
C[0,1] - de Haan, Laurens; Lin, Tao
The structure of extreme value distributions in in
finite-dimensional space is well known. We characterize the domain of
attraction of such extreme-value distributions in the framework of Giné
Hahn and Vatan. We intend to use the result for statistical applications.
1.
