127 research outputs found
One-dimensional random walks with self-blocking immigration
We consider a system of independent one-dimensional random walkers where new
particles are added at the origin at fixed rate whenever there is no older
particle present at the origin. A Poisson ansatz leads to a semi-linear lattice
heat equation and predicts that starting from the empty configuration the total
number of particles grows as . We confirm this prediction
and also describe the asymptotic macroscopic profile of the particle
configuration.Comment: Revised version; in particular, details of the proof of the lower
bound have been worked out more explicitl
Brownian Web and Oriented Percolation: Density Bounds
In a recent work, we proved that under diffusive scaling, the collection of
rightmost infinite open paths in a supercritical oriented percolation
configuration on the space-time lattice Z^2 converges in distribution to the
Brownian web. In that proof, the FKG inequality played an important role in
establishing a density bound, which is a part of the convergence criterion for
the Brownian web formulated by Fontes et al (2004). In this note, we illustrate
how an alternative convergence criterion formulated by Newman et al (2005) can
be verified in this case, which involves a dual density bound that can be
established without using the FKG inequality. This alternative approach is in
some sense more robust. We will also show that the spatial density of the
collection of rightmost infinite open paths starting at time 0 decays
asymptotically in time as c/\sqrt{t} for some c>0.Comment: 12 pages. This is a proceeding article for the RIMS workshop
"Applications of Renormalization Group Methods in Mathematical Sciences",
held at Kyoto University from September 12th to 14th, 2011. Submitted to the
RIMS Kokyuroku serie
Disorder relevance for the random walk pinning model in dimension 3
We study the continuous time version of the random walk pinning model, where
conditioned on a continuous time random walk Y on Z^d with jump rate \rho>0,
which plays the role of disorder, the law up to time t of a second independent
random walk X with jump rate 1 is Gibbs transformed with weight e^{\beta
L_t(X,Y)}, where L_t(X,Y) is the collision local time between X and Y up to
time t. As the inverse temperature \beta varies, the model undergoes a
localization-delocalization transition at some critical \beta_c>=0. A natural
question is whether or not there is disorder relevance, namely whether or not
\beta_c differs from the critical point \beta_c^{ann} for the annealed model.
In Birkner and Sun [BS09], it was shown that there is disorder irrelevance in
dimensions d=1 and 2, and disorder relevance in d>=4. For d>=5, disorder
relevance was first proved by Birkner, Greven and den Hollander [BGdH08]. In
this paper, we prove that if X and Y have the same jump probability kernel,
which is irreducible and symmetric with finite second moments, then there is
also disorder relevance in the critical dimension d=3, and
\beta_c-\beta^{ann}_c is at least of the order e^{-C(\zeta)\rho^{-\zeta}},
C(\zeta)>0, for any \zeta>2. Our proof employs coarse graining and fractional
moment techniques, which have recently been applied by Lacoin [L09] to the
directed polymer model in random environment, and by Giacomin, Lacoin and
Toninelli [GLT09] to establish disorder relevance for the random pinning model
in the critical dimension. Along the way, we also prove a continuous time
version of Doney's local limit theorem [D97] for renewal processes with
infinite mean.Comment: 36 pages, revised version following referee's comments. Change of
title. Added a monotonicity result (Theorem 1.3) on the critical point shift
shown to us by the referee
One-dimensional Voter Model Interface Revisited
We consider the voter model on Z, starting with all 1's to the left of the
origin and all 0's to the right of the origin. It is known that if the
associated random walk kernel p has zero mean and a finite r-th moment for any
r>3, then the evolution of the boundaries of the interface region between 1's
and 0's converge in distribution to a standard Brownian motion (B_t)_{t>0}
under diffusive scaling of space and time. This convergence fails when p has an
infinite r-th moment for any r<3, due to the loss of tightness caused by a few
isolated 1's appearing deep within the regions of all 0's (and vice versa) at
exceptional times. In this note, we show that as long as p has a finite second
moment, the measure-valued process induced by the rescaled voter model
configuration is tight, and converges weakly to the measure-valued process
1_{x0.Comment: revised versio
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
Subdiffusivity of a random walk among a Poisson system of moving traps on
We consider a random walk among a Poisson system of moving traps on . In earlier work [DGRS12], the quenched and annealed survival probabilities
of this random walk have been investigated. Here we study the path of the
random walk conditioned on survival up to time in the annealed case and
show that it is subdiffusive. As a by-product, we obtain an upper bound on the
number of so-called thin points of a one-dimensional random walk, as well as a
bound on the total volume of the holes in the random walk's range
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