36 research outputs found
The contact process in a dynamic random environment
We study a contact process running in a random environment in
where sites flip, independently of each other, between blocking and nonblocking
states, and the contact process is restricted to live in the space given by
nonblocked sites. We give a partial description of the phase diagram of the
process, showing in particular that, depending on the flip rates of the
environment, survival of the contact process may or may not be possible for
large values of the birth rate. We prove block conditions for the process that
parallel the ones for the ordinary contact process and use these to conclude
that the critical process dies out and that the complete convergence theorem
holds in the supercritical case.Comment: This version corrects a mistake in the original statement and proof
of Theorem 1(c). Published in at http://dx.doi.org/10.1214/08-AAP528 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations
We consider a branching-selection system in with particles
which give birth independently at rate 1 and where after each birth the
leftmost particle is erased, keeping the number of particles constant. We show
that, as , the empirical measure process associated to the system
converges in distribution to a deterministic measure-valued process whose
densities solve a free boundary integro-differential equation. We also show
that this equation has a unique traveling wave solution traveling at speed
or no such solution depending on whether or , where is the
asymptotic speed of the branching random walk obtained by ignoring the removal
of the leftmost particles in our process. The traveling wave solutions
correspond to solutions of Wiener-Hopf equations.Comment: Published in at http://dx.doi.org/10.1214/10-AOP601 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Extreme statistics of non-intersecting Brownian paths
We consider finite collections of non-intersecting Brownian paths on the
line and on the half-line with both absorbing and reflecting boundary
conditions (corresponding to Brownian excursions and reflected Brownian
motions) and compute in each case the joint distribution of the maximal height
of the top path and the location at which this maximum is attained. The
resulting formulas are analogous to the ones obtained in [MFQR13] for the joint
distribution of and , where is the
Airy process, and we use them to show that in the three cases the joint
distribution converges, as , to the joint distribution of
and . In the case of non-intersecting Brownian
bridges on the line, we also establish small deviation inequalities for the
argmax which match the tail behavior of . Our proofs are based on
the method introduced in [CQR13,BCR15] for obtaining formulas for the
probability that the top line of these line ensembles stays below a given
curve, which are given in terms of the Fredholm determinant of certain
"path-integral" kernels.Comment: Minor corrections, improved exposition. To appear in Electron. J.
Proba
Multiplicative functionals on ensembles of non-intersecting paths
The purpose of this article is to develop a theory behind the occurrence of
"path-integral" kernels in the study of extended determinantal point processes
and non-intersecting line ensembles. Our first result shows how determinants
involving such kernels arise naturally in studying ratios of partition
functions and expectations of multiplicative functionals for ensembles of
non-intersecting paths on weighted graphs. Our second result shows how Fredholm
determinants with extended kernels (as arise in the study of extended
determinantal point processes such as the Airy_2 process) are equal to Fredholm
determinants with path-integral kernels. We also show how the second result
applies to a number of examples including the stationary (GUE) Dyson Brownian
motion, the Airy_2 process, the Pearcey process, the Airy_1 and Airy_{2->1}
processes, and Markov processes on partitions related to the z-measures.Comment: 32 pages, 1 figur
Continuum statistics of the Airy2 process
We develop an exact determinantal formula for the probability that the
Airy process is bounded by a function on a finite interval. As an
application, we provide a direct proof that \sup(\aip(x)-x^2) is distributed
as a GOE random variable. Both the continuum formula and the GOE result have
applications in the study of the end point of an unconstrained directed polymer
in a disordered environment. We explain Johansson's [Joh03] observation that
the GOE result follows from this polymer interpretation and exact results
within that field. In a companion paper [MQR11] these continuum statistics are
used to compute the distribution of the endpoint of directed polymers.Comment: More details added, some minor mistakes correcte
Renormalization fixed point of the KPZ universality class
The one dimensional Kardar-Parisi-Zhang universality class is believed to
describe many types of evolving interfaces which have the same characteristic
scaling exponents. These exponents lead to a natural renormalization/rescaling
on the space of such evolving interfaces. We introduce and describe the
renormalization fixed point of the Kardar-Parisi-Zhang universality class in
terms of a random nonlinear semigroup with stationary independent increments,
and via a variational formula. Furthermore, we compute a plausible formula the
exact transition probabilities using replica Bethe ansatz. The semigroup is
constructed from the Airy sheet, a four parameter space-time field which is the
Airy2 process in each of its two spatial coordinates. Minimizing paths through
this field describe the renormalization group fixed point of directed polymers
in a random potential. At present, the results we provide do not have
mathematically rigorous proofs, and they should at most be considered
proposals.Comment: 19 pages, 2 figures; updated and expanded version with addition
author (Remenik
Exact formulas for random growth with half-flat initial data
We obtain exact formulas for moments and generating functions of the height
function of the asymmetric simple exclusion process at one spatial point,
starting from special initial data in which every positive even site is
initially occupied. These complement earlier formulas of E. Lee [J. Stat. Phys.
140 (2010) 635-647] but, unlike those formulas, ours are suitable in principle
for asymptotics. We also explain how our formulas are related to divergent
series formulas for half-flat KPZ of Le Doussal and Calabrese [J. Stat. Mech.
2012 (2012) P06001], which we also recover using the methods of this paper.
These generating functions are given as a series without any apparent Fredholm
determinant or Pfaffian structure. In the long time limit, formal asymptotics
show that the fluctuations are given by the Airy marginals.Comment: Published at http://dx.doi.org/10.1214/15-AAP1099 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org