169 research outputs found
T. E. Harris' contributions to interacting particle systems and percolation
Interacting particle systems and percolation have been among the most active
areas of probability theory over the past half century. Ted Harris played an
important role in the early development of both fields. This paper is a bird's
eye view of his work in these fields, and of its impact on later research in
probability theory and mathematical physics.Comment: Published in at http://dx.doi.org/10.1214/10-AOP593 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Finitely dependent coloring
We prove that proper coloring distinguishes between block-factors and
finitely dependent stationary processes. A stochastic process is finitely
dependent if variables at sufficiently well-separated locations are
independent; it is a block-factor if it can be expressed as an equivariant
finite-range function of independent variables. The problem of finding
non-block-factor finitely dependent processes dates back to 1965. The first
published example appeared in 1993, and we provide arguably the first natural
examples. More precisely, Schramm proved in 2008 that no stationary 1-dependent
3-coloring of the integers exists, and conjectured that no stationary
k-dependent q-coloring exists for any k and q. We disprove this by constructing
a 1-dependent 4-coloring and a 2-dependent 3-coloring, thus resolving the
question for all k and q.
Our construction is canonical and natural, yet very different from all
previous schemes. In its pure form it yields precisely the two finitely
dependent colorings mentioned above, and no others. The processes provide
unexpected connections between extremal cases of the Lovasz local lemma and
descent and peak sets of random permutations. Neither coloring can be expressed
as a block-factor, nor as a function of a finite-state Markov chain; indeed, no
stationary finitely dependent coloring can be so expressed. We deduce
extensions involving d dimensions and shifts of finite type; in fact, any
non-degenerate shift of finite type also distinguishes between block-factors
and finitely dependent processes
How likely is an i.i.d. degree sequence to be graphical?
Given i.i.d. positive integer valued random variables D_1,...,D_n, one can
ask whether there is a simple graph on n vertices so that the degrees of the
vertices are D_1,...,D_n. We give sufficient conditions on the distribution of
D_i for the probability that this be the case to be asymptotically 0, {1/2} or
strictly between 0 and {1/2}. These conditions roughly correspond to whether
the limit of nP(D_i\geq n) is infinite, zero or strictly positive and finite.
This paper is motivated by the problem of modeling large communications
networks by random graphs.Comment: Published at http://dx.doi.org/10.1214/105051604000000693 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Integrals, Partitions, and Cellular Automata
We prove that where
is the decreasing function that satisfies , for . When
is an integer and we deduce several combinatorial results. These
include an asymptotic formula for the number of integer partitions not having
consecutive parts, and a formula for the metastability thresholds of a
class of threshold growth cellular automaton models related to bootstrap
percolation.Comment: Revised version. 28 pages, 2 figure
A contact process with mutations on a tree
Consider the following stochastic model for immune response. Each pathogen
gives birth to a new pathogen at rate . When a new pathogen is born,
it has the same type as its parent with probability . With probability
, a mutation occurs, and the new pathogen has a different type from all
previously observed pathogens. When a new type appears in the population, it
survives for an exponential amount of time with mean 1, independently of all
the other types. All pathogens of that type are killed simultaneously. Schinazi
and Schweinsberg (2006) have shown that this model on behaves rather
differently from its non-spatial version. In this paper, we show that this
model on a homogeneous tree captures features from both the non-spatial version
and the version. We also obtain comparison results between this model
and the basic contact process on general graphs
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