319 research outputs found
Averaged large deviations for random walk in a random environment
In his 2003 paper, Varadhan proves the averaged large deviation principle for
the mean velocity of a particle taking a nearest-neighbor random walk in a
uniformly elliptic i.i.d. environment on with , and
gives a variational formula for the corresponding rate function . Under
Sznitman's transience condition (T), we show that is strictly convex and
analytic on a non-empty open set , and that the true velocity of
the particle is an element (resp. in the boundary) of when the
walk is non-nestling (resp. nestling). We then identify the unique minimizer of
Varadhan's variational formula at any velocity in .Comment: 14 pages. In this revised version, I state and prove all of the
results under Sznitman's (T) condition instead of Kalikow's condition. Also,
I rewrote many parts of Section 1, streamlined some of the proofs in Section
2, fixed some typos, and improved the wording here and there. Accepted for
publication in Annales de l'Institut Henri Poincar
The stochastic encounter-mating model
We propose a new model of permanent monogamous pair formation in zoological
populations with multiple types of females and males. According to this model,
animals randomly encounter members of the opposite sex at their so-called
firing times to form temporary pairs which then become permanent if mating
happens. Given the distributions of the firing times and the mating preferences
upon encounter, we analyze the contingency table of permanent pair types in
three cases: (i) definite mating upon encounter; (ii) Poisson firing times; and
(iii) Bernoulli firing times. In the first case, the contingency table has a
multiple hypergeometric distribution which implies panmixia. The other two
cases generalize the encounter-mating models of Gimelfarb (1988) who gives
conditions that he conjectures to be sufficient for panmixia. We formulate
adaptations of his conditions and prove that they not only characterize
panmixia but also allow us to reduce the model to the first case by changing
its underlying parameters. Finally, when there are only two types of females
and males, we provide a full characterization of panmixia, homogamy and
heterogamy.Comment: 27 pages. We shortened the abstract, added Section 1.1 (Overview),
and updated reference
Differing averaged and quenched large deviations for random walks in random environments in dimensions two and three
We consider the quenched and the averaged (or annealed) large deviation rate
functions and for space-time and (the usual) space-only RWRE on
. By Jensen's inequality, . In the space-time case,
when , and are known to be equal on an open set
containing the typical velocity . When , we prove that and
are equal only at . Similarly, when d=2+1, we show that
on a punctured neighborhood of . In the space-only case, we provide a
class of non-nestling walks on with d=2 or 3, and prove that
and are not identically equal on any open set containing
whenever the walk is in that class. This is very different from the known
results for non-nestling walks on with .Comment: 21 pages. In this revised version, we corrected our computation of
the variance of for (page 11 of the old version, after
(2.31)). We also added details explaining precisely how the space-only case
is handled, by mapping the appropriate objects to the space-time setup (see
pages 14--17 in the new version). Accepted for publication in Communications
in Mathematical Physics
Large deviations for random walk in a space--time product environment
We consider random walk on in a space--time
product environment . We take the point of view of the
particle and focus on the environment Markov chain
where denotes the shift on . Conditioned on the particle having
asymptotic mean velocity equal to any given , we show that the empirical
process of the environment Markov chain converges to a stationary process
under the averaged measure. When and is
sufficiently close to the typical velocity, we prove that averaged and quenched
large deviations are equivalent and when conditioned on the particle having
asymptotic mean velocity , the empirical process of the environment Markov
chain converges to under the quenched measure as well. In
this case, we show that is a stationary Markov process
whose kernel is obtained from the original kernel by a Doob -transform.Comment: Published in at http://dx.doi.org/10.1214/08-AOP400 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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