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Asymptotic extinction probability of an absorbing
random walk in random environment
AUGU
S
Stefan Junk, Prof. Dr. Nina Gantert
Center for Mathematical Sciences, Technische Universität München
Chair of Probability Theory
Abstract
A Random Walk in Random Environment (RWRE) is a process in which the transition probabilities of a markov chain on Z are chosen randomly and then the joint probability
measure of environment and random walk is considered. Motivated by a model from statistical mechanics we consider a version of RWRE where the process can also die out. In a
first step towards describing the distribution of the RWRE conditioned on survival the asymptotical rate with which the probability for survival goes to zero is determined. A number
of cases is given in which this can be estimated and it is shown that there are many different possible rates. For this a potential function is introduced on fixed environments and
we consider valleys in which the process will spend a long time.
Random Walks in Random Environments
Valleys of the Potential
Let Ω be the set of all transition probabilities on Z:
The event {τ > n} depends on properties of the environment as well as the random
walk. In order to separate those it is helpful to introduce for some fixed ω ∈ Ω a
potential function:
Ω := ω = (ωx− , ωxo , ωx+ )x∈Z ∀x ∈ Z : ωx+ , ωxo , ωx− ≥ 0, ωx+ + ωxo + ωx− = 1
One can interpret ω ∈ Ω as the transition probabilities of a markov chain
(Xn )n∈N on Z started in zero.
This is refered to as an environment.
 Px
for x > 0

i=0 ln ρi
0
for x = 0
V (x) :=
 P0
− i=x ln ρi for x < 0
When the process (Xn )n∈N is at position
x ∈ Z at some point in time, in the next
step it moves to the right with probability
ωx+ , to the left with probability ωx− and
stays with probability ωxo = 1 − ωx+ − ωx− .
This probability measure is denoted Pω .
The space Ω is then given a suitable σ-Algebra F such that the mapping
ω 7→ Pω (B)
is F-measurable for each B in the σ-algebra of Pω . We now choose the environment
randomly according to some probability measure P on (Ω, F) and consider the joint
measure P := P × Pω . That is,
ωi−
where ρi := + for i ∈ Z
ωi
We are interested in parts of the
environment where the potential
forms a valley in which the
random walk will stay for a long
time
We say there is a valley
at (a, b, c) if b has minimal
potential in I = [a, c], a has
maximal potential in [a, b] and
c has maximal potential in [b, c].
Its depth is denoted by H := min{V (b) − V (a), V (c) − V (b)}.
The following technical lemma contains the crucial information:
Z
P(A × B) =
Pω (B)P (dω)
A
where A is an event of the environment and B an event on the random walk. P
is called the annealed measure and this can be interpreted this as first averaging
over the environment and then considering the behavior of a random walk on typical
environments.
The main difficulty in dealing with RWREs is that (Xn )n∈N is no longer a markov chain
with respect to P: When given information on the whole evolution of the process up
to some time one may get additional information on the environment the process sees
in the future based on which parts one has already seen.
Physical Motivation
In statistical mechanics one can model the behavior of polymer folding in a solution
by considering a simple random walk (Xn )n∈N on Zd . For very long polymers the
distribution of the random walk can be used to determine certain physical properties.
One is for example interested in the distance between the end points of a polymer,
that is kXn − X0 k2 .
One can take small differences in the density of the solution into account by adapting
this model: One considers percolation on Z2 , that is edges are open with probability
p independently of each other. Whenever the random walk from before tries to use
a closed edge, it instead stays at the same site. It is known that for p > 21 there is
a unique open cluster that connects to infinity (P-almost surely) and the process is
assumed to start in this cluster.
However now the process (Xn )n∈N has a high probability of being at a site with few
outgoing edges, which corresponds to an unfavourable part of the solution. To counter
this one introduces a possibility for extinction: Whenever the RWRE stays at the same
site, it dies with a fixed probability r ∈ (0, 1). Now one can again try to determine
the distribution of kXn − X0 k2 conditioned on survival up to time n.
Adaption of the Model
Let τ be the extinction time, that is the first time n such that the RWRE stays at the
same place and where ξn = 1. Here the (ξn )n≥1 are i.i.d. Bernoulli random variables
with sucess probability r ∈ (0, 1).
We restrict to the case d = 1 which allows to use a potential on the environment, see
next box. We assume that P is a product measure so that (ω x )x∈Z are i.i.d. random
variables. Moreover P is assumed to be uniformly elliptic, that is there exists some
0 > 0 such that P-almost surely: ωo+ ≥ 0 and ωo− ≥ 0 . Under our assumptions it is
easy to show that
lim P(τ > n) = 0
n→∞
As a first step towards computing the distribtions mentioned in the previous box we
have examined in the asymptotical rate with which this probability goes to zero.
Lemma. Assume ω satisfies ωxo = 0 for all x ∈ [a, c] and that there is a valley of
depth H at (a, 0, c). Denote T the first time the random walk started in zero hits
either a or c.Then there are constants γ1 , γ2 such that
n
γ2
n
exp −γ1 ln(2(c − a)) H ≤ Pω (T > n) ≤ exp −
e
(c − a)4 eH
Results
The lemma in the previous box suggests that if there is a valley of length Θ(ln n) and
depth Θ(ln n), then the probability of staying in it until time n decays slower than
polynomially in n. On the other hand using large deviations techniques it is relatively
easy to show that under suitable conditions the probability of having such a valley goes
to zero with polynomial rate. In order to get a lower bound it is therefore enough to
assure that the process survives whenever it starts in such a valley.
The results show that this lower bound is also exact. More precisely consider
1 +
1
1 −
1
o
−
o
+
p(n) := min P ωo ≤ , ωo > ωo +
, P ωo ≤ , ωo > ωo +
n
n
n
n
With this we can formulate the main result:
Theorem. Assume p(∞) := min{P (ωoo = 0, ωo+ > 12 ), P (ωoo = 0, ωo− > 12 )} > 0.
Then there is a C1 ∈ (0, ∞) such that
lim −
n→∞
ln P(τ > n)
= C1
ln n
If p(n0 ) is zero for some n0 we have that for some C2 ∈ (0, 1]:
C2 ≤ lim inf
n→∞
ln(− ln P(τ > n))
ln(− ln P(τ > n))
≤ lim sup
≤1
ln n
ln n
n→∞
The case when p(n) goes to zero but stays positive for all n is more delicate. The
results in this case show that the extinction probability can decay with a wide variety
of different rates.
References
1. O. Zeitouni: Lecture Notes on Random Walks in Random Environments; in: Lecture Notes in Mathematics 1837, Lectures on
Probability Theory and Statistics (2004), Seiten 191-308
2. A. Giacometti, A. Maritan, H. Nakanishi: Statistical mechanics of random paths on disordered lattices; in: Journal of Statistical
Physics 75, Nummern 3-4 (1994), Seiten 669-706x
3. N. Gantert, S. Popov, M. Vachkovskaia: Survival time of random walk in random environment among soft obstacles; in: Electronic
Journal of Probability 14 (2009), Seiten 569-593