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1.
M.S. Bernabei 《Probability Theory and Related Fields》2001,119(3):410-432
The Central Limit Theorem for a model of discrete-time random walks on the lattice ℤν in a fluctuating random environment was proved for almost-all realizations of the space-time nvironment, for all ν > 1 in
[BMP1] and for all ν≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for
ν≥ 3 is finite. In the present paper we consider the cases ν = 1,2 and prove the Central Limit Theorem as T→∞ for the random correction to the first two cumulants. The rescaling factor for theaverage is for ν = 1 and (ln T), for ν=2; for the covariance it is , ν = 1,2.
Received: 25 November 1999 / Revised version: 7 June 2000 / Published online: 15 February 2001 相似文献
2.
We show that an i.i.d. uniformly colored scenery on ℤ observed along a random walk path with bounded jumps can still be reconstructed
if there are some errors in the observations. We assume the random walk is recurrent and can reach every point with positive
probability. At time k, the random walker observes the color at her present location with probability 1−δ and an error Y
k
with probability δ. The errors Y
k
, k≥0, are assumed to be stationary and ergodic and independent of scenery and random walk. If the number of colors is strictly
larger than the number of possible jumps for the random walk and δ is sufficiently small, then almost all sceneries can be
almost surely reconstructed up to translations and reflections.
Received: 3 February 2002 / Revised version: 15 January 2003 Published online: 28 March 2003
Mathematics Subject Classification (2000): 60K37, 60G50
Key words or phrases:Scenery reconstruction – Random walk – Coin tossing problems 相似文献
3.
We consider biased random walk on supercritical percolation clusters in ℤ2. We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random
walk has speed zero, while if the bias is small enough, the speed of the random walk is positive.
Received: 20 November 2002 / Revised version: 17 January 2003
Published online: 15 April 2003
Research supported by Microsoft Research graduate fellowship.
Research partially supported by the DFG under grant SPP 1033.
Research partially supported by NSF grant #DMS-0104073 and by a Miller Professorship at UC Berkeley.
Mathematics Subject Classification (2000): 60K37; 60K35; 60G50
Key words or phrases: Percolation – Random walk 相似文献
4.
We study models of continuous time, symmetric, ℤd-valued random walks in random environments. One of our aims is to derive estimates on the decay of transition probabilities
in a case where a uniform ellipticity assumption is absent. We consider the case of independent conductances with a polynomial
tail near 0 and obtain precise asymptotics for the annealed return probability and convergence times for the random walk confined
to a finite box. 相似文献
5.
Eric David Belsley 《Probability Theory and Related Fields》1998,112(4):493-533
When run on any non-bipartite q-distance regular graph from a family containing graphs of arbitrarily large diameter d, we show that d steps are necessary and sufficient to drive simple random walk to the uniform distribution in total variation distance, and
that a sharp cutoff phenomenon occurs. For most examples, we determine the set on which the variation distance is achieved,
and the precise rate at which it decays.
The upper bound argument uses spectral methods – combining the usual Cauchy-Schwarz bound on variation distance with a bound
on the tail probability of a first-hitting time, derived from its generating function.
Received: 2 April 1997 / Revised version: 10 May 1998 相似文献
6.
A.E. Kyprianou 《Probability Theory and Related Fields》2000,116(3):405-419
We discuss the construction of stopping lines in the branching random walk and thus the existence of a class of supermartingales
indexed by sequences of stopping lines. Applying a method of Lyons (1997) and Lyons, Pemantle and Peres (1995) concerning
size biased branching trees, we establish a relationship between stopping lines and certain stopping times. Consequently we
develop conditions under which these supermartingales are also martingales. Further we prove a generalization of Biggins'
Martingale Convergence Theorem, Biggins (1977a) within this context.
Received: 21 December 1998 / Revised version: 28 June 1999 / Published online: 7 February 2000 相似文献
7.
8.
Remco van der Hofstad Frank den Hollander Wolfgang König 《Probability Theory and Related Fields》2003,125(4):483-521
In this paper we present a new and flexible method to show that, in one dimension, various self-repellent random walks converge
to self-repellent Brownian motion in the limit of weak interaction after appropriate space-time scaling. Our method is based
on cutting the path into pieces of an appropriately scaled length, controlling the interaction between the different pieces,
and applying an invariance principle to the single pieces. In this way, we show that the self-repellent random walk large
deviation rate function for the empirical drift of the path converges to the self-repellent Brownian motion large deviation
rate function after appropriate scaling with the interaction parameters. The method is considerably simpler than the approach
followed in our earlier work, which was based on functional analytic arguments applied to variational representations and
only worked in a very limited number of situations.
We consider two examples of a weak interaction limit: (1) vanishing self-repellence, (2) diverging step variance. In example
(1), we recover our earlier scaling results for simple random walk with vanishing self-repellence and show how these can be
extended to random walk with steps that have zero mean and a finite exponential moment. Moreover, we show that these scaling
results are stable against adding self-attraction, provided the self-repellence dominates. In example (2), we prove a conjecture
by Aldous for the scaling of self-avoiding walk with diverging step variance. Moreover, we consider self-avoiding walk on
a two-dimensional horizontal strip such that the steps in the vertical direction are uniform over the width of the strip and
find the scaling as the width tends to infinity.
Received: 6 March 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003
Mathematics Subject Classification (2000): 60F05, 60F10, 60J55, 82D60
Key words or phrases: Self-repellent random walk and Brownian motion – Invariance principles – Large deviations – Scaling limits – Universality 相似文献
9.
Alain-Sol Sznitman 《Journal of the European Mathematical Society》2000,2(2):93-143
This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s
condition, we show a central limit theorem for random walks in random environment on ℤ
d
, when d≥2. We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the
natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed
in [25] plays an important role in the investigation of both problems. This article also improves the previous work of the
author [24], concerning estimates of probabilities of slowdowns for walks which are neutral or biased to the right.
Received May 31, 1999 / final version received January 18, 2000?Published online April 19, 2000 相似文献
10.
Silke W.W. Rolles 《Probability Theory and Related Fields》2006,135(2):216-264
Let G be a finite tree. It is shown that edge-reinforced random walk on ℤ×G with large initial weights is recurrent. This includes recurrence on multi-level ladders of arbitrary width. For edge-reinforced
random walk on {0,1, . . . ,n}×G, it is proved that asymptotically, with high probability, the normalized edge local times decay exponentially in the distance
from the starting level. The estimates are uniform in n. They are used in the recurrence proof. 相似文献
11.
Michel Talagrand 《Probability Theory and Related Fields》2001,119(2):187-212
We give a completely rigorous proof that the replica-symmetric solution holds at high enough temperature for the random K-sat problem. The most notable feature of this problem is that the order parameter of the system is a function and not a number.
Received: 21 April 1998 / Revised version: 24 April 2000 / Published online: 21 December 2000 相似文献
12.
We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional
and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed
into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting
of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give
sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are
applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise.
Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001 相似文献
13.
We present an upper bound O(n
2
) for the mixing time of a simple random walk on upper triangular matrices. We show that this bound is sharp up to a constant,
and find tight bounds on the eigenvalue gap. We conclude by applying our results to indicate that the asymmetric exclusion
process on a circle indeed mixes more rapidly than the corresponding symmetric process.
Received: 25 January 1999 / Revised version: 17 September 1999 / Published online: 14 June 2000 相似文献
14.
Ekaterina Vl. Bulinskaya 《Lithuanian Mathematical Journal》2011,51(3):310-321
The paper completes the investigation of limit distribution of the number of particles at the source of branching in the model
of critical catalytic branching random walk on
^dd N {{\mathbb Z}^d}\;d \in {\mathbb N} . Limit theorems of such kind were established only for d = 1, 2, 3, 4 under the assumption that, at the initial moment, there is a single particle at the source of branching. We prove their
analog for
d \geqslant 5 d \geqslant 5 . Moreover, in any dimension, we generalize the previous results by permitting the initial particle to start at an arbitrary
point of the lattice. 相似文献
15.
Paul Seignourel 《Probability Theory and Related Fields》2000,118(3):293-322
We present discrete schemes for processes in random media. We prove two results. The first one is the convergence of Sinai's
random walks in random environments to the Brox model. The second one is the convergence of random walks in media with random
“gates” to a continuous process in a Poisson potential. The proofs are based on the following idea: we consider the discrete
media as random potentials for continuous models.
Received: 6 May 1999 / Revised version: 18 October 1999 / Published online: 20 October 2000 相似文献
16.
E.G. Coffman Jr. George S. Lueker Joel Spencer Peter M. Winkler 《Probability Theory and Related Fields》2001,120(4):585-599
A random rectangle is the product of two independent random intervals, each being the interval between two random points
drawn independently and uniformly from [0,1]. We prove that te number C
n
of items in a maximum cardinality disjoint subset of n random rectangles satisfies
where K is an absolute constant. Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we are able to show that, for some absolute constat K,
Finally, for a certain distribution of random cubes we show that for some absolute constant K, the number Q
n
of items in a maximum cardinality disjoint subset of the cubes satisies
Received: 1 September 1999 / Revised version: 3 November 2000 / Published online: 14 June 2001 相似文献
17.
Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability
one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundaryΩ of the tree. The random subset Λ of the boundary consisting of all ends of the tree in which the population survives, called
the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric
boundary. Moreover, there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic. It is further shown that in all cases there is a distinguished probability measure μ supported by Ω such that the Hausdorff
dimension of Λ∩Ωμ, where Ωμ is the set of μ-generic points of Ω, converges to one half the Hausdorff dimension of Ωμ at the phase separation point. Exact formulas are obtained for the Hausdorff dimensions of Λ and Λ∩Ωμ, and it is shown that the log Hausdorff dimension of Λ has critical exponent 1/2 at the phase separation point.
Received: 30 June 1998 / Revised version: 10 March 1999 相似文献
18.
B. M. Hambly 《Probability Theory and Related Fields》2000,117(2):221-247
We consider natural Laplace operators on random recursive affine nested fractals based on the Sierpinski gasket and prove
an analogue of Weyl’s classical result on their eigenvalue asymptotics. The eigenvalue counting function N(λ) is shown to be of order λ
ds/2
as λ→∞ where we can explicitly compute the spectral dimension d
s
. Moreover the limit N(λ) λ
−ds/2
will typically exist and can be expressed as a deterministic constant multiplied by a random variable. This random variable
is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume
of the fractal.
Received: 22 January 1999 / Revised version: 2 September 1999 /?Published online: 30 March 2000 相似文献
19.
C. Boldrighini R. A. Minlos A. Pellegrinotti 《Probability Theory and Related Fields》1997,109(2):245-273
Summary We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ
t
(x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(X
t
+1= y|X
t
= x,ξ
t
=η) =P
0( y−x)+ c(y−x;η(x)). We assume that the variables {ξ
t
(x):(t,x) ∈ℤν+1} are i.i.d., that both P
0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P
0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X
t
, and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X
t
and a corresponding correction of order to the C.L.T.. Proofs are based on some new L
p
estimates for a class of functionals of the field.
Received: 4 January 1996/In revised form: 26 May 1997 相似文献
20.
Tokuzo Shiga 《Probability Theory and Related Fields》1997,108(3):417-439
Summary. We study the exponential decay rate of the survival probability up to time t>0 of a random walker moving in Zopf;
d
in a temporally and spatially fluctuating random environment. When the random walker has a speed parameter κ>0, we investigate
the influence of κ on the exponential decay rate λ(d,κ). In particular we prove that for any fixed d≥1, λ(d,κ) behaves like as logκ as κ↘0.
Received: 21 May 1996 / In revised form: 2 February 1997 相似文献