arXiv subjects

Jonathon Peterson

Publications and source records attributed to Jonathon Peterson.

34 records · Page 2Linked to original sources

arXiv2016

Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $κ>0$ that determines the fluctuations of the process. When $0<κ<2$, the averaged distributions of the hitting times of the random walk converge to a $κ$-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for almost every fixed environment, the distributions of the hitting times (centered and scaled in any manner) converge to a non-degenerate distribution. We show, however, that the quenched distributions do have a limit in the weak sense. That is, the quenched distributions of the hitting times -- viewed as a random probability measure -- converge in distribution to a random probability measure, which has interesting stability properties. Our results generalize both the averaged limiting distribution and the non-existence of quenched limiting distributions.

math.PR↗

arXiv2016

Weak weak quenched limits for the path-valued processes of hitting times and positions of a transient, one-dimensional random walk in a random environment

In this article we continue the study of the quenched distributions of transient, one-dimensional random walks in a random environment. In a previous article we showed that while the quenched distributions of the hitting times do not converge to any deterministic distribution, they do have a weak weak limit in the sense that - viewed as random elements of the space of probability measures - they converge in distribution to a certain random probability measure (we refer to this as a weak weak limit because it is a weak limit in the weak topology). Here, we improve this result to the path-valued process of hitting times. As a consequence, we are able to also prove a weak weak quenched limit theorem for the path of the random walk itself.

math.PR↗

arXiv2016

Large deviations and slowdown asymptotics for one-dimensional excited random walks

We study the large deviations of one-dimensional excited random walks. We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions. When the excited random walk is transient with positive speed $v_0$, then the large deviation rate function for the position of the excited random walk is zero on the interval $[0,v_0]$ and so probabilities such as $P(X_n < nv)$ for $v \in (0,v_0)$ decay subexponentially. We show that rate of decay for such slowdown probabilities is polynomial of the order $n^{1-δ/2}$, where $δ>2$ is the expected total drift per site of the cookie environment.

math.PR↗

arXiv2016

Strict monotonicity properties in one-dimensional excited random walks

We consider one-dimensional excited random walks with finitely many cookies at each site. There are certain natural monotonicity results that are known for the excited random walk under some partial orderings of the cookie environments. We improve these monotonicity results to be strictly monotone under a partial ordering of cookie environments introduced by Holmes and Salisbury. While the self-interacting nature of the excited random walk makes a direct coupling proof difficult, we show that there is a very natural coupling of the associated branching process from which the monotonicity results follow.

math.PR↗

arXiv2016

Extreme slowdowns for one-dimensional excited random walks

We study the asymptotics of the probabilities of extreme slowdown events for transient one-dimensional excited random walks. That is, if $\{X_n\}_{n\geq 0}$ is a transient one-dimensional excited random walk and $T_n = \min\{ k: \, X_k = n\}$, we study the asymptotics of probabilities of the form $P(X_n \leq n^γ)$ and $P(T_{n^γ} \geq n )$ with $γ< 1$. We show that there is an interesting change in the rate of decay of these extreme slowdown probabilities when $γ< 1/2$.

math.PR↗

arXiv2016

Strong transience of one-dimensional random walk in a random environment

A transient stochastic process is considered strongly transient if conditioned on returning to the starting location, the expected time it takes to return the the starting location is finite. We characterize strong transience for a one-dimensional random walk in a random environment. We show that under the quenched measure transience is equivalent to strong transience, while under the averaged measure strong transience is equivalent to ballisticity (transience with non-zero limiting speed).

math.PR↗

arXiv2016

A probabilistic proof of a binomial identity

We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.

math.PR↗

arXiv2016

Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment

We consider a system of independent random walks in a common random environment. Previously, a hydrodynamic limit for the system of RWRE was proved under the assumption that the random walks were transient with positive speed. In this paper we instead consider the case where the random walks are transient but with a sublinear speed of the order $n^κ$ for some $κ\in (0,1)$ and prove a quenched hydrodynamic limit for the system of random walks with time scaled by $n^{1/κ}$ and space scaled by $n$. The most interesting feature of the hydrodynamic limit is that the influence of the environment does not average out under the hydrodynamic scaling; that is, the asymptotic particle density depends on the specific environment chosen. The hydrodynamic limit for the system of RWRE is obtained by first proving a hydrodynamic limit for a system of independent particles in a directed trap environment.

math.PR↗

arXiv2016

Excited random walks with non-nearest neighbor steps

Let $W$ be an integer valued random variable satisfying $E[W] =: δ\geq 0$ and $P(W<0)>0$, and consider a self-interacting random walk that behaves like a simple symmetric random walk with the exception that on the first visit to any integer $x\in \mathbb{Z}$ the size of the next step is an independent random variable with the same distribution as $W$. We show that this self-interacting random walk is recurrent if $δ\leq 1$ and transient if $δ>1$. This is a special case of our main result which concerns the recurrence and transience of excited random walks (or cookie random walks) with non-nearest neighbor jumps.

math.PR↗

arXiv2016

Excited random walks with Markovian cookie stacks

We consider a nearest-neighbor random walk on $\mathbb{Z}$ whose probability $ω_x(j)$ to jump to the right from site $x$ depends not only on $x$ but also on the number of prior visits $j$ to $x$. The collection $(ω_x(j))_{x\in\mathbb{Z},n\ge 0}$ is sometimes called the "cookie environment" due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the transition probabilities according to the "strength" of the cookie eaten. We assume that the cookie stacks are i.i.d. and that the cookie "strengths" within the stack $(ω_x(j))_{j\ge 0}$ at site $x$ follow a finite state Markov chain. Thus, the environment at each site is dynamic, but it evolves according to the local time of the walk at each site rather than the original random walk time. The model admits two different regimes, critical or non-critical, depending on whether the expected probability to jump to the right (or left) under the invariant measure for the Markov chain is equal to $1/2$ or not. We show that in the non-critical regime the walk is always transient, has non-zero linear speed, and satisfies the classical central limit theorem. The critical regime allows for a much more diverse behavior. We give necessary and sufficient conditions for recurrence/transience and ballisticity of the walk in the critical regime as well as a complete characterization of limit laws under the averaged measure in the transient case. The setting considered in this paper generalizes the previously studied model with periodic cookie stacks. Our results on ballisticity and limit theorems are new even for the periodic model.

math.PR↗

arXiv2016

Functional limit laws for recurrent excited random walks with periodic cookie stacks

We consider one-dimensional excited random walks (ERWs) with periodic cookie stacks in the recurrent regime. We prove functional limit theorems for these walks which extend the previous results of D. Dolgopyat and E. Kosygina for excited random walks with "boundedly many cookies per site." In particular, in the non-boundary recurrent case the rescaled excited random walk converges in the standard Skorokhod topology to a Brownian motion perturbed at its extrema (BMPE). While BMPE is a natural limiting object for excited random walks with boundedly many cookies per site, it is far from obvious why the same should be true for our model which allows for infinitely many "cookies" at each site. Moreover, a BMPE has two parameters $α,β<1$ and the scaling limits in this paper cover a larger variety of choices for $α$ and $β$ than can be obtained for ERWs with boundedly many cookies per site.

math.PR↗

arXiv2015

Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment

We consider a one dimensional random walk in a random environment (RWRE) with a positive speed $\lim_{n\to\infty}\frac{X_n}{n}=v_α>0$. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities $P_ω(X_n < xn)$ with $x \in (0,v_α)$ decay approximately like $\exp\{-n^{1-1/s}\}$ for a deterministic $s > 1$. More precisely, they showed that $n^{-γ} \log P_ω( X_n < x n)$ converges to $0$ or $-\infty$ depending on whether $γ> 1-1/s$ or $γ< 1-1/s$. In this paper, we improve on this by showing that $n^{-1+1/s} \log P_ω( X_n < x n)$ oscillates between $0$ and $-\infty$, almost surely. This had previously been shown by Gantert only in a very special case of random environments.

math.PR↗

arXiv2011

Quenched limits for transient, zero speed one-dimensional random walk in random environment

We consider a nearest-neighbor, one dimensional random walk $\{X_n\}_{n\geq0}$ in a random i.i.d. environment, in the regime where the walk is transient but with zero speed, so that $X_n$ is of order $n^s$ for some $s<1$. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible: There exist sequences $\{n_k\}$ and $\{x_k\}$ depending on the environment only, such that $X_{n_k}-x_k=o(\log n_k)^2$ (a localized regime). On the other hand, there exist sequences $\{t_m\}$ and $\{s_m\}$ depending on the environment only, such that $\log s_m/\log t_m\to s<1$ and $P_ω(X_{t_m}/s_m\leq x)\to1/2$ for all $x>0$ and $\to0$ for $x\leq0$ (a spread out regime).

math.PR↗

arXiv2010

Current fluctuations of a system of one-dimensional random walks in random environment

We study the current of particles that move independently in a common static random environment on the one-dimensional integer lattice. A two-level fluctuation picture appears. On the central limit scale the quenched mean of the current process converges to a Brownian motion. On a smaller scale the current process centered at its quenched mean converges to a mixture of Gaussian processes. These Gaussian processes are similar to those arising from classical random walks, but the environment makes itself felt through an additional Brownian random shift in the spatial argument of the limiting current process.

math.PR↗

arXiv2010

Maximal Displacement for Bridges of Random Walks in a Random Environment

It is well known that the distribution of simple random walks on $\bf{Z}$ conditioned on returning to the origin after $2n$ steps does not depend on $p= P(S_1 = 1)$, the probability of moving to the right. Moreover, conditioned on $\{S_{2n}=0\}$ the maximal displacement $\max_{k\leq 2n} |S_k|$ converges in distribution when scaled by $\sqrt{n}$ (diffusive scaling). We consider the analogous problem for transient random walks in random environments on $\bf{Z}$. We show that under the quenched law $P_ω$ (conditioned on the environment $ω$), the maximal displacement of the random walk when conditioned to return to the origin at time $2n$ is no longer necessarily of the order $\sqrt{n}$. If the environment is nestling (both positive and negative local drifts exist) then the maximal displacement conditioned on returning to the origin at time $2n$ is of order $n^{κ/(κ+1)}$, where the constant $κ>0$ depends on the law on environment. On the other hand, if the environment is marginally nestling or non-nestling (only non-negative local drifts) then the maximal displacement conditioned on returning to the origin at time $2n$ is at least $n^{1-\varepsilon}$ and at most $n/(\ln n)^{2-\varepsilon}$ for any $\varepsilon>0$. As a consequence of our proofs, we obtain precise rates of decay for $P_ω(X_{2n}=0)$. In particular, for certain non-nestling environments we show that $P_ω(X_{2n}=0) = \exp\{-Cn -C'n/(\ln n)^2 + o(n/(\ln n)^2) \}$ with explicit constants $C,C'>0$.

math.PR↗

arXiv2008

Limiting distributions and large deviations for random walks in random environments

This thesis concerns the study of random walks in random environments (RWRE). Since there are two levels of randomness for random walks in random environments, there are two different distributions for the random walk that can be studied. The quenched distribution is the law of the random walk conditioned on a given environment. The annealed distribution is the quenched law averaged over all environments. The main results of the thesis fall into two categories: quenched limiting distributions for one-dimensional, transient RWRE and annealed large deviations for multidimensional RWRE. The analysis of the quenched distributions for transient, one-dimensional RWRE falls into two separate cases. First, when an annealed central limit theorem holds, we prove that a quenched central limit theorem also holds but with a random (depending on the environment) centering. In contrast, when the annealed limit distribution is not Gaussian, we prove that there is no quenched limiting distribution for the RWRE. Moreover, we show that for almost every environment, there exist two random (depending on the environment) sequences of times, along which random walk has different quenched limiting distributions. While an annealed large deviation principle for multidimensional RWRE was known previously, very little qualitative information was available about the annealed large deviation rate function. We prove that if the law on environments is non-nestling, then the annealed large deviation rate function is analytic in a neighborhood of its unique zero (which is the limiting velocity of the RWRE).

math.PR↗