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Recurrence and transience for suspension flows  [PDF]
Godofredo Iommi,Thomas Jordan,Mike Todd
Mathematics , 2012,
Abstract: We study the thermodynamic formalism for suspension flows over countable Markov shifts with roof functions not necessarily bounded away from zero. We establish conditions to ensure the existence and uniqueness of equilibrium measures for regular potentials. We define the notions of recurrence and transience of a potential in this setting. We define the "renewal flow", which is a symbolic model for a class of flows with diverse recurrence features. We study the corresponding thermodynamic formalism, establishing conditions for the existence of equilibrium measures and phase transitions. Applications are given to suspension flows defined over interval maps having parabolic fixed points.
Recurrence and transience for the frog model on trees  [PDF]
Christopher Hoffman,Tobias Johnson,Matthew Junge
Mathematics , 2014,
Abstract: The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite $d$-ary tree. We prove the model undergoes a phase transition, finding it recurrent for $d=2$ and transient for $d\geq 5$. Simulations suggest strong recurrence for $d=2$, weak recurrence for $d=3$, and transience for $d\geq 4$. Additionally, we prove a 0-1 law for all $d$-ary trees, and we exhibit a graph on which a 0-1 law does not hold.
The dichotomy of recurrence and transience of semi-Levy processes  [PDF]
Makoto Maejima,Taisuke Takamune,Yohei Ueda
Mathematics , 2012,
Abstract: Semi-Levy process is an additive process with periodically stationary increments. In particular, it is a generalization of Levy process. The dichotomy of recurrence and transience of Levy processes is well known, but this is not necessarily true for general additive processes. In this paper, we prove the recurrence and transience dichotomy of semi-Levy processes. For the proof, we introduce a concept of semi-random walk and discuss its recurrence and transience properties. An example of semi-Levy process constructed from two independent Levy processes is investigated. Finally, we prove the laws of large numbers for semi-Levy processes.
Walking within growing domains: recurrence versus transience  [PDF]
Amir Dembo,Ruojun Huang,Vladas Sidoravicius
Mathematics , 2013,
Abstract: For normally reflected Brownian motion and for simple random walk on independently growing in time d-dimensional domains, d>=3, we establish a sharp criterion for recurrence versus transience in terms of the growth rate.
Monotone interaction of walk and graph: recurrence versus transience  [PDF]
Amir Dembo,Ruojun Huang,Vladas Sidoravicius
Mathematics , 2014,
Abstract: We consider recurrence versus transience for models of random walks on domains of $\mathbb{Z}^d$, in which monotone interaction enforces domain growth as a result of visits by the walk (or probes it sent), to the neighborhood of domain boundary.
An Overshoot Approach to Recurrence and Transience of Markov Processes  [PDF]
Bj?rn B?ttcher
Mathematics , 2010,
Abstract: We develop criteria for recurrence and transience of one-dimensional Markov processes which have jumps and oscillate between $+\infty$ and $-\infty$. The conditions are based on a Markov chain which only consists of jumps (overshoots) of the process into complementary parts of the state space. In particular we show that a stable-like process with generator $-(-\Delta)^{\alpha(x)/2}$ such that $\alpha(x)=\alpha$ for $x<-R$ and $\alpha(x)=\beta$ for $x>R$ for some $R>0$ and $\alpha,\beta\in(0,2)$ is transient if and only if $\alpha+\beta<2$, otherwise it is recurrent. As a special case this yields a new proof for the recurrence, point recurrence and transience of symmetric $\alpha$-stable processes.
Degrees of transience and recurrence and hierarchical random walks  [PDF]
D. A. Dawson,L. G. Gorostiza,A. Wakolbinger
Mathematics , 2004,
Abstract: The notion of degree and related notions concerning recurrence and transience for a class of L'evy processes on metric Abelian groups are studied. The case of random walks on a hierarchical group is examined with emphasis on the role of the ultrametric structure of the group and on analogies and differences with Euclidean random walks. Applications to separation of time scales and occupation times of multilevel branching systems are discussed.
From transience to recurrence with Poisson tree frogs  [PDF]
Christopher Hoffman,Tobias Johnson,Matthew Junge
Mathematics , 2015,
Abstract: Consider the following interacting particle system on the d-ary tree, known as the frog model: Initially, one particle is awake at the root and i.i.d. Poisson many particles are sleeping at every other vertex. Particles that are awake perform simple random walks, awakening any sleeping particles they encounter. We prove that there is a phase transition between transience and recurrence as the initial density of particles increases, and we give the order of the transition up to a logarithmic factor.
Recurrence and transience for long-range reversible random walks on a random point process  [PDF]
P. Caputo,A. Faggionato,A. Gaudilliere
Mathematics , 2008,
Abstract: We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.
Transience, Recurrence and Critical Behavior for Long-Range Percolation  [PDF]
Noam Berger
Mathematics , 2001, DOI: 10.1007/s002200200617
Abstract: We study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions $d=1,2$, where $x$ and $y$ a re connected with probability $\sim\beta/\|x-y\|^{-s}$. We show that when $d
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