Markov Paths, Loops and Fields

Jan, Yves Le

doi:10.1007/978-3-642-21216-1

Cited by 78 publications

(49 citation statements)

References 35 publications

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“…for all 1 i k as well as B 0 . This is because Dη′ N ,EN (1),>λ,ΠN (v, ∂V αN ∩ H >λ ) for v ∈ I >λ k is the same as the graph distance on I >λ k between v and ∂V αN ∩H >λ , and thus I >λ i for 0 i k can be determined (similar for the version of < −λ) and therefore all the other sets can be determined (see (33)). We denote , and such that…”

Section: This Implies That Conditioned On {Lmentioning

confidence: 99%

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Chemical Distances for Percolation of Planar Gaussian Free Fields and Critical Random Walk Loop Soups

Ding

2018

Commun. Math. Phys.

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We initiate the study on chemical distances of percolation clusters for level sets of twodimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a hard wall.where G VN (u, v) is the Green's function for simple random walk, i.e., the expected number of visits to v before reaching ∂V N for a simple random walk started at u. The first goal of the present paper is to study chemical distances (i.e., graph distances) on percolation clusters for level sets of GFFs. Precisely, for any λ ∈ R, we let H N,λ = {v ∈ V N : η N,v λ} be the λ-level set, i.e., the collection of all vertices

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Section: This Implies That Conditioned On {Lmentioning

confidence: 99%

“…Let (P t x,y (·)) x,y∈VN ,t>0 be the bridge probability measures of X conditioned on not killed until time t, and let (p t (x, y)) x,y∈VN ,t 0 be the transition probabilities of X. Then the measure µ on time-parametrized loops associated to X is, as defined in [33],…”

Section: Introductionmentioning

confidence: 99%

Chemical Distances for Percolation of Planar Gaussian Free Fields and Critical Random Walk Loop Soups

Ding

2018

Commun. Math. Phys.

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“…Such considerations have prompted an extensive analysis of more general versions of the random walk loop soup (see e.g. [20,28]). …”

Section: Introductionmentioning

confidence: 99%

Random walk loop soups and conformal loop ensembles

Brug¹,

Camia

Lis

2015

Probab. Theory Relat. Fields

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The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm-Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, "large" lattice loops from the random walk loop soup converge to "large" loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of "large" lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of "large" lattice loops converges to CLE. B Tim van de Brug

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“…The concept of permanental fields is motivated by [10] and [11], Chapter 9. The statement in (1.1) makes sense when the kernel u is bounded.…”

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confidence: 99%

“…Eisenbaum and Kaspi [3] show that an α-permanental process with kernel u exists whenever u is the potential density of a transient Markov process X in S. (This can also be done using loop soups. See [11], Chapters 2, 4, 5, for a study in the discrete symmetric case.) In [18], we give sufficient conditions for the continuity of α-permanental processes and use this, together with an isomorphism theorem of Eisenbaum and Kaspi, [3] to give sufficient conditions for the joint continuity of the local times of X.…”

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confidence: 99%

Permanental fields, loop soups and continuous additive functionals

Jan¹,

Marcus

Rosen

2015

Ann. Probab.

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A permanental field, $\psi=\{\psi(\nu),\nu\in {\mathcal{V}}\}$, is a particular stochastic process indexed by a space of measures on a set $S$. It is determined by a kernel $u(x,y)$, $x,y\in S$, that need not be symmetric and is allowed to be infinite on the diagonal. We show that these fields exist when $u(x,y)$ is a potential density of a transient Markov process $X$ in $S$. A permanental field $\psi$ can be realized as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of $X$, which we carefully construct. A Dynkin-type isomorphism theorem is obtained that relates $\psi$ to continuous additive functionals of $X$ (continuous in $t$), $L=\{L_t^{\nu},(\nu ,t)\in {\mathcal{V}}\times R_+\}$. Sufficient conditions are obtained for the continuity of $L$ on ${\mathcal{V}}\times R_+$. The metric on ${\mathcal{V}}$ is given by a proper norm.Comment: Published in at http://dx.doi.org/10.1214/13-AOP893 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Markov Paths, Loops and Fields

Cited by 78 publications

References 35 publications

Chemical Distances for Percolation of Planar Gaussian Free Fields and Critical Random Walk Loop Soups

Chemical Distances for Percolation of Planar Gaussian Free Fields and Critical Random Walk Loop Soups

Random walk loop soups and conformal loop ensembles

Permanental fields, loop soups and continuous additive functionals

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