From loop clusters and random interlacements to the free field

Lupu, Titus

doi:10.1214/15-aop1019

Cited by 96 publications

(258 citation statements)

References 22 publications

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“…With (1.4) and (1.5) we will be able to use Theorem 2.4 of [13] (recalled below at the beginning of the proof of Theorem 1.1). It refines Lupu's coupling between the Gaussian free field and the random interlacements on the cable graph, see Proposition 6.3 of [7].…”

Section: Coupling and Stochastic Dominationsupporting

confidence: 57%

“…We now proceed with some notation concerning random interlacements on the cable graph. Given a level u > 0, we consider on some probability space ( W , B, P I ) a continuous random field ( ℓ z,u ) z∈ E on E describing the local times with respect to the Lebesgue measure m on E of random interlacements at level u on E, see Section 6 of [7], or below (1.21) of [13], and assume that…”

Section: Coupling and Stochastic Dominationmentioning

confidence: 99%

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On coupling and “vacant set level set” percolation

Sznitman

2019

Electron. Commun. Probab.

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In this note we discuss vacant set level set percolation on a transient weighted graph. It interpolates between the percolation of the vacant set of random interlacements and the level set percolation of the Gaussian free field. We employ coupling and derive a stochastic domination from which we deduce in a rather general set-up a certain monotonicity property of the percolation function. In the case of regular trees this stochastic domination leads to a strict inequality between some eigenvalues related to Ornstein-Uhlenbeck semi-groups for which we have no direct analytical proof. It underpins a certain strict monotonicity property that has significant consequences for the percolation diagram. It is presently open whether a similar looking diagram holds in the case of Z d , d ≥ 3.

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Section: Coupling and Stochastic Dominationsupporting

confidence: 57%

Section: Coupling and Stochastic Dominationmentioning

confidence: 99%

On coupling and “vacant set level set” percolation

Sznitman

2019

Electron. Commun. Probab.

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“…Proposition 2.1 (Strong Markov property, [Lup16a]). Let A be a random compact subset of G, with finitely many connected components and optional for the metric graph GFFφ.…”

Section: Preliminaries On the Metric Graphmentioning

confidence: 99%

“…In this article, we make the above heuristic description of the FPS exact by approximating the continuum GFF by metric graph GFF-s. A metric graph is obtained by taking a discrete electrical network and replacing each edge by a continuous line segment of length proportional to the resistance (inverse of the conductance) of the edge. On the metric graph, one can define a Gaussian free field by interpolating discrete GFF on vertices by conditional independent Brownian bridges inside the edges [Lup16a]. Such a field is pointwise defined, continuous, and still satisfies a domain Markov property, even when cutting the domain inside the edges.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, in Le Jan's isomorphism [LJ07,LJ11], the whole square of a discrete GFF is given by the occupation field of a Markov jump process loop-soup. The introduction of metric graphs as in [Lup16a] provides "polarized" versions of isomorphism theorems, where one has the additional property that the GFF has constant sign on each Markovian trajectory. More precisely, one considers a metric graph loop-soup and an independent Poisson point process of boundary-to-boundary metric graph excursions.…”

Section: Introductionmentioning

confidence: 99%

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The First Passage Sets of the 2D Gaussian Free Field: Convergence and Isomorphisms

Aru

Lupu²,

Sepúlveda

2020

Commun. Math. Phys.

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In a previous article, we introduced the first passage set (FPS) of constant level −a of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path along which the GFF is greater than or equal to −a. This description can be taken as a definition of the FPS for the metric graph GFF, and it justifies the analogy with the first hitting time of −a by a onedimensional Brownian motion. In the current article, we prove that the metric graph FPS converges towards the continuum FPS in the Hausdorff metric. This allows us to show that the FPS of the continuum GFF can be represented as a union of clusters of Brownian excursions and Brownian loops, and to prove that Brownian loop soup clusters admit a non-trivial Minkowski content in the gauge r → | log r| 1/2 r 2 . We also show that certain natural interfaces of the metric graph GFF converge to SLE4 processes.2010 Mathematics Subject Classification. 60G15; 60G60; 60J65; 60J67; 81T40.

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Random Flows Defined by Markov Loops

Jan

2019

Lecture Notes in Mathematics

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From loop clusters and random interlacements to the free field

Cited by 96 publications

References 22 publications

On coupling and “vacant set level set” percolation

On coupling and “vacant set level set” percolation

The First Passage Sets of the 2D Gaussian Free Field: Convergence and Isomorphisms

Random Flows Defined by Markov Loops

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