On scaling limits and Brownian interlacements

Sznitman, Alain-Sol

doi:10.1007/s00574-013-0025-7

Cited by 31 publications

(56 citation statements)

References 30 publications

(50 reference statements)

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“…Below we extend the renormalized Le Jan's isomorphism to the case of non-negative boundary conditions. For an analogous statement in dimension 3 see [Szn13]. In particular, the field L ctr (L D 1/2 ) + L ctr (Ξ D u ) has same law as 1 2 : (Φ + u) 2 :.…”

Section: Brownian Loop and Excursion Measuresmentioning

confidence: 90%

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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Section: Brownian Loop and Excursion Measuresmentioning

confidence: 90%

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

Aru

Lupu²,

Sepúlveda

2020

Commun. Math. Phys.

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“…By Equations 2.7 on p.564 and 2.21 on p.568 in [25] it follows that for K ⊂ R d compact ν(W * K ) = cap(K). Now we introduce the space of point measures or configurations, where δ is the usual Dirac measure:…”

Section: Brownian Interlacementsmentioning

confidence: 99%

“…We begin with the setup as in [25]. Let C = C(R; R d ) denote the continuous functions from R to R d and let C + = C(R + ; R d ) denote the continuous functions from R + to R d .…”

Section: Brownian Interlacementsmentioning

confidence: 99%

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Visibility in the vacant set of the Brownian interlacements and the Brownian excursion process

Elias¹,

Tykesson²

2019

ALEA

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We consider the Brownian interlacements model in Euclidean space, introduced by A.S. Sznitman in [25]. We give estimates for the asymptotics of the visibility in the vacant set. We also consider visibility inside the vacant set of the Brownian excursion process in the unit disc and show that it undergoes a phase transition regarding visibility to infinity as in [1]. Additionally, we determine the critical value and that there is no visibility to infinity at the critical intensity.

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“…In the continuous setting and dimension d ≥ 3, Brownian random interlacements bring the similar limit description for the Wiener sausage on the torus [29,39]. Continuous setting is very appropriate to capture the geometrical properties of the sausage in their full complexity, see [20].…”

Section: Introductionmentioning

confidence: 99%

Two-Dimensional Brownian Random Interlacements

Comets

Popov

2019

Potential Anal

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We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage on the torus around a late point. As such, it can be seen as a continuous analogue of discrete two-dimensional random interlacements [13]. At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of [12,13], as well as the results specific to the continuous case.

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On scaling limits and Brownian interlacements

Cited by 31 publications

References 30 publications

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

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

Visibility in the vacant set of the Brownian interlacements and the Brownian excursion process

Two-Dimensional Brownian Random Interlacements

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