The Brownian loop soup

By Gregory F. Lawler, Wendelin Werner. 2003

311 citations recorded by the source corpus.

Abstract.We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to ‘‘chronologically add Brownian loops’’ to simple curves in the plane.

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/facc85f04b4140b97f5a7a30a876090fb936abed

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Citations in the corpus

Euler Integrals for Commuting SLEs (2005, methodology): “tion with Fomin’s formulae), and uniform spanning trees (in relation with Wilson’s algorithm) provide important examples. In the continuous set-up, the theory of conformal restriction and loop-soups ([16, 21]) can also be connected to the problem of commuting SLEs ([8]). In the following section, as”
Random walk loop soup (2004, methodology): “The family of measure {� u D} clearly satisfy the restriction property, i.e., if D ′ ⊂ D then � u D′ is � u D restricted to curves lying in D ′ . It is also shown in [ 3 ] that the family satisfies a conformal invariance property, i.e., if f : D → D ′ is a conformal transformation, then”
SLE and the free field: partition functions and couplings (2007, methodology): “This gives another justification to introduce the (rooted) loop measure ([26]) (and also to normalize it this way): μ r = ∫ ∞”
Reversibility of chordal SLE (2008, background): “is the probability that in a loop soup [7] in H with intensity λ, there is no loop that intersects both K1(t1) and K2(t2).”
Random planar curves and Schramm-Loewner evolutions (2003, background): “ the restriction property. The second description goes as follows: Consider an SLE κ with κ≤ 8/3 and add to this path a certain cloud Brownian loops (this Poisson cloud 90 of loops is also studied in [98]). For a well-tuned density d(κ) of the loops, one constructs the restriction measure correspond”
The configurational measure on mutually avoiding SLE paths (2006, methodology): “2 Brownian loop measure We review the Brownian loop measure as introduced in [14]; see also [8, Section 5.”
Topics in Occupation Times and Gaussian Free Fields (2012, methodology): “The following notion introduced by Lawler-Werner [16] is convenient to handle computations with μ.”
2D growth processes: SLE and Loewner chains (2006, methodology): “a discrete analog of the 3dBessel process). In the continuum, such a systematic removal of loops is impossible, because Brownian motion makes loops at all scales. But one can work the other way round [97]. The loop soup companion of SLE 2, known to describe the continuum limit of loop erased random ”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, background): “, the usual normalization) of the measure μ, as in [20]; note that there can be some confusion about a factor 2 in the definition, related to whether one keeps track of the orientation of the Brownian loops or not.”
Commutation relations for Schramm‐Loewner evolutions (2004, methodology): “other (simply connected) domains are obtained by conformal equivalence. For general background on SLE, see [8, 13]. Also, we will use freely results on the restriction property and the loop soup (see [5, 7, 11]). Consider the family of ODEs, indexed by zin H: ∂tgt(z) = 2 gt(z)−Wt with initial condit”
The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$ (2016, background): “The Brownian loop measure is a natural conformally invariant measure on the unrooted loops defined in [13].”