TY - JOUR

T1 - Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos

AU - Camia, Federico

AU - Gandolfi, Alberto

AU - Peccati, Giovanni

AU - Reddy, Tulasi Ram

N1 - Funding Information:
Part of this paper was written while Giovanni Peccati was visiting the Division of Science of New York University at Abu Dhabi, in February 2018. This author wishes to heartily thank Federico Camia and Alberto Gandolfi for their kind hospitality and support. Giovanni Peccati is also supported by the FNR grant FoRGES (R-AGR-3376-10) at Luxembourg University. Federico Camia thanks Wei Wu for interesting discussions on the GMC and related topics. The authors thank the associate editor and four anonymous referees for their constructive remarks and useful comments and suggestions, in particular for a question that led to an improvement of Theorem and for providing a shorter proof of Lemma .
Funding Information:
Part of this paper was written while Giovanni Peccati was visiting the Division of Science of New York University at Abu Dhabi, in February 2018. This author wishes to heartily thank Federico Camia and Alberto Gandolfi for their kind hospitality and support. Giovanni Peccati is also supported by the FNR grant FoRGES (R-AGR-3376-10) at Luxembourg University. Federico Camia thanks Wei Wu for interesting discussions on the GMC and related topics. The authors thank the associate editor and four anonymous referees for their constructive remarks and useful comments and suggestions, in particular for a question that led to an improvement of Theorem?1.1 and for providing a shorter proof of Lemma?A.1.

PY - 2021/2

Y1 - 2021/2

N2 - We study fields reminiscent of vertex operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia et al. (Nucl Phys B 902:483–507, 2016), we take a (massless or massive) Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. For domains conformally equivalent to a disk, the sum diverges logarithmically due to the small loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener–Itô chaos expansion of linear functionals of vertex-like fields. Our methods apply to other variants of the model in which, for example, Brownian loops are replaced by disks.

AB - We study fields reminiscent of vertex operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia et al. (Nucl Phys B 902:483–507, 2016), we take a (massless or massive) Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. For domains conformally equivalent to a disk, the sum diverges logarithmically due to the small loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener–Itô chaos expansion of linear functionals of vertex-like fields. Our methods apply to other variants of the model in which, for example, Brownian loops are replaced by disks.

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U2 - 10.1007/s00220-020-03932-9

DO - 10.1007/s00220-020-03932-9

M3 - Article

AN - SCOPUS:85100626239

VL - 381

SP - 889

EP - 945

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -