Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos

Abstract

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.

New Recipes for Brownian Loop Soups

“…Either of these can be given a physical interpretation similar to the one described above: if the Brownian loops themselves (for the winding operator) or their outer boundaries (for the layering operator) represent a domain wall or charged object across which some quantity (like the value of a scalar field, or the electric field) changes discontinuously, then these operators are counting the value of that field at the point z (this is also known as a height model). The layering and winding vertex operators in the infinite intensity limit have been studied in [12,13].…”

“…Given that the variance of the probability distribution exists, there exists a limit in which the correlators in the full plane become those of free field vertex operators. Consider now taking β i → 0 and λ → ∞ with the product λβ 2 i fixed (this limit is discussed in detail in [12]). The Taylor expansion of the characteristic function is…”

Conformal Measure Ensembles and Planar Ising Magnetization: A Review

“…Proposition 2.2 of [18] implies that, for any f ∈ C ∞ 0 (D), the number of elements in the sum defining Φ a D,ε (f ) remains finite as a ↓ 0; hence for every f ∈ C ∞ 0 (D), by Theorem 4.3, Φ a D,ε (f ) converges in distribution to Φ 0 D,ε (f ) as a ↓ 0. This, combined with the tightness of Φ a D,ε and with Lemma A.4 of [10], shows that, as k → ∞, Φ a k D,ε converges to Φ 0 D,ε in distribution in the topology induced by • H −α (D) . Therefore, i:diam(C i )≤ε η i µ a k i also converges in distribution in the topology induced by • H −α (D) to some Xε ∈ H −α (D), and Φ a k D converges to Φ0 D = Φ 0 D,ε + Xε .…”

“…To see why (4.6) holds, following [10], one can assume without loss of generality that α ≥ 1 is an integer. For such an α and for every f ∈ C ∞ 0 (D), one has that (−∆) α f ∈ C ∞ 0 (D), and consequently (−∆) α f = i (−∆) α f, u i L 2 (D) u i , where the series converges in L 2 (D).…”

2D Percolation as a Conformal Field Theory

“…The proofs of Theorem 1.7 and Corollary 1.8, presented in Section 3, use ideas and tools from [11,[14][15][16], as well as properties of the conformal measure ensemble for critical percolation whose existence was conjectured in [21] and proved in [12].…”