Gaussian Multiplicative Chaos and KPZ Duality

Abstract

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ 2 beyond the transition phase (i.e. γ 2 > 2d) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and to understand the duality relation in Liouville quantum gravity. Contents

Liouville heat kernel: Regularity and bounds

“…Indeed, physicists have long conjectured that LQG (which is parametrized by a constant γ) is the limit of random planar maps weighted by a 2d statistical physics system at critical temperature, usually described by a conformal field theory with central charge c 1. These conjectures were made more explicit in a recent work [14] and in particular they provide a geometrical and probabilistic framework for the celebrated KPZ relation (first derived in [31] in the so-called light cone gauge and then in [9,11] within the framework of the conformal gauge): see [5,6,12,14,36] for rigorous probabilistic formulations of the KPZ relation. In this geometrical point of view, (critical) LQG corresponds to studying a conformal field theory with central charge c 1 (called the matter field in the physics literature) in an independent random geometry which can be described formally by a Riemannian metric tensor of the form…”

On exact scaling log-infinitely divisible cascades

“…Also, fine continuity properties of the critical measure µ are analyzed in [5]. Similar properties are conjectured to hold for log-infinitely divisible cascades, and some of them have been established in the log-gaussian case [3,10,11,4]. Relation (1.13) can be obtained from Bacry and Muzy construction by writing, for any c ∈ (0, 1), the almost sure relation for 0 < ≤ 1 1] ; this defines the process (ω ,x ) x∈[0,T ] , obviously independent of Λ(V T (0) ∩ V T (cT )), and which can be shown to have the same distribution as (Λ(V T T (x)) x∈[0,T ] via Fourier transform, and implies (1.1) (see Figure 4a).…”

“…Higher dimensional versions have been built as well (see [18,9,34]). In particular, in dimension 2 and in the Gaussian case, they are closely related to the validity of the so-called KPZ formula and its dual version in Liouville quantum gravity (see [12] and [35], as well as [3]). …”

KPZ formula derived from Liouville heat kernel

“…Recently, a rigorous version of the KPZ formula has been established in [3,4,10,11,27] (see also [5] in the context of one-dimensional Mandelbrot's cascades), which we recall now. Usually, on a metric space, one defines the Hausdorff dimension of a given set A by studying the diameters of open sets required to cover this set.…”

“…To go around the difficulty of formulating the notion of Hausdorff dimension in the absence of a well-defined distance, different notions of Hausdorff dimensions have been suggested. For instance, a measure-based notion was formulated in [4,27]. Instead of working with the diameter of open sets covering the set A, one works with the measure of small Euclidean balls covering this set (see Subsection 3.3 for more details).…”