Liouville quantum gravity and KPZ

By Bertrand Duplantier, Scott Sheffield. 2008

529 citations recorded by the source corpus.

AbstractConsider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫D∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on D is the weak limit as ε→0 of the measures $$\varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz,$$ where dz is Lebesgue measure on D and hε(z) denotes the mean value of h on the circle of radius ε centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819–826, 1988) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

Subfields: Condensed Matter Physics, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/02edb65979fcef284b50db2eb6a078f810f8dd73

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Citations in the corpus

Conformal weldings of random surfaces: SLE and the quantum gravity zipper (2010, methodology): “the circle @B "(z) and the limit represents weak convergence (on compact subsets) in the space of measures on D. (The limit exists almost surely, at least if "is restricted to powers of two [DS11a].) We interpret h as the area measure of a random surface conformally parameterized by D. Wh”
Recurrence of planar graph limits (2012, unknown): “(See [21], [3], [15], [5] and the references within.”
Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions (2014, background): “We refer to [8] and to the survey article [11] for more details on this topic.”
The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, background): “e has been a new wave of interest in multiplicative chaos, partly due to its important connections to Stochastic Loewner Evolution [69, 3], to quantum gravity and scaling limits of random planar maps [28, 53, 12, 54, 55, 56, 24], as well as to models in ﬁnance and turbulence [64, Section 5]. In Sect”
Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012, methodology): “Define then the analog of the rooted random measure in [30] (also called the “Peyrière probability measure” in this context [45]),”
Gaussian multiplicative chaos and applications: A review (2013, methodology): “We will explain the geometrical KPZ formulae rigorously proved in [56, 128] (see also [27, 17]).”
Scaling limits of random trees and planar maps (2011, methodology): “More recently, graphs on surfaces have been used in physics as discrete models of random geometry in the so-called two-dimensional quantum gravity; see in particular the book [3] (a different mathematical approach to quantum gravity using the Gaussian free field appears in the work of Duplantier and”
Uniqueness and universality of the Brownian map (2011, unknown): “tic 3-point function has been computed by Bouttier and Guitter [5]. However, the extension of these calculations to higher values of kseems a diﬃcult problem. As a ﬁnal remark, Duplantier and Sheﬃeld [10] recently developed a mathematical approach to two-dimensional quantum gravity based on the Gaus”
An elementary approach to Gaussian multiplicative chaos (2015, methodology): “relations. This is the particular case of the theory when d = 2 and h is the massless Gaussian free ﬁeld or GFF (see [15, 2]) with appropriate boundary conditions. The paper by Duplantier and Sheﬃeld [4] constructed the volume measure µ in this particular case using arguments restriced to the case o”
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, unknown): “20 applied for κ = 16/κ′ ∈ [2, 4) that we can construct a coupling of η′ with a whole-plane SLEκ′(ρ) process η̃′ from ∞ to 0 such that the left and right boundaries of η̃′ are almost surely equal to η, η.”
Random geometry on the sphere (2014, background): “It is worth mentioning that a different mathematical approach to two-dimensional quantum gravity relying on the Gaussian free field has been given by Duplantier and Sheffield [24].”
The Brownian map is the scaling limit of uniform random plane quadrangulations (2011, background): “We also mention that a purely continuous approach to quantum gravity, called Liouville quantum gravity, has been developed in parallel in the physics literature, and that the mathematical grounds for this theory are starting to emerge after a seminal work by Duplantier and Sheffield, see [10] and re”
Uniqueness of critical Gaussian chaos (2017, background): “Duplantier’s and Sheffield’s paper [14] gives uniqueness results for particular approximations of the 2-dimensional GFF.”
Liouville Quantum Gravity on the Riemann Sphere (2014, background): “In particular, in this geometrical and probabilistic context, a precise mathematical formulation of the KPZ scaling relations can be given [5,24,53].”
Scaling limits of random planar maps with large faces (2011, unknown): “This approach would still be different from the one developed recently by Duplantier and Sheffield [10] as we are focusing more on the metric aspects of planar maps, rather than on the conformal invariance properties that are intrinsic to [10].”
Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding (2016, methodology): “tes Lebesgue measure on D. Since hdoes not take values at points, it is necessary to use a regularization procedure in order to make sense of this expression rigorously. This has been accomplished in [DS11], for example, by considering the approximation 22=2eh(z)dzwhere h (z) denotes the average of”