Quantum Loewner Evolution

Abstract

What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the {\em dielectric breakdown model} $\eta$-DBM, a generalization of DLA in which particle locations are sampled from the $\eta$-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider $\eta$-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter $\gamma \in [0,2]$. In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE$({\gamma }^2,\eta )$. QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion ${\nu }_t$ derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of ${\nu }_t$ using an SPDE. For each $\gamma \in (0,2]$, there are two or three special values of $\eta$ for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of ${\nu }_t$. We also explain discrete versions of our construction that relate DLA to loop-erased random walk and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot machine reels) facilitates explicit calculation. We propose QLE$(2,1)$ as a scaling limit for DLA on a random spanning-tree-decorated planar map, and QLE$(8/3,0)$ as a scaling limit for the Eden model on a random triangulation. We propose using QLE$(8/3,0)$ to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE$(8/3,0)$, up to a fixed time, as a metric ball in a random metric space.

132 pages, approximately 100 figures and computer simulations

References in corpus (6)

• The topological structure of scaling limits of large planar maps
• Two-Dimensional Critical Percolation: The Full Scaling Limit
• KPZ in one dimensional random geometry of multiplicative cascades
• Duality and KPZ in Liouville Quantum Gravity
• Shocks and finite-time singularities in Hele-Shaw flow
• Dimensions, Maximal Growth Sites and Optimization in the Dielectric Breakdown Model

Cited by in corpus (16)

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• Almost sure multifractal spectrum of SLE
• The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph
• Upper bounds on Liouville first passage percolation and Watabiki's prediction
• Large deviations of Schramm-Loewner evolutions: A survey
• Subsequential scaling limits for Liouville graph distance
• One-dimensional scaling limits in a planar Laplacian random growth model
• Nesting statistics in the O(n) loop model on random planar maps
• The Loewner-Kufarev Energy and Foliations by Weil-Petersson Quasicircles
• Large deviations of radial SLE${}_{\infty }$
• Distances on the ${\mathrm{CLE}}_4$, critical Liouville quantum gravity and $3/2$-stable maps
• Loewner chains and evolution families on parallel slit half-planes
• SLE scaling limits for a Laplacian random growth model