Quantum Loewner Evolution

By Jason Miller, Scott Sheffield. 2013

144 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Materials Chemistry, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5e24b411814895e838096a433daaed01d5a9d425

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Citations in the corpus

Uniqueness of critical Gaussian chaos (2017, background): “This kind of approximations might be useful for example in studying fields that have been considered in [24].”
The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, background): “During the last 15 years there has been a new wave of interest in multiplicative chaos, partly due to its important connections to Stochastic Loewner Evolution [82, 4], to quantum gravity and scaling limits of random planar maps [32, 63, 13, 64, 65, 66, 27], as well as to models in finance and turbu”
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, result): “S14], and the results from this paper are extensively cited there. The results in this article (including results 5 about reversibility and duality) also feature prominently in a program announced in [MS16g] and carried out in [MS15c, MS15a, MS15b, MS16b, MS16c] to construct the metric space structu”
Liouville Quantum Gravity on the Riemann Sphere (2014, background): “…problem have been achieved in [12] and [45] in the special case γ = 8 / 3: recall that for this value of γ (and in the context µ = 0) the work [45] constructs a random growth process which is conjectured to be the growth of balls of a metric space formally corresponding to the ”tangent plane”…”
Scaling limits for the peeling process on random maps (2014, methodology): “The same variant has been considered by Miller and Sheffield [27] and served as a motivation for the construction of Quantum Loewner Evolutions.”
Random geometry on the sphere (2014, background): “pective, but the paper [45] gives a number of conjectures that would relate large random planar maps and the Brownian map to the Gaussian free eld approach. The very recent work of Miller and Sheeld [42] seems promising in this respect. Acknowledgements. I thank Nicolas Curien and Igor Kortchemski f”
Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding (2016, methodology): “al welding [She16], conformal loop ensembles [She09,SW12], and the mating of trees in innite and nite volume settings [DMS14,MS15c], as well as the above mentioned works on quantum Loewner evolution [MS16e] and TBM [MS15a]. We also cite foundational works by many other authors on Liouville quantum g”