{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/quantum-loewner-evolution","title":"Quantum Loewner Evolution","arxiv_id":"1312.5745","date":"2013-12-19","proceeding":null,"authors":["Jason Miller","Scott Sheffield"],"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.","url_abs":"http://arxiv.org/abs/1312.5745v1","url_pdf":"http://arxiv.org/pdf/1312.5745v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"quantum-loewner-evolution","repo_url":"https://github.com/jaumededios/LQG_Simulations","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}