Quantum Loewner evolution
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 dielectric breakdown model η-DBM, a generalization of DLA in which particle locations are sampled fr...
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| Format: | Article |
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Duke University Press
2018
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| Online Access: | http://hdl.handle.net/1721.1/116004 https://orcid.org/0000-0002-5951-4933 |
| _version_ | 1826205333910978560 |
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| author | Miller, Jason P. Sheffield, Scott Roger |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Miller, Jason P. Sheffield, Scott Roger |
| author_sort | Miller, Jason P. |
| collection | MIT |
| description | 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 dielectric breakdown model η-DBM, a generalization of DLA in which particle locations are sampled from the η th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider η-DBM on random graphs known or believed to conve rge in law to a Liouville quantum gravity (LQG) surface with parameter γ e [0,2]. In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ 2 ,η). QLE is defined in terms of the radial Loewner equation like radial stochastic Loewner evolution, except that it is driven by a measure-valued diffusion v t derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of v t using a stochastic partial differential equation. For each γ e [0, 2], there are two or three special values of η for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of v t . We also explain discrete versions of our construction that relate DLA to loop-erased random walks 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. |
| first_indexed | 2024-09-23T13:11:03Z |
| format | Article |
| id | mit-1721.1/116004 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T13:11:03Z |
| publishDate | 2018 |
| publisher | Duke University Press |
| record_format | dspace |
| spelling | mit-1721.1/1160042022-10-01T13:37:59Z Quantum Loewner evolution Miller, Jason P. Sheffield, Scott Roger Massachusetts Institute of Technology. Department of Mathematics Miller, Jason P. Sheffield, Scott Roger 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 dielectric breakdown model η-DBM, a generalization of DLA in which particle locations are sampled from the η th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider η-DBM on random graphs known or believed to conve rge in law to a Liouville quantum gravity (LQG) surface with parameter γ e [0,2]. In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ 2 ,η). QLE is defined in terms of the radial Loewner equation like radial stochastic Loewner evolution, except that it is driven by a measure-valued diffusion v t derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of v t using a stochastic partial differential equation. For each γ e [0, 2], there are two or three special values of η for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of v t . We also explain discrete versions of our construction that relate DLA to loop-erased random walks 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. 2018-05-30T20:41:12Z 2018-05-30T20:41:12Z 2016-10 2015-10 2018-05-30T15:29:14Z Article http://purl.org/eprint/type/JournalArticle 0012-7094 http://hdl.handle.net/1721.1/116004 Miller, Jason, and Scott Sheffield. “Quantum Loewner Evolution.” Duke Mathematical Journal 165, 17 (November 2016): 3241–3378 © 2016 Duke University Press https://orcid.org/0000-0002-5951-4933 http://dx.doi.org/10.1215/00127094-3627096 Duke Mathematical Journal Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Duke University Press arXiv |
| spellingShingle | Miller, Jason P. Sheffield, Scott Roger Quantum Loewner evolution |
| title | Quantum Loewner evolution |
| title_full | Quantum Loewner evolution |
| title_fullStr | Quantum Loewner evolution |
| title_full_unstemmed | Quantum Loewner evolution |
| title_short | Quantum Loewner evolution |
| title_sort | quantum loewner evolution |
| url | http://hdl.handle.net/1721.1/116004 https://orcid.org/0000-0002-5951-4933 |
| work_keys_str_mv | AT millerjasonp quantumloewnerevolution AT sheffieldscottroger quantumloewnerevolution |