Macdonald processes
Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters q,t ∈ [0,1). We prove several results about these processes, which include the following. (1) We explicitly evalua...
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| Format: | Article |
| Language: | en_US |
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Springer-Verlag
2015
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| Online Access: | http://hdl.handle.net/1721.1/92818 https://orcid.org/0000-0002-2913-5238 |
| _version_ | 1826217578303848448 |
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| author | Borodin, Alexei Corwin, Ivan |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Borodin, Alexei Corwin, Ivan |
| author_sort | Borodin, Alexei |
| collection | MIT |
| description | Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters q,t ∈ [0,1). We prove several results about these processes, which include the following. (1) We explicitly evaluate expectations of a rich family of observables for these processes. (2) In the case t = 0, we find a Fredholm determinant formula for a q-Laplace transform of the distribution of the last part of the Macdonald-random partition. (3) We introduce Markov dynamics that preserve the class of Macdonald processes and lead to new “integrable” 2d and 1d interacting particle systems. (4) In a large time limit transition, and as q goes to 1, the particles of these systems crystallize on a lattice, and fluctuations around the lattice converge to O’Connell’s Whittaker process that describe semi-discrete Brownian directed polymers. (5) This yields a Fredholm determinant for the Laplace transform of the polymer partition function, and taking its asymptotics we prove KPZ universality for the polymer (free energy fluctuation exponent 1/3 and Tracy-Widom GUE limit law). (6) Under intermediate disorder scaling, we recover the Laplace transform of the solution of the KPZ equation with narrow wedge initial data. (7) We provide contour integral formulas for a wide array of polymer moments. (8) This results in a new ansatz for solving quantum many body systems such as the delta Bose gas. |
| first_indexed | 2024-09-23T17:05:54Z |
| format | Article |
| id | mit-1721.1/92818 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T17:05:54Z |
| publishDate | 2015 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/928182022-09-29T23:39:26Z Macdonald processes Borodin, Alexei Corwin, Ivan Massachusetts Institute of Technology. Department of Mathematics Borodin, Alexei Corwin, Ivan Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters q,t ∈ [0,1). We prove several results about these processes, which include the following. (1) We explicitly evaluate expectations of a rich family of observables for these processes. (2) In the case t = 0, we find a Fredholm determinant formula for a q-Laplace transform of the distribution of the last part of the Macdonald-random partition. (3) We introduce Markov dynamics that preserve the class of Macdonald processes and lead to new “integrable” 2d and 1d interacting particle systems. (4) In a large time limit transition, and as q goes to 1, the particles of these systems crystallize on a lattice, and fluctuations around the lattice converge to O’Connell’s Whittaker process that describe semi-discrete Brownian directed polymers. (5) This yields a Fredholm determinant for the Laplace transform of the polymer partition function, and taking its asymptotics we prove KPZ universality for the polymer (free energy fluctuation exponent 1/3 and Tracy-Widom GUE limit law). (6) Under intermediate disorder scaling, we recover the Laplace transform of the solution of the KPZ equation with narrow wedge initial data. (7) We provide contour integral formulas for a wide array of polymer moments. (8) This results in a new ansatz for solving quantum many body systems such as the delta Bose gas. National Science Foundation (U.S.) (Grant DMS-1056390) National Science Foundation (U.S.). Partnerships for International Research (Grant OISE-07-30136) National Science Foundation (U.S.) (Grant DMS-1208998) Clay Mathematics Institute (Research Fellowship) Microsoft Research (Schramm Memorial Fellowship) 2015-01-13T15:28:10Z 2015-01-13T15:28:10Z 2013-03 2012-11 Article http://purl.org/eprint/type/JournalArticle 0178-8051 1432-2064 http://hdl.handle.net/1721.1/92818 Borodin, Alexei, and Ivan Corwin. “Macdonald Processes.” Probability Theory and Related Fields 158, no. 1–2 (March 30, 2013): 225–400. https://orcid.org/0000-0002-2913-5238 en_US http://dx.doi.org/10.1007/s00440-013-0482-3 Probability Theory and Related Fields Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Springer-Verlag arXiv |
| spellingShingle | Borodin, Alexei Corwin, Ivan Macdonald processes |
| title | Macdonald processes |
| title_full | Macdonald processes |
| title_fullStr | Macdonald processes |
| title_full_unstemmed | Macdonald processes |
| title_short | Macdonald processes |
| title_sort | macdonald processes |
| url | http://hdl.handle.net/1721.1/92818 https://orcid.org/0000-0002-2913-5238 |
| work_keys_str_mv | AT borodinalexei macdonaldprocesses AT corwinivan macdonaldprocesses |