On the stationary solutions of random polymer models and their zero-temperature limits

[2104.03458] On the stationary solutions of random polymer models and their zero-temperature limits Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2104.03458 (math) [Submitted on 8 Apr 2021 ( v1 ), last revised 13 May 2022 (this version, v3)] Title: On the stationary solutions of random polymer models and their zero-temperature limits Authors: David A. Croydon , Makiko Sasada View a PDF of the paper titled On the stationary solutions of random polymer models and their zero-temperature limits, by David A. Croydon and 1 other authors View PDF HTML (experimental) Abstract: We derive stationary measures for certain zero-temperature random polymer models, which we believe are new in the case of the zero-temperature limit of the beta random polymer (that has been called the river delta model). To do this, we apply techniques developed for understanding the stationary measures of the corresponding positive-temperature random polymer models (and some deterministic integrable systems). More specifically, the article starts with a survey of results for the four basic beta-gamma models (i.e. the inverse-gamma, gamma, inverse-beta and beta random polymers), highlighting how the maps underlying the systems in question can each be reduced to one of two basic bijections, and that through an `independence preservation' property, these bijections characterise the associated stationary measures. We then derive similar descriptions for the corresponding zero-temperature maps, whereby each is written in terms of one of two bijections. One issue with this picture is that, unlike in the positive-temperature case, the change of variables required is degenerate in general, and so whilst the argument yields stationary solutions, it does not provide a complete characterisation of them. On the other hand, this degeneracy does allow us to explain the appearance of atoms in the stationary measures of certain zero-temperature models. We also derive from our viewpoint various links between random polymer models, some of which recover known results, some of which are novel, and some of which lead to further questions. Subjects: Probability (math.PR) ; Mathematical Physics (math-ph) MSC classes: 82D60 (primary), 37K60, 37L40, 60E05, 60K35, 82B23 Cite as: arXiv:2104.03458 [math.PR] (or arXiv:2104.03458v3 [math.PR] for this version) https://doi.org/10.48550/arXiv.2104.03458 Focus to learn more arXiv-issued DOI via DataCite Journal reference: J Stat Phys 188, 23 (2022) Related DOI : https://doi.org/10.1007/s10955-022-02947-x Focus to learn more DOI(s) linking to related resources Submission history From: David Croydon [ view email ] [v1] Thu, 8 Apr 2021 01:30:21 UTC (27 KB) [v2] Mon, 17 Jan 2022 04:57:57 UTC (28 KB) [v3] Fri, 13 May 2022 07:21:31 UTC (182 KB) Full-text links: Access Paper: View a PDF of the paper titled On the stationary solutions of random polymer models and their zero-temperature limits, by David A. 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