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Dynamic polymers: invariant measures and ordering by noise

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Abstract

We develop a dynamical approach to infinite volume directed polymer measures in random environments. We define polymer dynamics in \(1+1\) dimension as a stochastic gradient flow on polymers pinned at the origin, for energy involving quadratic nearest neighbor interaction and local interaction with random environment. We prove existence and uniqueness of the solution, continuity of the flow, the order-preserving property with respect to the coordinatewise partial order, and the invariance of the asymptotic slope. We establish ordering by noise which means that if two initial conditions have distinct slopes, then the associated solutions eventually get ordered coordinatewise. This, along with the shear-invariance property and existing results on static infinite volume polymer measures, allows to prove that for a fixed asymptotic slope and almost every realization of the environment, the polymer dynamics has a unique invariant distribution given by a unique infinite volume polymer measure, and, moreover, One Force—One Solution principle holds. We also prove that every polymer measure is concentrated on paths with well-defined asymptotic slopes and give an estimate on deviations from straight lines.

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Acknowledgements

Yuri Bakhtin is grateful to the National Science Foundation for partial support via grant DMS-1811444. Both authors thank the anonymous referee for the comments that helped to improve the paper significantly.

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  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY, 10012, USA

    Yuri Bakhtin & Hong-Bin Chen

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  1. Yuri Bakhtin
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Bakhtin, Y., Chen, HB. Dynamic polymers: invariant measures and ordering by noise. Probab. Theory Relat. Fields 183, 167–227 (2022). https://doi.org/10.1007/s00440-021-01099-5

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