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Directed Polymers on Infinite Graphs

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Abstract

We study the directed polymer model for general graphs (beyond \({\mathbb {Z}}^d\)) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an \(L^2\) region, and of very strong disorder, in terms of properties of the graph and of the random walk. We study in some detail (biased) random walk on various trees including the Galton–Watson trees, and provide a range of other examples that illustrate counter-examples to intuitive extensions of the \({\mathbb {Z}}^d\)/SRW result.

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Acknowledgements

We thank the authors of [38] for a their comments and for sending us a preliminary draft of their work. This project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (Grant Agreement No. 692452).

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  1. Weizmann Institute of Science, Rehovot, Israel

    Clément Cosco, Inbar Seroussi & Ofer Zeitouni

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  1. Clément Cosco
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  2. Inbar Seroussi
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Correspondence to Ofer Zeitouni.

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Communicated by H. Spohn

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Cosco, C., Seroussi, I. & Zeitouni, O. Directed Polymers on Infinite Graphs. Commun. Math. Phys. 386, 395–432 (2021). https://doi.org/10.1007/s00220-021-04034-w

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