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Conditioned Stochastic Particle Systems and Integrable Quantum Spin Systems

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From Particle Systems to Partial Differential Equations II

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 129))

Abstract

We consider from a microscopic perspective large deviation properties of several stochastic interacting particle systems, using their mapping to integrable quantum spin systems. A brief review of recent work is given and several new results are presented: (i) For the general disordered symmetric exclusion process (SEP) on some finite lattice conditioned on no jumps into some absorbing sublattice and with initial Bernoulli product measure with density \(\rho \) we prove that the probability \(S_{\rho (t)}\) of no absorption event up to microscopic time \(t\) can be expressed in terms of the generating function for the particle number of a SEP with particle injection and empty initial lattice. Specifically, for the symmetric simple exclusion process on \({\mathbb Z}\) conditioned on no jumps into the origin we obtain the explicit first and second order expansion in \(\rho \) of \(S_{\rho (t)}\) and also to first order in \(\rho \) the optimal microscopic density profile under this conditioning. (ii) For the disordered ASEP on the finite torus conditioned on a very large current we show that the effective dynamics that optimally realizes this rare event does not depend on the disorder, except for the time scale. (iii) For annihilating and coalescing random walkers we obtain the generating function of the number of annihilated particles up to time \(t\), which turns out to exhibit some universal features.

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Notes

  1. 1.

    To clarify our terminology: The symmetric simple exclusion process (SSEP) has nearest neighbour jumps only on a regular lattice such as \({\mathbb Z}^d\), whereas the general symmetric exclusion process (SEP) can have jumps between any pair of nodes on any graph. This terminology comes from physical intuition and is not mathematically precise, since the set of links on which jumps have non-zero rate may just as well be used to define an underlying graph on which one then would again have only nearest neighbour jumps so that the process could also be called SSEP. However, we will only consider the SSEP on \({\mathbb Z}\) with bond-independent rates. This removes any ambiguity of notation. Moreover, we shall always speak of sites and lattices rather than of nodes and graphs.

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Acknowledgments

The author thanks B. Meerson, P.L. Krapivsky, V. Popkov and Andreas Schadschneider for inspiring discussions and the Galileo Galilei Institute for Theoretical Physics for hospitality. Partial support by DFG and the INFN during the completion of this work is gratefully acknowledged.

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  1. Institute of Complex Systems II, Forschungszentrum Jülich GmbH, 52425, Jülich, Germany

    Gunter M. Schütz

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  1. Gunter M. Schütz
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  1. Centre of Mathematics, University of Minho, Braga, Portugal

    Patrícia Gonçalves

  2. Centre of Mathematics, University of Minho, Braga, Portugal

    Ana Jacinta Soares

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Schütz, G.M. (2015). Conditioned Stochastic Particle Systems and Integrable Quantum Spin Systems. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations II. Springer Proceedings in Mathematics & Statistics, vol 129. Springer, Cham. https://doi.org/10.1007/978-3-319-16637-7_15

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