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Untangling of Trajectories and Integrable Systems of Interacting Particles: Exact Results and Universal Laws

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Abstract

This review gives a survey of some results about systems of interacting particles and the laws characterizing their behavior on large scales, which are common for a number of phenomena unified under the notion of the Kardar–Parisi–Zhang universality class.

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Notes

  1. By typical we mean the choice of parameters away from the points of phase transitions.

  2. Our example of the model of vicious walkers, or rather its restriction on processes with a constant number of particles is strictly speaking not a Markov chain, since some of the processes drop out from consideration in course of time, violating the probability conservation. Within this section, however, the the probability conservation is not important.

  3. Above, we used the summation over permutations of the ends of the paths that, however, play exactly the same role as the heads.

  4. The prefix q-Hahn was first used when applied to a model with probabilities jumps (27) in the work [47].

  5. Strictly speaking, the sign \( \subset \) denoting a subset, is not used quite correctly in this case, since the sites in \(({\mathbf{x}},{\mathbf{t}})\) may coincide. Nevertheless, we will use it to denote the fact that all points \(({\mathbf{x}},{\mathbf{t}})\) are on the boundary \(\mathcal{B}\).

  6. That is a particle can jump forward with probability \(p\) if at the moment there are no particles with lower numbers in the same site or if all such particles also have decided to jump.

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  1. Bogoliubov Laboratory of Theoretical Physics, Joined Institute for Nuclear Research, 141980, Dubna, Russia

    A. M. Povolotsky

  2. National Research University Higher School of Economics, 101000, Moscow, Russia

    A. M. Povolotsky

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In memory of my Teacher and senior friend Vyacheslav Borisovich Priezzhev

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Povolotsky, A.M. Untangling of Trajectories and Integrable Systems of Interacting Particles: Exact Results and Universal Laws. Phys. Part. Nuclei 52, 239–273 (2021). https://doi.org/10.1134/S1063779621020040

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