Electronic Journal of Probability

Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations

Volker Betz and Lorenzo Taggi

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We consider nearest neighbour spatial random permutations on $\mathbb Z ^{d}$. In this case, the energy of the system is proportional to the sum of all cycle lengths, and the system can be interpreted as an ensemble of edge-weighted, mutually self-avoiding loops. The constant of proportionality, $\alpha $, is the order parameter of the model. Our first result is that in a parameter regime of edge weights where it is known that a single self-avoiding loop is weakly space filling, long cycles of spatial random permutations are still exponentially unlikely. For our second result, we embed a self-avoiding walk into a background of spatial random permutations, and condition it to cover a macroscopic distance. For large values of $\alpha $ (where long cycles are very unlikely) we show that this walk collapses to a straight line in the scaling limit, and give bounds on the fluctuations that are almost sufficient for diffusive scaling. For proving our results, we develop the concepts of spatial strong Markov property and iterative sampling for spatial random permutations, which may be of independent interest. Among other things, we use them to show exponential decay of correlations for large values of $\alpha $ in great generality.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 74, 37 pp.

Received: 15 June 2018
Accepted: 27 May 2019
First available in Project Euclid: 3 July 2019

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Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

self-avoiding walk random spatial permutations

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Betz, Volker; Taggi, Lorenzo. Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations. Electron. J. Probab. 24 (2019), paper no. 74, 37 pp. doi:10.1214/19-EJP328. https://projecteuclid.org/euclid.ejp/1562119474

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