Electronic Journal of Probability

Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup

Caio Alves and Artem Sapozhnikov

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It has been recently understood [9, 24, 30] that for a general class of percolation models on $\mathbb{Z} ^{d}$ satisfying suitable decoupling inequalities, which includes i.a. Bernoulli percolation, random interlacements and level sets of the Gaussian free field, large scale geometry of the unique infinite cluster in strongly percolative regime is qualitatively the same; in particular, the random walk on the infinite cluster satisfies the quenched invariance principle, Gaussian heat-kernel bounds and local CLT.

In this paper we consider the random walk loop soup on $\mathbb{Z} ^{d}$ in dimensions $d\geq 3$. An interesting aspect of this model is that despite its similarity and connections to random interlacements and the Gaussian free field, it does not fall into the above mentioned general class of percolation models, since the required decoupling inequalities are not valid.

We identify weaker (and more natural) decoupling inequalities and prove that (a) they do hold for the random walk loop soup and (b) all the results about the large scale geometry of the infinite percolation cluster proved for the above mentioned class of models hold also for models that satisfy the weaker decoupling inequalities. Particularly, all these results are new for the vacant set of the random walk loop soup. (The range of the random walk loop soup has been addressed by Chang [6] by a model specific approximation method, which does not apply to the vacant set.)

Finally, we prove that the strongly supercritical regime for the vacant set of the random walk loop soup is non-trivial. It is expected, but open at the moment, that the strongly supercritical regime coincides with the whole supercritical regime.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 110, 34 pp.

Received: 13 August 2018
Accepted: 8 September 2019
First available in Project Euclid: 3 October 2019

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Digital Object Identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35] 60G55: Point processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Random walk loop soup percolation decoupling inequality long-range correlations Poisson point process random walk

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Alves, Caio; Sapozhnikov, Artem. Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup. Electron. J. Probab. 24 (2019), paper no. 110, 34 pp. doi:10.1214/19-EJP360. https://projecteuclid.org/euclid.ejp/1570068174

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  • [1] Alves, C. and Popov, S.: Conditional decoupling of random interlacements. ALEA Lat. Am. J. Probab. Math. Stat. 15(2), (2018), 1027–1063.
  • [2] van den Berg, J. and Keane, M.: On the continuity of the percolation probability function. Conference in modern analysis and probability (New Haven, Conn., 1982), Contemp. Math., 26, Amer. Math. Soc., Providence, RI (1984), 61–65.
  • [3] van de Burg, T., Camia, F. and Lis, M.: Random walk loop soups and conformal loop ensembles. Probab. Theory Relat. Fields 166, (2016), 553–584.
  • [4] Burton, R. M. and Keane, M.: Density and uniqueness in percolation. Comm. Math. Phys. 121, (1989), 501–505.
  • [5] Camia, F.: Scaling limits, Brownian loops, and conformal fields. Advances in disordered systems, random processes and some applications, 205–269, Cambridge Univ. Press, Cambridge, 2017.
  • [6] Chang, Y.: Supercritical loop percolation on $\mathbb{Z} ^{d}$ for $d\geq 3$. Stochastic Process. Appl. 127(10), (2017), 3159–3186.
  • [7] Chang, Y. and Sapozhnikov, A.: Phase transition in loop percolation. Probab. Theory Related Fields 164(3-4), (2016), 979–1025.
  • [8] Drewitz, A., Prévost, A. and Rodriguez, P.-F.: Geometry of Gaussian free field sign clusters and random interlacements, arXiv:1811.05970.
  • [9] Drewitz, A., Ráth, B. and Sapozhnikov. A.: On chemical distances and shape theorems in percolation models with long-range correlations. J. Math. Phys. 55(8), (2014), 083307, 30 pp.
  • [10] Drewitz, A., Ráth, B. and Sapozhnikov. A.: Local percolative properties of the vacant set of random interlacements with small intensity. Ann. Inst. H. Poincaré Probab. Statist. 50(4), (2014), 1165–1197.
  • [11] Drewitz, A., Ráth, B. and Sapozhnikov. A.: An introduction to random interlacements. SpringerBriefs in Mathematics. Springer, Cham, 2014.
  • [12] Grimmett, G. R.: Percolation. Springer-Verlag, Berlin, 1999, second edition.
  • [13] Lawler, G. F.: Intersections of random walks. Birkhäuser, Basel, 1991.
  • [14] Lawler, G. F. and Limic, V.: Random walk: a modern introduction. Cambridge University Press, Cambridge, 2010.
  • [15] Lawler, G. F. and Trujillo Ferreras, J. A.: Random walk loop soup. Trans. Amer. Math. Soc. 359(2), (2007), 767–787.
  • [16] Lawler, G. F. and Werner, W.: The Brownian loop soup. Probab. Theory Related Fields 128(4), (2004), 565–588.
  • [17] Le Jan, Y.: Markov paths, loops and fields. Lecture Notes in Mathematics, vol. 2026, Springer, Heidelberg, 2011, Lectures from the 38th Probability Summer School held in Saint-Flour, 2008.
  • [18] Le Jan, Y. and Lemaire, S.: Markovian loop clusters on graphs. Illinois J. Math. 57(2), (2013), 525–558.
  • [19] Lupu, T.: From loop clusters and random interlacement to the free field. Ann. Probab. 44(3), (2016), 2117–2146.
  • [20] Lupu, T.: Convergence of the two-dimensional random walk loop soup clusters to CLE. J. Eur. Math. Soc. 21(4), (2019), 1201–1227.
  • [21] Lupu, T.: Loop percolation on discrete half-plane. Electron. Commun. Probab. 21(30), (2016).
  • [22] Popov, S. and Ráth, B.: On decoupling inequalities and percolation of excursion sets of the Gaussian free field. J. Stat. Phys. 159(2), (2015), 312–320.
  • [23] Popov, S. and Teixeira, A.: Soft local times and decoupling of random interlacements. J. Eur. Math. Soc. 17(10), (2015), 2545–2593.
  • [24] Procaccia, E., Rosenthal, R. and Sapozhnikov, A.: Quenched invariance principle for simple random walk on clusters in correlated percolation models. Probab. Theory Related Fields 166(3-4), (2016), 619–657.
  • [25] Ráth, B.: A short proof of the phase transition for the vacant set of random interlacements. Electron. Commun. Probab. 20(3), (2015), 11 pp.
  • [26] Ráth, B. and Sapozhnikov, A.: On the transience of random interlacements. Electron. Commun. Probab. 16, (2011), 379–391.
  • [27] Ráth. B. and Valesin, D.: Percolation on the stationary distributions of the voter model. Ann. Probab. 45(3), (2017), 1899–1951.
  • [28] Rodriguez, P.-F.: Decoupling inequalities for the Ginzburg-Landau $\nabla \varphi $ models, arXiv:1612.02385.
  • [29] Rodriguez, P.-F. and Sznitman, A.-S.: Phase transition and level-set percolation for the Gaussian free field, Comm. Math. Phys. 320(2), (2013), 571–601.
  • [30] Sapozhnikov, A.: Random walks on infinite percolation clusters in models with long-range correlations. Ann. Probab. 45(3), (2017), 1842–1898.
  • [31] Sapozhnikov, A. and Shiraishi, D.: On Brownian motion, simple paths, and loops. Probab. Theory Related Fields 172(3-4), (2018), 615–662.
  • [32] Sheffield, S. and Werner, W.: Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Ann. of Math. (2) 176(3), (2012), 1827–1917.
  • [33] Symanzik, K.: Euclidean quantum field theory. Scuola internazionale di Fisica “Enrico Fermi” XLV, (1969), 152–223.
  • [34] Sznitman, A.-S.: Vacant set of random interlacements and percolation. Ann. Math. 171(2), (2010), 2039–2087.
  • [35] Sznitman, A.-S.: Decoupling inequalities and interlacement percolation on $G\times \mathbb{Z} $. Invent. Math. 187(3), (2012), 645–706.
  • [36] Sznitman, A.-S.: Topics in occupation times and Gaussian free fields, Zürich Lectures in Advanced Mathematics, European Mathematical Society, 2012.
  • [37] Teixeira, T.: On the uniqueness of the infinite cluster of the vacant set of random interlacements. Ann. Appl. Probab. 19(1), (2009), 454–466.
  • [38] Teixeira, A.: On the size of a finite vacant cluster of random interlacements with small intensity. Probab. Theory Related Fields 150(3-4), (2011), 529–574.