# zbMATH — the first resource for mathematics

Disconnection and level-set percolation for the Gaussian free field. (English) Zbl 1337.60246
The paper under review is a study of the level-set percolation of the Gaussian free field on $$Z^d$$ for $$d\geq 3$$ and it aims to derive the upper and lower bounds on the probability that a box of large side-length gets disconnected from the boundary of a larger homothetic box by the excursion-set of the Gaussian free field below the level $$\alpha$$.
By results of J. Bricmont et al. [J. Stat. Phys. 48, No. 5–6, 1249–1268 (1987; Zbl 0962.82520)] and P.-F. Rodriguez and A.-S. Sznitman [Commun. Math. Phys. 320, No. 2, 571–601 (2013; Zbl 1269.82028)], there is a critical value $$0\leq h^* < \infty$$ such that the level set $$E^{\geq \alpha}=\{x\in Z^d: \phi_x\geq \alpha\}$$ has only finite connected components a.s. for $$\alpha > h_*$$ and the level set $$E^{\geq \alpha}$$ has a unique infinite connected component a.s. for $$\alpha < h_*$$, where $$\phi = (\phi_x)_{x\in Z^d}$$ is the canonical process in the canonical law of the discrete Gaussian free field. There is another critical value $$h_{**} =\inf \{\alpha \in \mathbb{R}: \lim_L \inf \operatorname{P}[B_L \overset{\geq \alpha}{\longleftrightarrow} \partial B_{2L}] =0\}$$. Define the event $$A_N$$ as the complement of the event $$B_N \overset{\geq \alpha}{\longleftrightarrow} \partial S_N=\{x\in Z^d: |x|_{\infty} =|MN|\}$$. In the strongly non-percolative regime $$\alpha > h_{**}$$ of the excursion set $$E^{\geq \alpha}$$, one has $$\lim_N \operatorname{P}[A_N]=1$$.
The main result of the paper is Theorem 5.5 that there is a critical value $$\overline{h} = \sup \{h\in\mathbb R: \text{for all }h>\alpha > \beta$$, $$\phi$$ strongly percolates at levels $$\alpha, \beta$$

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G60 Random fields 60G15 Gaussian processes 60G50 Sums of independent random variables; random walks 60F10 Large deviations 82B43 Percolation
Full Text:
##### References:
 [1] R. J. Adler and J. E. Taylor, Random Fields and Geometry, Springer, Berlin, 2007. · Zbl 1149.60003 [2] K. S. Alexander, J. T. Chayes and L. Chayes, The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation, Commun. Math. Phys., 131 (1990), 1-150. · Zbl 0698.60098 [3] E. Bolthausen, J.-D. Deuschel and O. Zeitouni, Entropic repulsion for the lattice free field, Commun. Math. Phys., 170 (1995), 417-443. · Zbl 0821.60040 [4] E. Bolthausen and J. D. Deuschel, Critical large deviations for Gaussian fields in the phase transition regime, Ann. Probab., 21 (1993), 1876-1920. · Zbl 0801.60018 [5] J. Bricmont, J. L. Lebowitz and C. Maes, Percolation in strongly correlated systems: the massless Gaussian field, J. Stat. Phys., 48 (1987), 1249-1268. · Zbl 0962.82520 [6] M. Campanino and L. Russo, An upper bound on the critical percolation probability for the three-dimensional cubic lattice, Ann. Probab., 13 (1985), 478-491. · Zbl 0567.60096 [7] R. Cerf, Large deviations for three dimensional supercritical percolation, Astérisque, 267 , Société Mathématique de France, 2000. · Zbl 0962.60002 [8] J. D. Deuschel and D. W. Stroock, Large deviations, Academic Press, Boston, 1989. · Zbl 0705.60029 [9] A. Drewitz, B. Ráth and A. Sapozhnikov, Local percolative properties of the vacant set of random interlacements with small intensity, Ann. Inst. Henri Poincaré Probab. Stat., 50 (2014), 1165-1197. · Zbl 1319.60180 [10] A. Drewitz, B. Ráth and A. Sapozhnikov, On chemical distances and shape theorems in percolation models with long-range correlations, J. Math. Phys., 55 (2014), 30. · Zbl 1301.82027 [11] A. Drewitz and P.-F. Rodriguez, High-dimensional asymptotics for percolation of Gaussian free field level sets, Electron. J. Probab., 20 (2015), no.,47, 1-39. · Zbl 1321.60207 [12] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes, Walter de Gruyter, Berlin, 1994. · Zbl 0838.31001 [13] O. Garet, Percolation transition for some excursion sets, Electronic. J. Probab., 9 (2004), 244-292. · Zbl 1065.60147 [14] G. Giacomin, Aspects of statistical mechanics of random surfaces, Notes of lectures at IHP (Fall 2001), version of February 24, 2003, available at http://www.proba.jussieu.fr/pageperso/giacomin/pub/IHP.ps, 2003. [15] G. Grimmett, Percolation, Second edition, Springer, Berlin, 1999. [16] G. F. Lawler, Intersections of random walks, Birkhäuser, Basel, 1991. · Zbl 1228.60004 [17] J. L. Lebowitz and H. Saleur, Percolation in strongly correlated systems, Phys. A, 138 (1986), 194-205. · Zbl 0666.60110 [18] X. Li and A. S. Sznitman, Large deviations for occupation time profiles of random interlacements, Probab. Theory Related Fields, 161 (2015), 309-350. · Zbl 1314.60078 [19] X. Li and A. S. Sznitman, A lower bound for disconnection by random interlacements, Electron. J. Probab., 19 (2014), 126. · Zbl 1355.60035 [20] T. Liggett, R. H. Schonmann and A. M. Stacey, Domination by product measures, Ann. Probab., 25 (1997), 71-95. · Zbl 0882.60046 [21] T. Lupu, From loop clusters and random interlacement to the free field, preprint, available at arXiv: arXiv: arXiv:1402.0298 · Zbl 1348.60141 [22] V. Marinov, Percolation in correlated systems, PhD thesis, Rutgers University, available online at: http://www.books.google.com/books?sisbn=0549701338, 2007. [23] S. Popov and B. Ráth, On decoupling inequalities and percolation of excursion sets of the Gaussian free field, J. Stat. Phys., 159 (2015), 312-320. · Zbl 1328.82026 [24] S. Popov and A. Teixeira, Soft local times and decoupling of random interlacements, to appear in J. Eur. Math. Soc., also available at arXiv: arXiv: arXiv:1212.1605 · Zbl 1329.60342 [25] S. Port and C. Stone, Brownian motion and classical Potential Theory, Academic Press, New York, 1978. · Zbl 0413.60067 [26] P.-F. Rodriguez and A. S. Sznitman, Phase transition and level-set percolation for the Gaussian free field, Commun. Math. Phys., 320 (2013), 571-601. · Zbl 1269.82028 [27] F. Spitzer, Principles of random walk, Springer, Berlin, second edition, 2001. · Zbl 0979.60002 [28] A. S. Sznitman, Vacant set of random interlacements and percolation, Ann. Math., 171 (2010), 2039-2087. · Zbl 1202.60160 [29] A. S. Sznitman, Decoupling inequalities and interlacement percolation on $$G \times Z$$, Invent. Math., 187 (2012), 645-706. · Zbl 1277.60183 [30] A. S. Sznitman, An isomorphism theorem for random interlacements, Electron. Commun. Probab., 17 (2012), 1-9. · Zbl 1247.60135 [31] A. Teixeira, On the size of a finite vacant cluster of random interlacements with small intensity, Probab. Theory Related Fields, 150 (2011), 529-574. · Zbl 1231.60117 [32] Á. Timár, Boundary connectivity via graph theory, Proc. Amer. Math. Soc., 141 (2012), 475-480. · Zbl 1259.05049 [33] D. Windisch, On the disconnection of a discrete cylinder by a biased random walk, Ann. Appl. Probab., 18 (2008), 1441-1490. · Zbl 1148.60028 [34] W. Woess, Random walks on infinite graphs and groups, Cambridge University Press, 2000. · Zbl 0951.60002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.