The Annals of Applied Probability

Eigenvalue versus perimeter in a shape theorem for self-interacting random walks

Marek Biskup and Eviatar B. Procaccia

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We study paths of time-length $t$ of a continuous-time random walk on $\mathbb{Z}^{2}$ subject to self-interaction that depends on the geometry of the walk range and a collection of random, uniformly positive and finite edge weights. The interaction enters through a Gibbs weight at inverse temperature $\beta$; the “energy” is the total sum of the edge weights for edges on the outer boundary of the range. For edge weights sampled from a translation-invariant, ergodic law, we prove that the range boundary condensates around an asymptotic shape in the limit $t\to\infty$ followed by $\beta\to\infty$. The limit shape is a minimizer (unique, modulo translates) of the sum of the principal harmonic frequency of the domain and the perimeter with respect to the first-passage percolation norm derived from (the law of) the edge weights. A dense subset of all norms in $\mathbb{R}^{2}$, and thus a large variety of shapes, arise from the class of weight distributions to which our proofs apply.

Article information

Ann. Appl. Probab., Volume 28, Number 1 (2018), 340-377.

Received: June 2016
Revised: March 2017
First available in Project Euclid: 3 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 49Q10: Optimization of shapes other than minimal surfaces [See also 90C90]

Interacting polymer random environment shape theorem first-passage percolation Dirichlet eigenvalue perimeter


Biskup, Marek; Procaccia, Eviatar B. Eigenvalue versus perimeter in a shape theorem for self-interacting random walks. Ann. Appl. Probab. 28 (2018), no. 1, 340--377. doi:10.1214/17-AAP1307.

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  • [1] Alexander, K., Chayes, J. T. and Chayes, L. (1990). The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Comm. Math. Phys. 131 1–50.
  • [2] Asselah, A. and Shapira, B. (2015). Boundary of the range I: Typical behaviour. arXiv:1507.01031.
  • [3] Asselah, A. and Shapira, B. (2016). Boundary of the range II: Lower tails. arXiv:1601.03957.
  • [4] Auffinger, A., Damron, M. and Hanson, J. (2015). 50 years of first passage percolation. arXiv:1511.03262.
  • [5] Becker, M. and König, W. (2012). Self-intersection local times of random walks: Exponential moments in subcritical dimensions. Probab. Theory Related Fields 154 585–605.
  • [6] Berestycki, N. and Yadin, A. (2013). Condensation of random walks and the Wulff crystal. arXiv:1305.0139.
  • [7] Biskup, M., Fukushima, R. and König, W. (2016). Eigenvalue fluctuations for lattice Anderson Hamiltonians. SIAM J. Math. Anal. 48 2674–2700.
  • [8] Biskup, M. and König, W. (2016). Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails. Comm. Math. Phys. 341 179–218.
  • [9] Biskup, M., Louidor, O., Procaccia, E. B. and Rosenthal, R. (2015). Isoperimetry in two-dimensional percolation. Comm. Pure Appl. Math. 68 1483–1531.
  • [10] Biskup, M. and Proccacia, E. B. (2016). Shapes of drums with lowest base frequency under non-isotropic perimeter constraints. arXiv:1603.03871.
  • [11] Bodineau, T. (1999). The Wulff construction in three and more dimensions. Comm. Math. Phys. 207 197–229.
  • [12] Bodineau, T., Ioffe, D. and Velenik, Y. (2000). Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41 1033–1098.
  • [13] Bolthausen, E. (1994). Localization of a two-dimensional random walk with an attractive path interaction. Ann. Probab. 22 875–918.
  • [14] Cerf, R. (2000). Large deviations for three dimensional supercritical percolation. Astérisque 267 vi+177.
  • [15] Cerf, R. and Pisztora, Á. (2000). On the Wulff crystal in the Ising model. Ann. Probab. 28 947–1017.
  • [16] den Hollander, F. (2009). Random Polymers. Lectures from the 37th Probability Summer School Held in Saint-Flour, 2007. Lecture Notes in Math. 1974. Springer, Berlin.
  • [17] Dobrushin, R., Kotecký, R. and Shlosman, S. (1992). Wulff Construction: A Global Shape from Local Interaction. Translations of Mathematical Monographs 104. Amer. Math. Soc., Providence, RI.
  • [18] Donsker, M. D. and Varadhan, S. R. S. (1979). On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32 721–747.
  • [19] Durrett, R. and Liggett, T. M. (1981). The shape of the limit set in Richardson’s growth model. Ann. Probab. 9 186–193.
  • [20] Fan, K. (1949). On a theorem of Weyl concerning eigenvalues of linear transformations. I. Proc. Natl. Acad. Sci. USA 35 652–655.
  • [21] Häggström, O. and Meester, R. (1995). Asymptotic shapes for stationary first passage percolation. Ann. Probab. 23 1511–1522.
  • [22] Ioffe, D. and Schonmann, R. H. (1998). Dobrushin–Kotecký–Shlosman theorem up to the critical temperature. Comm. Math. Phys. 199 117–167.
  • [23] Kesten, H. (1986). Aspects of first passage percolation. In École d’été de probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 125–264. Springer, Berlin.
  • [24] König, W. (2016). The Parabolic Anderson Model: Random Walk in Random Potential. Birkhäuser/Springer, Cham.
  • [25] Povel, T. (1999). Confinement of Brownian motion among Poissonian obstacles in ${\mathbf{R}}^{d}$, $d\ge3$. Probab. Theory Related Fields 114 177–205.
  • [26] Sznitman, A.-S. (1991). On the confinement property of two-dimensional Brownian motion among Poissonian obstacles. Comm. Pure Appl. Math. 44 1137–1170.
  • [27] Sznitman, A.-S. (1998). Brownian Motion, Obstacles and Random Media. Springer, Berlin.
  • [28] van den Berg, M., Bolthausen, E. and den Hollander, F. (2001). Moderate deviations for the volume of the Wiener sausage. Ann. of Math. (2) 153 355–406.
  • [29] van der Hofstad, R., König, W. and Mörters, P. (2006). The universality classes in the parabolic Anderson model. Comm. Math. Phys. 267 307–353.