The Annals of Probability

Capacity of the range of random walk on $\mathbb{Z}^{4}$

Amine Asselah, Bruno Schapira, and Perla Sousi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the scaling limit of the capacity of the range of a random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-Gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in ’86 [Comm. Math. Phys. 104 (1986) 471–507] for the volume of the range in dimension two.

Article information

Source
Ann. Probab., Volume 47, Number 3 (2019), 1447-1497.

Dates
Received: January 2017
Revised: March 2018
First available in Project Euclid: 2 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1556784024

Digital Object Identifier
doi:10.1214/18-AOP1288

Mathematical Reviews number (MathSciNet)
MR3945751

Zentralblatt MATH identifier
07067274

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G50: Sums of independent random variables; random walks

Keywords
Capacity Green kernel law of large numbers central limit theorem

Citation

Asselah, Amine; Schapira, Bruno; Sousi, Perla. Capacity of the range of random walk on $\mathbb{Z}^{4}$. Ann. Probab. 47 (2019), no. 3, 1447--1497. doi:10.1214/18-AOP1288. https://projecteuclid.org/euclid.aop/1556784024


Export citation

References

  • [1] Aizenman, M. (1985). The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory. Comm. Math. Phys. 97 91–110.
  • [2] Albeverio, S. and Zhou, X. Y. (1996). Intersections of random walks and Wiener sausages in four dimensions. Acta Appl. Math. 45 195–237.
  • [3] Asselah, A. and Schapira, B. (2017). Moderate deviations for the range of a transient random walk: Path concentration. Ann. Sci. Éc. Norm. Supér. (4) 50 755–786.
  • [4] Asselah, A., Schapira, B. and Sousi, P. (2018). Capacity of the range of random walk on $\mathbb{Z}^{d}$. Trans. Amer. Math. Soc. 370 7627–7645.
  • [5] Asselah, A., Schapira, B. and Sousi, P. (2018). Strong law of large numbers for the capacity of the Wiener sausage in dimension four. Probab. Theory Related Fields 173 813–858.
  • [6] Brydges, D. C. and Spencer, T. (1985). Self-avoiding random walk and the renormalisation group. In Applications of Field Theory to Statistical Mechanics (Sitges, 1984). Lecture Notes in Physics 216 189–198. Springer, Berlin.
  • [7] Burdzy, K. and Lawler, G. F. (1990). Nonintersection exponents for Brownian paths. I. Existence and an invariance principle. Probab. Theory Related Fields 84 393–410.
  • [8] Chang, Y. (2017). Two observations on the capacity of the range of simple random walks on $\mathbb{Z}^{3}$ and $\mathbb{Z}^{4}$. Electron. Commun. Probab. 22 Paper No. 25, 9.
  • [9] Chang, Y. and Sapozhnikov, A. (2016). Phase transition in loop percolation. Probab. Theory Related Fields 164 979–1025.
  • [10] Duplantier, B. (1998). Random walks and quantum gravity in two dimensions. Phys. Rev. Lett. 81 5489–5492.
  • [11] Duplantier, B. and Kwon, K.-H. (1988). Conformal invariance and intersections of random walks. Phys. Rev. Lett. 61 2514–2517.
  • [12] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics 31. Cambridge Univ. Press, Cambridge.
  • [13] Dvoretzky, A. and Erdős, P. (1951). Some problems on random walk in space. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950 353–367. Univ. California Press, Berkeley, CA.
  • [14] Dvoretzky, A., Erdős, P. and Kakutani, S. (1950). Double points of paths of Brownian motion in $n$-space. Acta Sci. Math. (Szeged) 12 75–81.
  • [15] Erhard, D. and Poisat, J. (2016). Asymptotics of the critical time in Wiener sausage percolation with a small radius. ALEA Lat. Am. J. Probab. Math. Stat. 13 417–445.
  • [16] Felder, G. and Fröhlich, J. (1985). Intersection properties of simple random walks: A renormalization group approach. Comm. Math. Phys. 97 111–124.
  • [17] Fernández, R., Fröhlich, J. and Sokal, A. D. (1992). Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Springer, Berlin.
  • [18] Jain, N. and Orey, S. (1968). On the range of random walk. Israel J. Math. 6 373–380.
  • [19] Jain, N. C. and Pruitt, W. E. (1971). The range of transient random walk. J. Anal. Math. 24 369–393.
  • [20] Khoshnevisan, D. (2003). Intersections of Brownian motions. Expo. Math. 21 97–114.
  • [21] Lawler, G. F. (1980). A self-avoiding random walk. Duke Math. J. 47 655–693.
  • [22] Lawler, G. F. (1982). The probability of intersection of independent random walks in four dimensions. Comm. Math. Phys. 86 539–554.
  • [23] Lawler, G. F. (1985). Intersections of random walks in four dimensions. II. Comm. Math. Phys. 97 583–594.
  • [24] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
  • [25] Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge.
  • [26] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 275–308.
  • [27] Le Gall, J.-F. (1985). Sur le temps local d’intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan. In Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math. 1123 314–331. Springer, Berlin.
  • [28] Le Gall, J.-F. (1986). Propriétés d’intersection des marches aléatoires. I. Convergence vers le temps local d’intersection. Comm. Math. Phys. 104 471–507.
  • [29] Le Gall, J.-F. (1988). Fluctuation results for the Wiener sausage. Ann. Probab. 16 991–1018.
  • [30] Le Gall, J.-F. (1994). Exponential moments for the renormalized self-intersection local time of planar Brownian motion. In Séminaire de Probabilités, XXVIII. Lecture Notes in Math. 1583 172–180. Springer, Berlin.
  • [31] Le Gall, J.-F. and Rosen, J. (1991). The range of stable random walks. Ann. Probab. 19 650–705.
  • [32] Madras, N. and Slade, G. (2013). The Self-Avoiding Walk. Birkhäuser/Springer, New York.
  • [33] Pemantle, R., Peres, Y. and Shapiro, J. W. (1996). The trace of spatial Brownian motion is capacity-equivalent to the unit square. Probab. Theory Related Fields 106 379–399.
  • [34] Ráth, B. and Sapozhnikov, A. (2012). Connectivity properties of random interlacement and intersection of random walks. ALEA Lat. Am. J. Probab. Math. Stat. 9 67–83.
  • [35] Symanzik, K. (1969). Euclidean quantum field theory. In Local Quantum Theory (R. Jost, ed.) 152–226. Academic Press, New York.
  • [36] Sznitman, A.-S. (2010). Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 2039–2087.
  • [37] van den Berg, M., Bolthausen, E. and den Hollander, F. (2004). On the volume of the intersection of two Wiener sausages. Ann. of Math. (2) 159 741–782.
  • [38] van den Berg, M., Bolthausen, E. and den Hollander, F. (2018). Torsional rigidity for regions with a Brownian boundary. Potential Anal. 48 375–403.