×

Shape transition under excess self-intersections for transient random walk. (English) Zbl 1202.60151

The paper extends the previous line of research of the present author on random walks in a random scenery. Special attention is paid to large deviations for two-fold self-intersection local times of a transient random walk in \(d\)-dimensions, with \(d\geq 3\). For a simple random walk on \(\mathbb{Z}^d\) started at the origin, the basic technical input is the \(q\)-norm i.e. the sum of \(q\)-th power of the local times which, for integer \(q\), can be written in terms of the \(q\)-fold self-intersection local times of that walk. The shape transition is described as the parameter \(q\) crosses the critical value \(q_c= d/(d-2)\). That is quantified in terms of two different sites exploration strategy by a transient random walk. A central limit theorem is established for the \(q\)-norm of the local times in \(d\geq 4\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
82C22 Interacting particle systems in time-dependent statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML

References:

[1] M. Aizenman. The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory. Comm. Math. Phys. 97 (1985) 91-110. · Zbl 0573.60076 · doi:10.1007/BF01206180
[2] A. Asselah. Large deviations principle for the self-intersection local times for simple random walk in dimension 5 or more. Preprint, 2007. Available at:
[3] A. Asselah. Large deviations for the Self-intersection local times for simple random walk in dimension d =3. Probab. Theory Related Fields 141 (2008) 19-45. · Zbl 1135.60340 · doi:10.1007/s00440-007-0078-x
[4] A. Asselah and F. Castell. A note on random walk in random scenery. Ann. Inst. H. Poincaré 43 (2007) 163-173. · Zbl 1112.60088 · doi:10.1016/j.anihpb.2006.01.004
[5] A. Asselah and F. Castell. Random walk in random scenery and self-intersection local times in dimensions d \geq 5. Probab. Theory Related Fields 138 (2007) 1-32. · Zbl 1116.60057 · doi:10.1007/s00440-006-0014-5
[6] M. Becker and W. König. Moments and distribution of the local times of a transient random walk on \Bbb Z d . J. Theoret. Probab. 22 (2009) 365-374. · Zbl 1175.60043 · doi:10.1007/s10959-008-0168-4
[7] P. Billingsley. Probability and Measure . Wiley, New York, 1979. · Zbl 0411.60001
[8] E. Bolthausen and U. Schmock. On self-attracting d -dimensional random walks. Ann. Probab. 25 (1997) 531-572. · Zbl 0873.60008 · doi:10.1214/aop/1024404411
[9] D. C. Brydges and G. Slade. The diffusive phase of a model of self-interacting walks. Probab. Theory Related Fields 103 (1995) 285-315. · Zbl 0832.60096 · doi:10.1007/BF01195476
[10] X. Chen. Limit laws for the energy of a charged polymer. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 638-672. · Zbl 1178.60024 · doi:10.1214/07-AIHP120
[11] X. Chen. Random walk intersections: Large deviations and some related topics. 2009. · Zbl 1192.60002
[12] A. Dvoretzky and P. Erdös. Some problems on random walk in space. In Proc. Berkeley Symposium 1951 353-367. Univ. California Press, Berkeley, 1951. · Zbl 0044.14001
[13] S. Edwards. The statistical mechanics of polymers with excluded volume. Proc. Phys. Sci 85 (1965) 613-624. · Zbl 0125.23205 · doi:10.1088/0370-1328/85/4/301
[14] G. Felder and J. Frölich. Intersection properties of simple random walks: A renormalization group approach. Comm. Math. Phys. 97 (1985) 111-124. · Zbl 0573.60065 · doi:10.1007/BF01206181
[15] K. Fleischmann, P. Mörters and V. Wachtel. Moderate deviations for random walk in random scenery. Stochastic Process. Appl. 118 (2008) 1768-1802. · Zbl 1157.60020 · doi:10.1016/j.spa.2007.11.001
[16] N. C. Jain and W. E. Pruitt. The range of transient random walk. J. Anal. Math. 24 (1971) 369-393. · Zbl 0249.60038 · doi:10.1007/BF02790380
[17] G. Lawler. Intersection of Random Walks. Probability and Its Applications . Birkhäuser, Boston, MA, 1991. · Zbl 0735.60071
[18] J.-F. Le Gall. Propriétés d’intersection des marches aléatoires. Comm. Math. Phys. 104 (1985) 471-507.
[19] J.-F. Le Gall. Sur le temps local d’intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan. Séminaire de probabilités de Strasbourg 19 (1985) 314-331. · Zbl 0563.60072
[20] J.-F. Le Gall and J. Rosen. The range of stable random walks. Ann. Probab. 19 (1991) 650-705. · Zbl 0729.60066 · doi:10.1214/aop/1176990446
[21] M. J. Westwater. On Edwards’ model for long polymer chains. Comm. Math. Phys. 72 (1980) 131-174. · Zbl 0431.60100 · doi:10.1007/BF01197632
[22] S. R. S. Varadhan. Appendix to Euclidean quantum field theory by K.Symanzik. In Local Quantum Field Theory . R. Jost (Ed.). Academic Press, New York, 1966.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.