On the physical relevance of random walks: an example of random walks on a randomly oriented lattice

Random walks on general graphs play an important role in the understanding of the general theory of stochastic processes. Beyond their fundamental interest in probability theory, they arise also as simple models of physical systems. A brief survey of the physical relevance of the notion of random walk on both undirected and directed graphs is given followed by the exposition of some recent results on random walks on randomly oriented lattices. It is worth noticing that general undirected graphs are associated with (not necessarily Abelian) groups while directed graphs are associated with (not necessarily Abelian) $C^*$-algebras. Since quantum mechanics is naturally formulated in terms of $C^*$-algebras, the study of random walks on directed lattices has been motivated lately by the development of the new field of quantum information and communication.

[1]  Palle E.T. Jorgensen,et al.  Iterated Function Systems and Permutation Representations of the Cuntz Algebra , 1996 .

[2]  The Infinite Self-Avoiding Walk in High Dimensions , 1989 .

[3]  M. Westwater.  On Edwards' model for long polymer chains , 1980 .

[4]  J. Preskill.  Quantum computing: pro and con , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[5]  Gordon Slade,et al.  Self-avoiding walk in five or more dimensions I. The critical behaviour , 1992 .

[6]  N. Madras,et al.  THE SELF-AVOIDING WALK , 2006 .

[7]  T. W. Körner,et al.  The Pleasures of Counting: Subtle is the Lord , 1996 .

[8]  K. Symanzik.  Euclidean Quantum field theory , 1969 .

[9]  M. Aizenman,et al.  Topological anomalies in the n dependence of the n-states Potts lattice gauge theory , 1984 .

[10]  D. Vere-Jones.  Markov Chains , 1972, Nature.

[11]  Proof of the Conjecture that the Planar Self-Avoiding Walk has Root Mean Square Displacement Exponent 3/4 , 2001, math/0108077.

[12]  E. Bolthausen,et al.  A Central Limit Theorem for Convolution Equations and Weakly Self-Avoiding Walks , 2001, math/0103218.

[13]  NONCOMMUTATIVE GEOMETRY OF TILINGS AND GAP LABELLING , 1994, cond-mat/9403065.

[14]  S. Edwards.  The statistical mechanics of polymers with excluded volume , 1965 .

[15]  The flow space of a directed G-graph , 1993 .

[16]  Gordon Slade,et al.  A new inductive approach to the lace expansion for self-avoiding walks , 1997 .

[17]  G. Pólya.  Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz , 1921 .

[18]  Thomas Spencer,et al.  Self-avoiding walk in 5 or more dimensions , 1985 .

[19]  A Connection between Multiresolution Wavelet Theory of Scale N and Representations of the Cuntz Algebra O N , 1996 .

[20]  J. Glimm,et al.  Quantum Physics: A Functional Integral Point of View , 1981 .

[21]  J. Renault,et al.  Graphs, Groupoids, and Cuntz–Krieger Algebras , 1997 .

[22]  F. Koukiou,et al.  The Hausdorff dimension of the two-dimensional Edwards' random walk , 1989 .

[23]  Random walks on randomly oriented lattices , 2001, math/0111305.

[24]  Gelu Popescu,et al.  Isometric dilations for infinite sequences of noncommuting operators , 1989 .

[25]  Andrew Lesniewski,et al.  Noncommutative Geometry , 1997 .

[26]  Wolfgang Krieger,et al.  A class ofC*-algebras and topological Markov chains , 1980 .

[27]  Une preuve « standard » du principe d'invariance de Stoll , 1997 .

[28]  I. Raeburn,et al.  CUNTZ-KRIEGER ALGEBRAS OF DIRECTED GRAPHS , 1998 .

[29]  P. Forcrand,et al.  Critical behaviour of the Edwards random walk in two dimensions: a case where the fractal and Hausdorff dimensions are not equal , 1988 .

[30]  J. Fröhlich,et al.  The random walk representation of classical spin systems and correlation inequalities , 1982 .

[31]  W. Woess.  Random walks on infinite graphs and groups, by Wolfgang Woess, Cambridge Tracts , 2001 .

[32]  Martin Gennis,et al.  Explorations in Quantum Computing , 2001, Künstliche Intell..

[33]  T. B. Grimley.  The Configuration of Real Polymer Chains , 1951 .

[34]  Alan D. Sokal,et al.  Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory , 1992 .

[35]  A. Holevo.  Statistical structure of quantum theory , 2001 .