×

Phase transition in the exit boundary problem for random walks on groups. (English. Russian original) Zbl 1328.60117

Funct. Anal. Appl. 49, No. 2, 86-96 (2015); translation from Funkts. Anal. Prilozh. 49, No. 2, 7-20 (2015).
Summary: We describe the full exit boundary of random walks on homogeneous trees, in particular, on free groups. This model exhibits a phase transition; namely, the family of Markov measures under study loses ergodicity as a parameter of the random walk changes.{ }The problem under consideration is a special case of the problem of describing the invariant (central) measures on branching graphs, which covers a number of problems in combinatorics, representation theory, and probability and was fully stated in a series of recent papers by the first author. On the other hand, in the context of the theory of Markov processes, close problems were discussed as early as the 1960s by E. B. Dynkin [Usp. Mat. Nauk 24, No. 4(148), 89–152 (1969; Zbl 0185.45602)].

MSC:

60G50 Sums of independent random variables; random walks
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J50 Boundary theory for Markov processes
05C81 Random walks on graphs

Citations:

Zbl 0185.45602
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A. M. Vershik, “Intrinsic metric on graded graphs, standardness, and invariant measures,” Zap. Nauchn. Sem. POMI, 421 (2014), 58-67; English transl.: J. Math. Sci. (N. Y.), 200:6 (2014), 677-681. · Zbl 1336.28014
[2] A. M. Vershik, “The problem of describing central measures on the path spaces of graded graphs,” Funkts. Anal. Prilozhen., 48:4 (2014), 26-46; English transl.: Funct. Anal. Appl., 48:4 (2014), 256-271.
[3] A. M. Vershik, “Equipped graded graphs, projective limits of simplices, and their boundaries,” Zap. Nauchn. Sem. POMI, 432 (2015), 83-104. · Zbl 1323.05127
[4] A. M. Vershik, “Dynamic theory of growth in groups: Entropy, boundaries, examples,” Uspekhi Mat. Nauk, 55:4(334) (2000), 59-128; English transl.: Russian Math. Surveys, 55:4 (2000), 667-733. · Zbl 0991.37005
[5] A. M. Vershik and P. P. Nikitin, “Traces on infinite-dimensional Brauer algebras,” Funkts. Anal. Prilozhen., 40:3 (2006), 3-11; English transl.: Functional Anal. Appl., 40:3 (2006). · Zbl 1137.20011
[6] V. A. Kaimanovich and A. M. Vershik, “Random walks on discrete groups, boundary and entropy,” Ann. Probab., 11:3 (1983), 457-490. · Zbl 0641.60009
[7] R. I. Grigorchuk, “Symmetric random walks on discrete groups,” Uspekhi Mat. Nauk, 32:6(198) (1977), 217-218. · Zbl 0375.60008
[8] E. B. Dynkin and A. A. Yushkevich, Markov processes: Theorems and Problems, Plenum Press, New York, 1969. · Zbl 0073.34801
[9] E. B. Dynkin, “The space of exits of a Markov process,” Uspekhi Mat. Nauk, 24:4(148) (1969), 89-152. · Zbl 0201.19801
[10] Dynkin, E. B., Entrance and exit spaces for a Markov process, 507-512 (1971), Paris · Zbl 0265.60068
[11] E. B. Dynkin, “The initial and final behaviour of trajectories of Markov processes,” Uspekhi Mat. Nauk, 26:4(160) (1971), 153-172.
[12] A. Figà-Talamanca and C. Nebbia, Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees, London Mathematical Society Lecture Note Series, vol. 162, Cambridge University Press, Cambridge, 1991. · Zbl 1154.22301
[13] S. V. Kerov, “The boundary of Young lattice and random Young tableaux,” in: DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 24, Amer. Math. Soc., Providence, RI, 1996, 133-158. · Zbl 0856.05008
[14] S. Kerov, A. Okounkov, and G. Olshansky, “The boundary of the Young graph with Jack edge multiplicities,” Internat. Math. Res. Notices, No. 4, 1998, 173-199. · Zbl 0960.05107
[15] F. M. Goodman and S. V. Kerov, “The Martin boundary of the Young-Fibonacci lattice,” J. Algebraic Combin., 11:1 (2000), 17-48. · Zbl 0959.06003
[16] B. Levit and S. A. Molchanov, “Invariant Markov chains on a free group with a finite number of generators,” Vestnik Mosk. Univ., No. 6, 1971, 80-88. · Zbl 0226.60082
[17] M. Pagliacci, “Heat and wave equation on homogeneous trees,” Boll. Un. Mat. Ital. Ser. VII, A7:1 (1993), 37-45. · Zbl 0798.05066
[18] S. Helgason, “Eigenspaces of the Laplacian, integral representations and irreducibility,” J. Funct. Anal., 17 (1974), 328-353. · Zbl 0303.43021
[19] Y. Guivar’h, Ji Lizhen, and J. C. Taylor, Compactifications of Symmetric Spaces, Progress in Math., vol. 156, Birkhauser, Boston, 1998. · Zbl 1053.31006
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.