zbMATH — the first resource for mathematics

Self-avoiding walk on nonunimodular transitive graphs. (English) Zbl 1448.60187
Summary: We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length \(n\) is comparable to the \(n\) th power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All of these results apply in particular to the product \(T_k\times\mathbb{Z}^d\) of a \(k\)-regular tree \((k\geq 3)\) with \(\mathbb{Z}^d \), for which these results were previously only known for large \(k\).

60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
60G50 Sums of independent random variables; random walks
05C30 Enumeration in graph theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
Full Text: DOI Euclid arXiv
[1] Bauerschmidt, R., Brydges, D. C. and Slade, G. (2015). Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: A renormalisation group analysis. Comm. Math. Phys. 337 817-877. · Zbl 1318.60049
[2] Bauerschmidt, R., Duminil-Copin, H., Goodman, J. and Slade, G. (2012). Lectures on self-avoiding walks. In Probability and Statistical Physics in Two and More Dimensions. Clay Math. Proc. 15 395-467. Amer. Math. Soc., Providence, RI. · Zbl 1317.60125
[3] Benjamini, I. (2016). Self avoiding walk on the seven regular triangulation. Preprint. Available at arXiv:1612.04169.
[4] Benjamini, I. and Curien, N. (2012). Ergodic theory on stationary random graphs. Electron. J. Probab. 17 no. 93. · Zbl 1278.05222
[5] Diestel, R. and Leader, I. (2001). A conjecture concerning a limit of non-Cayley graphs. J. Algebraic Combin. 14 17-25. · Zbl 0985.05020
[6] Domb, C. and Joyce, G. (1972). Cluster expansion for a polymer chain. J. Phys. C, Solid State Phys. 5 956.
[7] Duminil-Copin, H., Glazman, A., Hammond, A. and Manolescu, I. (2016). On the probability that self-avoiding walk ends at a given point. Ann. Probab. 44 955-983. · Zbl 1347.60131
[8] Duminil-Copin, H. and Hammond, A. (2013). Self-avoiding walk is sub-ballistic. Comm. Math. Phys. 324 401-423. · Zbl 1277.82027
[9] Duminil-Copin, H. and Smirnov, S. (2012). The connective constant of the honeycomb lattice equals \(\sqrt{2+\sqrt{2}} \). Ann. of Math. (2) 175 1653-1665. · Zbl 1253.82012
[10] Gilch, L. A. and Müller, S. (2017). Counting self-avoiding walks on free products of graphs. Discrete Math. 340 325-332. · Zbl 1351.05193
[11] Grimmett, G. and Li, Z. (2013). Self-avoiding walks and the Fisher transformation. Electron. J. Combin. 20 Paper 47. · Zbl 1295.05123
[12] Grimmett, G. R., Holroyd, A. E. and Peres, Y. (2014). Extendable self-avoiding walks. Ann. Inst. Henri Poincaré D 1 61-75. · Zbl 1285.05163
[13] Grimmett, G. R. and Li, Z. (2014). Strict inequalities for connective constants of transitive graphs. SIAM J. Discrete Math. 28 1306-1333. · Zbl 1305.05102
[14] Grimmett, G. R. and Li, Z. (2015). Bounds on connective constants of regular graphs. Combinatorica 35 279-294. · Zbl 1374.05123
[15] Grimmett, G. R. and Li, Z. (2016). Cubic graphs and the golden mean. Available at arXiv:1610.00107. · Zbl 1429.05048
[16] Grimmett, G. R. and Li, Z. (2017). Connective constants and height functions for Cayley graphs. Trans. Amer. Math. Soc. 369 5961-5980. · Zbl 1362.05059
[17] Grimmett, G. R. and Li, Z. (2017). Self-avoiding walks and amenability. Electron. J. Combin. 24 Paper 4.38. · Zbl 1376.05068
[18] Grimmett, G. R. and Li, Z. (2017). Self-avoiding walks and connective constants Available at arXiv:1704.05884.
[19] Grimmett, G. R. and Li, Z. (2018). Locality of connective constants. Discrete Math. 341 3483-3497. · Zbl 1397.05076
[20] Hammersley, J. M. and Morton, K. W. (1954). Poor man’s Monte Carlo. J. Roy. Statist. Soc. Ser. B 16 23-38; discussion 61-75. · Zbl 0055.36901
[21] Hammersley, J. M. and Welsh, D. J. A. (1962). Further results on the rate of convergence to the connective constant of the hypercubical lattice. Q. J. Math. 13 108-110. · Zbl 0123.00304
[22] Hara, T. (2008). Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36 530-593. · Zbl 1142.82006
[23] Hara, T. and Slade, G. (1992). The lace expansion for self-avoiding walk in five or more dimensions. Rev. Math. Phys. 4 235-327. · Zbl 0755.60054
[24] Hara, T. and Slade, G. (1992). Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys. 147 101-136. · Zbl 0755.60053
[25] Hutchcroft, T. (2016). Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters. C. R. Math. Acad. Sci. Paris 354 944-947. · Zbl 1351.60128
[26] Hutchcroft, T. (2017). Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs. Available at arXiv:1711.02590.
[27] Hutchcroft, T. (2018). The Hammersley-Welsh bound for self-avoiding walk revisited. Electron. Commun. Probab. 23 Paper No. 5. · Zbl 1388.60162
[28] Kesten, H. (1964). On the number of self-avoiding walks. II. J. Math. Phys. 5 1128-1137. · Zbl 0161.37402
[29] Li, Z. (2016). Positive speed self-avoiding walks on graphs with more than one end. Preprint. Available at arXiv:1612.02464.
[30] Lyons, R. and Peres, Y. (2016). Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics 42. Cambridge Univ. Press, New York.
[31] Madras, N. and Slade, G. (2013). The Self-Avoiding Walk. Modern Birkhäuser Classics. Birkhäuser/Springer, New York. Reprint of the 1993 original. · Zbl 1254.01051
[32] Madras, N. and Wu, C. C. (2005). Self-avoiding walks on hyperbolic graphs. Combin. Probab. Comput. 14 523-548. · Zbl 1139.82026
[33] Nachmias, A. and Peres, Y. (2012). Non-amenable Cayley graphs of high girth have \(p_c<p_u\) and mean-field exponents. Electron. Commun. Probab. 17 Paper No. 57. · Zbl 1302.82056
[34] Nienhuis, B. (1982). Exact critical point and critical exponents of \({\text{O}}(n)\) models in two dimensions. Phys. Rev. Lett. 49 1062-1065.
[35] Pak, I. and Smirnova-Nagnibeda, T. (2000). On non-uniqueness of percolation on nonamenable Cayley graphs. C. R. Acad. Sci., Sér. 1 Math. 330 495-500. · Zbl 0947.43003
[36] Soardi, P. M. and Woess, W. (1990). Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Z. 205 471-486. · Zbl 0693.43001
[37] Timár, Á. (2006). Percolation on nonunimodular transitive graphs. Ann. Probab. 34 2344-2364.
[38] Yamamoto, K. (2017). An upper bound for the critical probability on the cartesian product graph of a regular tree and a line. Preprint. Available at arXiv:1705.06873.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.