Probability on Trees: An Introductory Climb

1. Preface 2. Basic Definitions and a Few Highlights 3. Galton-Watson Trees 4. General percolation on a connected graph 5. The First-Moment Method 6. Quasi-independent Percolation 7. The Second Moment Method 8. Electrical Networks 9. Infinite Networks 10. The Method of Random Paths 11. Transience of Percolation Clusters 12. Subperiodic Trees 13. The Random Walks \({\rm RW}_\lambda\) 14. Capacity 15. Intersection-Equivalence 16. Reconstruction for the Ising Model on a Tree 17. Unpredictable Paths in Z and EIT inZ2 18. Tree-Indexed Processes 19. Recurrence for Tree-Indexed Markov Chains 20. Dynamical Percolation 21. Stochastic Domination Between Trees

[1]  C. Nash-Williams,et al.  Random walk and electric currents in networks , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  D. Freedman,et al.  Random distribution functions , 1963 .

[3]  H. Kesten,et al.  A Limit Theorem for Multidimensional Galton-Watson Processes , 1966 .

[4]  H. Kesten,et al.  Additional Limit Theorems for Indecomposable Multidimensional Galton-Watson Processes , 1966 .

[5]  Kai Lai Chung.  Probabilistic approach in potential theory to the equilibrium problem , 1973 .

[6]  J. Hammersley.  Postulates for Subadditive Processes , 1974 .

[7]  F. Spitzer.  Markov Random Fields on an Infinite Tree , 1975 .

[8]  J. Kingman.  The First Birth Problem for an Age-dependent Branching Process , 1975 .

[9]  J. Biggins.  Chernoff's theorem in the branching random walk , 1977, Journal of Applied Probability.

[10]  J. A. Cavender.  Taxonomy with confidence , 1978 .

[11]  J. Snell,et al.  A branching process showing a phase transition , 1979, Journal of Applied Probability.

[12]  H. Kesten.  The critical probability of bond percolation on the square lattice equals 1/2 , 1980 .

[13]  J. Hawkes,et al.  Trees Generated by a Simple Branching Process , 1981 .

[14]  G. Lawler.  The probability of intersection of independent random walks in four dimensions , 1982 .

[15]  H. Poincaré,et al.  Percolation ? , 1982 .

[16]  Terry Lyons.  A Simple Criterion for Transience of a Reversible Markov Chain , 1983 .

[17]  Rick Durrett,et al.  Oriented percolation in dimensions d ≥ 4: bounds and asymptotic formulas , 1983, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  J. Laurie Snell,et al.  Random Walks and Electrical Networks , 1984 .

[19]  G. Lawler.  Intersections of random walks in four dimensions. II , 1985 .

[20]  D. Thouless,et al.  A mean field spin glass with short-range interactions , 1986 .

[21]  S. Evans.  Multiple points in the sample paths of a Lévy process , 1987 .

[22]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[23]  Which sets contain multiple points of Brownian motion , 1988 .

[24]  R. Lyons.  The Ising model and percolation on trees and tree-like graphs , 1989 .

[25]  Thomas S. Salisbury,et al.  Capacity and energy for multiparameter Markov processes , 1989 .

[26]  I. Vajda.  Theory of statistical inference and information , 1989 .

[27]  R. Lyons.  Random Walks and Percolation on Trees , 1990 .

[28]  G. Grimmett,et al.  The supercritical phase of percolation is well behaved , 1990, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[29]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[30]  Y. Peres,et al.  Random walks on a tree and capacity in the interval , 1992 .

[31]  Russell Lyons,et al.  Random walk in a random environment and rst-passage percolation on trees , 1992 .

[32]  Russell Lyons,et al.  Random Walks, Capacity and Percolation on Trees , 1992 .

[33]  Yu Zhang,et al.  Random walk on the infinite cluster of the percolation model , 1993 .

[34]  Leonard J. Schulman,et al.  Signal propagation, with application to a lower bound on the depth of noisy formulas , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[35]  Yuval Peres,et al.  Markov chains indexed by trees , 1994 .

[36]  Gordon Slade,et al.  Mean-Field Behaviour and the Lace Expansion , 1994 .

[37]  Domination Between Trees and Application to an Explosion Problem , 2004, math/0404044.

[38]  Paolo M. Soardi,et al.  Potential Theory on Infinite Networks , 1994 .

[39]  Russell Lyons,et al.  Conceptual proofs of L log L criteria for mean behavior of branching processes , 1995 .

[40]  Random walks and the growth of groups , 1995 .

[41]  R. Pemantle,et al.  Martin capacity for Markov chains , 1995, math/0404054.

[42]  Critical Random Walk in Random Environment on Trees , 1995 .

[43]  J. Ruiz,et al.  On the purity of the limiting gibbs state for the Ising model on the Bethe lattice , 1995 .

[44]  Tree-Indexed Processes , 1995, math/0404100.

[45]  Mike Steel,et al.  Five surprising properties of parsimoniously colored trees , 1995 .

[46]  Robin Pemantle,et al.  Galton-Watson Trees with the Same Mean Have the Same Polar Sets , 1995, math/0404053.

[47]  L. Schulman,et al.  Information theory and noisy computation , 1995, Proceedings of 1995 IEEE International Symposium on Information Theory.

[48]  Random walks on the lamplighter group , 1996 .

[49]  Arbres et Grandes Déviations , 1996 .

[50]  Yuval Peresy,et al.  Dynamical Percolation , 1996 .

[51]  Ãgoston Pisztora,et al.  Surface order large deviations for Ising, Potts and percolation models , 1996 .

[52]  D. Ioffe.  On the extremality of the disordered state for the Ising model on the Bethe lattice , 1996 .

[53]  Ãgoston Pisztora,et al.  ON THE CHEMICAL DISTANCE FOR SUPERCRITICAL BERNOULLI PERCOLATION , 1996 .

[54]  Y. Peres.  Intersection-equivalence of Brownian paths and certain branching processes , 1996 .

[55]  Itai Benjamini,et al.  Every Graph with a Positive Cheeger Constant Contains a Tree with a Positive Cheeger Constant , 1997 .

[56]  Geoffrey Grimmett,et al.  Percolation and disordered systems , 1997 .

[57]  R. Schonmann,et al.  Domination by product measures , 1997 .

[58]  Robin Pemantle,et al.  Unpredictable paths and percolation , 1998 .

[59]  Energy and Cutsets in Innite Percolation Clusters , 1998 .

[60]  P. Marchal.  The Best Bounds in a Theorem of Russell Lyons , 1998 .

[61]  Elchanan Mossel,et al.  Nearest-neighbor walks with low predictability profile and percolation in 2 + ε dimensions , 1998 .

[62]  Jeffrey E. Steif,et al.  The number of infinite clusters in dynamical percolation , 1998 .

[63]  Unpredictable nearest neighbor processes , 1998 .

[64]  Quansheng Liu.  Sur certaines martingales de Mandelbrot , 1999 .

[65]  Y. Peres,et al.  Broadcasting on trees and the Ising model , 2000 .

[66]  Two . dimensional Brownian Motion and Harmonic Functions , 2022 .