Spectral Dimension of Trees with a Unique Infinite Spine
Using generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically distributed, then both the Hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex, provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n. We apply this new method to both critical non-generic trees and the attachment and grafting model, which is a special case of the vertex splitting model, resulting in a simplified proof for the values of the Hausdorff and spectral dimension for the former and novel results for the latter.
KeywordsRandom trees Randomly grown trees Spectral dimension Random walk Hausdorff dimension Simply generated trees Non-generic trees Galton-Watson process Vertex splitting model
The authors would like to thank Bergfinnur Durhuus, Georgios Giasemidis, Thordur Jonsson and John Wheater for discussions as well as the anonymous referee for useful comments which improved the manuscript. S.Ö.S. acknowledges hospitality at the University of Iceland. S.Z. acknowledges financial support of STFC grant ST/G000492/1. Furthermore, he would like to thank NORDITA for kind hospitality and financial support for a visit during which this work was initiated.
- 2.Ford, D.J.: Probabilities on cladograms: introduction to the alpha model. Preprint, math.PR/0511246 (2005)
- 3.David, F., Hagendorf, C., Wiese, K.J.: A growth model for RNA secondary structures. J. Stat. Mech. P04008 (2008). 0711.3421
- 5.Schaeffer, G.: Conjugaison d’arbres et cartes combinatoires aléatoires. PhD thesis, Université de Bordeaux I (1998) Google Scholar
- 7.Di Francesco, P.: 2D quantum gravity, matrix models and graph combinatorics. math-ph/0406013
- 10.Stefánsson, S.Ö.: Markov branching in the vertex splitting model. J. Stat. Mech. P04018 (2012). 1103.3445
- 11.David, F., Dukes, M., Jonsson, T., Stefánsson, S.Ö.: Random tree growth by vertex splitting. J. Stat. Mech. P04009 (2009). 0811.3183
- 16.Fujii, I., Kumagai, T.: Heat kernel estimates on the incipient infinite cluster for critical branching processes. In: Proceedings of German-Japanese Symposium in Kyoto 2006, vol. B6, pp. 85–95 (2008) Google Scholar
- 22.Coulhon, T.: Random walks and geometry on infinite graphs. In: Ambrosio, L., Cassano, F.S. (eds.) Lecture Notes on Analysis on Metric Spaces (2000) Google Scholar
- 32.Janson, S.: 2011, Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Preprint 1112.0510
- 37.Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York (1974) Google Scholar
- 39.Giasemidis, G., Wheater, J.F., Zohren, S.: Dynamical dimensional reduction in toy models of 4D causal quantum gravity. Preprint 1202.2710 (2012)
- 40.Giasemidis, G., Wheater, J.F., Zohren, S.: Multigraph models for causal quantum gravity and scale dependent spectral dimension. Preprint 1202.6322 (2012)