Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph

Grabner, Peter J.; Woess, Wolfgang

doi:10.1016/s0304-4149(97)00033-1

Cited by 57 publications

(86 citation statements)

References 17 publications

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“…More recently, Gluzman and Sornette [12,13] have reviewed the existence of log-periodic oscillations mirroring underlying scale hierarchies in several areas of physics, and have developed classifications of such things in terms of complex dimensions and the decay of the coefficients in the expansion of the periodic coefficient function. Grabner and Woess [14] (see also [15], §16) have given an elegant discussion of random walks on the Sierpiński lattice where similar phenomena are encountered.…”

Section: The General Casementioning

confidence: 98%

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On the distribution of family names

Reed

Hughes

2003

Physica A: Statistical Mechanics and its Applications

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We present a model for the distribution of family names that explains the power-law decay of the probability distribution for the number of people with a given family name. The model includes a description of the process of generation or importation of new names, and a description of the growth of the number of individuals with a name, and corresponds for a long-enduring culture to a Galton-Watson branching process killed at a random time. The exponent that characterizes the decay of the resulting distribution is determined by the characteristic rates for the creation of new names and for the growth of the population. The power-law decay is modulated by small-amplitude log-periodic oscillations. This is rigorously established for a particular form of the offspring distribution in the branching process, but arguments are presented to show that the phenomenon will occur under wide circumstances.

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Section: The General Casementioning

confidence: 98%

“…and so for large τ we arrive at the exponential density φ(t) = λe −λt (14) for the time that a name has been in existence.…”

Section: The Modelmentioning

confidence: 99%

On the distribution of family names

Reed

Hughes

2003

Physica A: Statistical Mechanics and its Applications

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“…Harris was unable to establish that L(·) is not a constant as soon as q p < 1 (for q p = 1, that is for F (x) = x p , it is straightforward to see that L(·) is constant) and this open issue has drawn the attention in the mathematical community (see for example [5,6], dealing precisely with the problem left open by Harris) and similar -sometimes strictly related -oscillatory behaviors have been pointed out repeatedly (e.g. [27,17,33,22,37] and [8] for further references: the stress is often on the nearly constant and nearly sinusoidal character of these oscillations). To our knowledge, establishing in general that L(·) is non constant is still an open problem for p > 2 (for p = 2 the non triviality of L(·) is established in [8] by exploiting results in [10]).…”

Section: The Galton Watson Processmentioning

confidence: 99%

Log-periodic Critical Amplitudes: A Perturbative Approach

Derrida

Giacomin

2013

J Stat Phys

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Abstract. Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated.Dedicated to Herbert Spohn, our colleague, friend and source of inspiration, on the occasion of his 65 th birthday

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“…Symmetrically self-similar graphs were constructed in [18] as a class of selfsimilar graphs such that the combinatorial substitution mentioned above can be applied. Sections 5 to 7 of the present paper are a generalisation and a further development of the asymptotic analysis in [12].…”

Section: Introductionmentioning

confidence: 97%

Asymptotics of the transition probabilities of the simple random walk on self-similar graphs

Krön

Teufl

2003

Trans. Amer. Math. Soc.

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Abstract. It is shown explicitly how self-similar graphs can be obtained as 'blow-up' constructions of finite cell graphsĈ. This yields a larger family of graphs than the graphs obtained by discretising continuous self-similar fractals.For a class of symmetrically self-similar graphs we study the simple random walk on a cell graphĈ, starting at a vertex v of the boundary ofĈ. It is proved that the expected number of returns to v before hitting another vertex in the boundary coincides with the resistance scaling factor.Using techniques from complex rational iteration and singularity analysis for Green functions, we compute the asymptotic behaviour of the n-step transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpiński graph are generalised to the class of symmetrically self-similar graphs, and at the same time the error term of the asymptotic expression is improved. Finally, we present a criterion for the occurrence of oscillating phenomena of the n-step transition probabilities.

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Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph

Cited by 57 publications

References 17 publications

On the distribution of family names

On the distribution of family names

Log-periodic Critical Amplitudes: A Perturbative Approach

Asymptotics of the transition probabilities of the simple random walk on self-similar graphs

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