Why study multifractal spectra?

Negative compressibility and nonequivalence of two statistical ensembles in the escape transition of a polymer chainWe consider a model of directed polymers on a regular tree with a disorder given by independent, identically distributed weights attached to the vertices. For suitable weight distributions this model undergoes a phase transition with respect to its localization behavior. We show that, for high temperatures, the free energy is supported by a random tree of positive exponential growth rate, which is strictly smaller than that of the full tree. The growth rate of the minimal supporting subtree decreases to zero as the temperature decreases to the critical value. At the critical value and all lower temperatures, a single polymer suffices to support the free energy. Our proofs rely on elegant martingale methods adapted from the theory of branching random walks.

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“…• The function f can be interpreted as the entropy of the system. Its rôle as a multifractal spectrum is highlighted in [Mör08].…”

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confidence: 99%

Minimal supporting subtrees for the free energy of polymers on disordered trees

Mörters

Ortgiese

2008

Journal of Mathematical Physics

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“…The prediction can however be reconciled with our results, if one moves to the appropriate metric, which in our case is again the random metric d. While for fixed intervals the ratio of lengths with respect to d and the Euclidean metric are typically bounded from zero and infinity, the optimal coverings implicit in the Hausdorff dimension above use random intervals for which these diameters are radically different. Indeed, given β the covering intervals I for the corresponding set have metric diameters given by their length to the power Φ ′ (0)/β (see for example [27]). As a result the multifractal spectrum in the intrinsic random metric becomes…”

Section: Physical Heuristicsmentioning

confidence: 99%

The Largest Fragment of a Homogeneous Fragmentation Process

2017

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“…Examples appear in the analysis literature in the work of Rand [36], Brown et al [8], Cawley and Mauldin [9], and Olsen [29] in the early 1990s, and more recently in the probability literature in the work of Arbeiter and Patzschke [2] in the case of random self-similar fractals, Perkins and Taylor [35] in the case of super Brownian motion, Mannersalo et al [25] and Anh et al [1] for products of stochastic processes, Berestycki [6] in the case of fragmentation processes, and Klenke and Mörters [17] in the case of intersection local time of Brownian motion. For some further examples of how multifractal spectra can improve our understanding about certain stochastic processes, see [26].…”

Section: Background On Multifractal Analysismentioning

confidence: 99%

Simultaneous Multifractal Analysis of the Branching and Visibility Measure on a Galton-Watson Tree

Kinnison

Mörters

2010

Advances in Applied Probability

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On the boundary of a Galton-Watson tree one can define the visibility measure by splitting mass equally between the children of each vertex, and the branching measure by splitting unit mass equally between all vertices in the nth generation and then letting n go to infinity. The multifractal structure of each of these measures is well studied. In this paper we address the question of a joint multifractal spectrum, i.e., we ask for the Hausdorff dimension of the boundary points which have an unusual local dimension for both these measures simultaneously. The resulting two-parameter spectrum exhibits a number of surprising new features, among them the emergence of a swallowtail shaped spectrum for the visibility measure in the presence of a nontrivial condition on the branching measure. MSC classification: 60J80, 28A80.Keywords: multifractal spectrum, two-parameter spectrum, mixed spectrum, two-dimensional multifractal analysis, random tree, self-similar fractal, branching process, percolation. MotivationMultifractal analysis provides a way of encapsulating complex information about the fractal nature of an object in a single curve, the multifractal spectrum. In this paper we show how a multifractal analysis can also offer deep insight into the relationship of two fractal objects in the form of a two-parameter multifractal spectrum. In particular we shall see that when one of the analysed measures fails to obey the 'multifractal formalism' such an analysis can lead to the discovery of new phenomena which are deeply rooted in the geometry of these measures.Our test case is the boundary of a Galton-Watson tree with nonzero offspring at every vertex. This set is, on the one hand, a familiar and well-studied object in probability and, on the other hand, it represents the symbolic dynamics of a class of self-similar random fractals and as such it is representative of the behaviour of a wider range of fractal objects with statistical self-similarity. Two natural measures can be defined on the boundary of a Galton-Watson tree, the visibility measure and the branching measure. Both have been studied separately from a multifractal point of view.The visibility measure is easily defined, by starting at the root of the tree with a unit mass and, recursively, at each vertex splitting it equally among the children. If the offspring distribution of the Galton-Watson tree is nondegenerate and satisfies some mild moment conditions, the visibility measure is multifractal and the multifractal formalism, see e.g. Falconer [11], applies, see Kinnison [16] for details. The branching measure represents the uniform measure on the boundary. It can be defined by taking the uniform distribution on the vertices in the nth generation and taking a limit as n → ∞. This measure is not multifractal if the offspring variable has zero probability of taking the value one, and otherwise it is only multifractal in a weaker sense. Only unusually large values of the upper local dimension are possible, and can be represented in a spectrum of hyp...

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Why study multifractal spectra?

Cited by 6 publications

References 20 publications

Minimal supporting subtrees for the free energy of polymers on disordered trees

Minimal supporting subtrees for the free energy of polymers on disordered trees

The Largest Fragment of a Homogeneous Fragmentation Process

Simultaneous Multifractal Analysis of the Branching and Visibility Measure on a Galton-Watson Tree

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