Related papers: Multifractality in the Tree of Life: A Branching-P…
By unifying three foundational principles of modern biology, we develop a mathematical framework to analyze the growing tree of life. Contrary to the static case, where the analogy between phylogenetic trees and the tree that grows in soil…
We extend a formal framework that previously derived time from the multifractal structure of biological lineages (Hudnall \& D'Souza, 2025). That work showed that time itself is multifractal -- not a universal background dimension, but an…
We study fragmentation of a random recursive tree into a forest by repeated removal of nodes. The initial tree consists of N nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly-selected…
Iterated Graph Systems (IGS) transplant ideas from fractal geometry into graph theory. Building on this framework, we extend Edge IGS from the primitive to the reducible setting. Within this broader context, we formulate rigorous…
We study locally constant skew-product maps over full shifts of finite symbols with arbitrary compact metric spaces as fiber spaces. We introduce a new criterion to determine the density of leaves of the strong unstable (and strong stable)…
Understanding the patterns and processes of diversification of life in the planet is a key challenge of science. The Tree of Life represents such diversification processes through the evolutionary relationships among the different taxa, and…
In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$.…
Tree structures are ubiquitous in data across many domains, and many datasets are naturally modelled by unobserved tree structures. In this paper, first we review the theory of random fragmentation processes [Bertoin, 2006], and a number of…
We consider the hierarchic tree Random Energy Model with continuous branching and calculate the moments of the corresponding partition function. We establish the multifractal properties of those moments. We derive formulas for the normal…
This study develops a comprehensive theoretical and computational framework for Random Nonlinear Iterated Function Systems (RNIFS), a generalization of classical IFS models that incorporates both nonlinearity and stochasticity. We establish…
Iterated function systems (IFS) can be a surprisingly useful tool for studying structure in data. Here we present results stemming from a 2013 computational study by the author using IFS. The results include fractal patterns that reveal…
Random spanning trees are among the most prominent determinantal point processes. We give four examples of random spanning trees on ladder-like graphs whose rungs form stationary renewal processes or regenerative processes of order two,…
We present a new class of multifractal process on R, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the…
Multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a nonsymmetric lattice of obstacles are investigated. Using the equivalence of the model to random walks on a locally…
We consider additive functionals $X_n(\phi)$ with small toll functions on split trees and a generalization of split trees, which we call fractional split trees, where the split vector does not need to sum up to 1. These additive functionals…
By decomposing the random walk path, we construct a multitype branching process with immigration in random environment for corresponding random walk with bounded jumps in random environment. Then we give two applications of the branching…
We investigate multifractal regularity for infinite conformal iterated function systems (cIFS). That is we determine to what extent the multifractal spectrum depends continuously on the cIFS and its thermodynamic potential. For this we…
Among the statistical models employed to approximate nonlinear interactions in biological and psychological processes, one prominent framework is that of cascades. Despite decades of empirical work using multifractal formalisms, a…
In this article, starting from a Gibbs capacity, we build a new random capacity by applying two simple operators, the first one introducing some redundancy and the second one performing a random sampling. Depending on the values of the two…
Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of…