Ultrametric and tree potential

Optimization of random high-dimensional functions: Structure and algorithms

Replica symmetry breaking postulates that near optima of spin glass Hamiltonians have an ultrametric structure. Namely, near optima can be associated to leaves of a tree, and the Euclidean distance between them corresponds to the distance along this tree. We survey recent progress towards a rigorous proof of this picture in the context of mixed $p$-spin spin glass models. We focus in particular on the following topics: $(i)$~The structure of critical points of the Hamiltonian; $(ii)$~The realization of the ultrametric tree as near optima of a suitable TAP free energy; $(iii)$~The construction …

The Phase Transition in the Ultrametric Ensemble and Local Stability of Dyson Brownian Motion

We study the ultrametric random matrix ensemble, whose independent entries have variances decaying exponentially in the metric induced by the tree topology on $\mathbb{N}$, and map out the entire localization regime in terms of eigenfunction localization and Poisson statistics. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model. In the simpler case of the Rosenzweig-Porter model, the analysis yields a complete characterization of the transition in the local statistics. The proofs are…

Branching trees I: Concatenation and infinite divisibility

The goal of this work is to decompose random populations with a genealogy in subfamilies of a given degree of kinship and to obtain a notion of infinitely divisible genealogies. We model the genealogical structure of a population by (equivalence classes of) ultrametric measure spaces (um-spaces) as elements of the Polish space U which we recall. In order to then analyze the family structure in this coding we introduce an algebraic structure on um-spaces (a consistent collection of semigroups). This allows us to obtain a path of collections of subfamilies of fixed kinship h (described as ultram…

Mutations on a Random Binary Tree with Measured Boundary

Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure (infinitely-many-allele model) on the skeleton of the tree. The partition of the boundary into distinct alleles is the so-called allelic partition. In this paper, we are interested in the infinite trees generated by supercritical, possibly time-inhomogeneous, binary branching processes, and in their boundary, which is the set of particles `co-existing at infinity'. We prove that…

保守乘性级联的精确拓扑：一种超度量转移算子亏格

Exact topology of conservative multiplicative cascades: An ultrametric transfer-operator genus

本研究精确预测保守随机置换级联的拓扑结构，闭式计算二维级联 excursion 集的欧拉特征，关联其与多重分形谱，为强非高斯场形态学提供基准。

Recovering a tree from the lengths of subtrees spanned by a randomly chosen sequence of leaves

Given an edge-weighted tree $T$ with $n$ leaves, sample the leaves uniformly at random without replacement and let $W_k$, $2 \le k \le n$, be the length of the subtree spanned by the first $k$ leaves. We consider the question, "Can $T$ be identified (up to isomorphism) by the joint probability distribution of the random vector $(W_2, \ldots, W_n)$?" We show that if $T$ is known {\em a priori} to belong to one of various families of edge-weighted trees, then the answer is, "Yes." These families include the edge-weighted trees with edge-weights in general position, the ultrametric edge-weighted …

Probabilistic Models for the (sub)Tree(s) of Life

The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths), culminating in the $β$-family of Aldous' branching models. We next introduce real trees (trees as metric spaces) and show how to study them through their contour, provided they are properly measured and ordered. We then focus on the reduced tree, or coalescent tree, which is the tree spanned by individuals/species alive at the same fixed time. We show how red…

平均双曲性的定量树相似度界

A quantitative tree-likeness bound from average hyperbolicity

本文建立了平均双曲性与树逼近误差之间的显式定量界，证明了有界相似函数的树表示误差不超过平均双曲性的立方根乘以常数，并探讨了最优依赖关系。