Given an edge-weighted tree T with n leaves, sample the leaves uniformly at random without replacement and let Wk, 2≤k≤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 (W2,…,Wn)?" We show that if T is known 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 trees, and certain families with equal weights on all edges such as (k+1)-valent and rooted k-ary trees for k≥2 and caterpillars.

We introduce a Voter Model variant, inspired by social evolution of musical preferences. In our model, agents have preferences over a set of songs and upon meeting update their own preferences incrementally towards those of the other agents they meet. Using the spectral gap of an associated Markov chain, we give a geometry dependent result on the asymptotic consensus time of the model.

In the compulsive gambler process there is a finite set of agents who meet pairwise at random times ($i$ and $j$ meet at times of a rate-$\nu_{ij}$ Poisson process) and, upon meeting, play an instantaneous fair game in which one wins the other's money. We introduce this process and describe some of its basic properties. Some properties are rather obvious (martingale structure; comparison with Kingman coalescent) while others are more subtle (an exchangeable over the money elements property, and a construction reminiscent of the Donnelly-Kurtz look-down construction). Several directions for possible future research are described. One -- where agents meet neighbors in a sparse graph -- is studied here, and another -- a continuous-space extension called the {\em metric coalescent} -- is studied in Lanoue (2014).

The Metric Coalescent (MC) is a measure-valued Markov Process generalizing the classical Kingman Coalescent. We show how the MC arises naturally from a discrete agent based model (Compulsive Gambler) of social dynamics and prove an existence and uniqueness theorem extending the MC to the space of all Borel probability measures on any locally compact Polish space.

To interpret interacting particle system style models as social dynamics, suppose each pair {i, j} of individuals in a finite population meet at random times of arbitrary specified rates nu(ij), and update their states according to some specified rule. The averaging process has real-valued states and the rule: upon meeting, the values X-i (t-), X-j (t-) are replaced by 1/2 (X-i (t-) + X-j (t-)), 1/2 (X-i (t-) + X-j (t-)). It is curious this simple process has not been studied very systematically. We provide an expository account of basic facts and open problems.

We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV. For a given \({n\in N\cup\left\{\omega\right\}}\) , these groups of germs are free abelian groups of rank r, for r ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin’s theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associated groups of germs. Thus, if m ≠ n the groups mV and nV cannot be isomorphic. This answers a question of Brin.