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Hannah Cairns

Publications and source records attributed to Hannah Cairns.

6 recordsLinked to original sources
arXiv2026

Symmetric Cooperative Motion in Higher Dimensions

We prove a distributional convergence result for a multidimensional version of symmetric cooperative motion which was introduced and studied in one dimension in \cite{HRW, SCM1}. Our approach relies on framing the associated recursive distributional equation as a discretization of the porous medium equation. A major challenge is to analyze the behaviour of finite difference schemes which approximate weak solutions of the porous medium equation with unbounded initial data. In overcoming this difficulty, we perform a detailed analysis of the probability mass function of symmetric cooperative motion, in which we introduce several new comparison arguments for the discrete process. Consequently, along the way, we establish a novel multidimensional convergence result for a finite difference scheme approximating the ZKB/Barenblatt solution of the porous medium equation, which is of independent interest.

math.PR↗
arXiv2026

Does freezing impede the growth of random recursive trees?

Uniform attachment with freezing is an extension of the classical model of random recursive trees, in which trees are recursively built by attaching new vertices to old ones. In the model of uniform attachment with freezing, vertices are allowed to freeze, in the sense that new vertices cannot be attached to already frozen ones. We study the impact of removing attachment and/or freezing steps on the height of the trees. We show in particular that removing an attachment step can increase the expected height, and that freezing cannot substantially decrease the height of random recursive trees. Our methods are based on coupling arguments.

math.PR↗
arXiv2024

Note about the existence and essential uniqueness of quadrature domains

This note is intended to explain the proof of two facts about quadrature domains: first, they are essentially unique if they exist; and second, they do exist for a large class of weight functions. The proofs roughly follow Sakai's "Solutions to the obstacle problem as Green potentials," but are presented at an easier level.

math.AP↗
arXiv2021

Some halting problems for abelian sandpiles are undecidable in dimension three

The abelian sandpile model is a simple combinatorial model for critical behaviour which has the "abelian property" that the order in which we make moves does not change the final outcome of the game. This might seem to restrict the model's computational ability, but we show that, given three dimensions to work with, the sandpile is able to simulate a Turing machine. We use that to prove the undecidability of three halting problems.

math.CO↗
arXiv2019

Hipster random walks

We introduce and study a family of random processes on trees we call hipster random walks, special instances of which we heuristically connect to the min-plus binary trees introduced by Robin Pemantle and studied by Auffinger and Cable (2017; arXiv:1709.07849), and to the critical random hierarchical lattice studied by Hambly and Jordan (2004). We prove distributional convergence for the processes by showing that their evolutions can be understood as a discrete analogues of certain convection-diffusion equations, then using a combination of coupling arguments and results from the numerical analysis literature on convergence of numerical approximations of PDEs.

math.PR↗