Nicholas Georgiou (University of Durham) – Balls-in-bins models with interaction

We study a family of balls-in-bins models with a power-law feedback and a local

interaction determined by an underlying graph on the bins. Specifically, for a fixed

graph on $d$ bins, and fixed positive real numbers $\beta_1, \dots, \beta_d$, at each time

step the model allocates a new ball to bin $i$ with probability proportional to

$U_i^{\beta_i}$, where $U_i$ is the total number of balls currently allocated to all bins

in the graph neighbourhood of bin $i$ (including bin $i$ itself).

In this talk, we focus attention on the case of a path graph on 3 bins, studying the

asymptotic behaviour of $X_n$, the vector of the number of balls allocated to each bin

after $n$ steps. Despite its apparent simplicity, the model exhibits a variety of

behaviours, depending on the parameters $\beta_1, \beta_2, \beta_3$. We analyse both the

symmetric ($\beta_i$ equal) and asymmetric ($\beta_i$ distinct) cases, presenting a

complete classification of the growth rates of the coordinates of $X_n$ when $\beta_i > 1$

for all $i$. In each case, we identify when the asymptotic behaviour is

deterministic, and when it is random.

Our analysis employs the method of stochastic approximation for the symmetric case,

semimartingale methods for the asymmetric case, and liberal use of L\'evy's extension of

the Borel--Cantelli lemma. This is joint work with Mikhail Menshikov and Vadim

Shcherbakov.