Probability in the North East day

14 June 2023

Organizers: Nic Freeman.

Venue: University of Sheffield. All talks will be held in LT 9 on Floor H of the Hicks Building. Lunch and coffee will be held in Room 15 on Floor I.

Attendence is free but registration is required for organisational purposes, by emailing Nic Freeman by Wednesday 7th June 2023.

Please indicate any special dietary requirements you may have.


Denis Denisov (University of Manchester)
Maksim Zhukovski (University of Sheffield)
Let $G$ be a graph with several vertices $v_1,..,v_r$ being roots. A $G$-extension of $u_1,..,u_r$ in a graph $H$ is a subgraph $G^*$ of $H$ such that there exists a bijection from $V(G)$ to $V(G^*)$ that maps $v_i$ to $u_i$ and preserves edges of $G$ with at least one non-root vertex. In particular, if $G$ is a single edge with a single root, then the number of $G$-extensions of $u$ in $H$ is the degree of $u$. It is well known that, in dense binomial random graphs, the maximum number of $G$-extensions obeys the law of large numbers. The talk is devoted to new results describing the limit distribution of the maximum number of $G$-extensions. To prove these results, we develop new bounds on the probability that none of a given finite set of events occur. Our inequalities allow us to distinguish between weakly and strongly dependent events in contrast to well-known Janson's and Suen's inequalities as well as Lovasz Local Lemma. These bounds imply a general result on the convergence of maxima of dependent random variables.
Tea and coffee
Linglong Yuan (University of Liverpool)

For large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an $\alpha$-stable law with $\alpha$ in (0,2], there are two different scenarios: either the deviation is realised via a collective behaviour with all summands contributing to the deviation (a Gaussian scenario), or a single summand is atypically large and contributes to the deviation (a one-big-jump scenario). Such results are known when alpha in (0,2) or when $\alpha=2$ and the random variables admit a moment of order $2+\delta$ for some $\delta>0$ with the right-tail probability regularly varying.

This talk will present new results extending the above known phenomenon to the last missing case including where $\alpha=2$ and the right tail is regularly varying with index -2. Threshold for the phase transition between the Gaussian and the one-big-jump regimes will be identified and applications will be discussed.

Nikolaos Fountoulakis(University of Birmingham)
We consider population games on a binomial random graph $G(n,p)$. These games are determined through a 2-player symmetric game with 2 strategies, played between the incident members of the vertex set. Players/vertices update their strategies synchronously: at each round, each player selects the strategy that is the best response to the current profile of strategies its neighbours play. We show that such a system reduces to generalised majority and minority dynamics. We show rapid convergence to unanimity for $p$ in a range that is determined by a certain quantity of the payoff matrix. In the presence of a bias among the pure Nash equilibria of the game, we determine a sharp threshold for $p$, above which the largest connected component reaches unanimity with high probability, and below which this does not happen. We also discuss the case where the game has more than 2 strategies. In particular, we consider payoff matrices with 3 strategies. We show convergence to unanimity after a bounded number of steps under certain conditions of the payoff matrix. (This is joint work with J. Chellig and C. Durbac.)