TBA

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.

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.

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.)