Júlia Komjáthi: Schramm’s locality conjecture for long-range percolation

Schramm’s locality conjecture says the following: consider a sequence G(n) of infinite, vertex transitive graphs that converge locally to a limiting graph G. For instance, a (Z/nZ)3 * Z2 converges to Z5 as n tends to infinity. In Bernoulli nearest neighbor percolation, we keep each edge with probability p, and pc(G) is then critical edge-retention probability for seeing an infinite component in the kept graph. Then, the locality conjecture says that pc( G(n)) converges to pc(G). Recently this conjecture have been solved by a sequence of papers. In this talk we prove part of the locality conjecture for long-range percolation. Long-range percolation is a model where each pair of vertices in G(n) may form an edge, but the probability of the edge decreases with the graph distance between the two vertices. Assuming that G(n) is a graph of polynomial ball-growth with at least quadratic asymptotic growth for all sufficiently large n, we show that the critical parameter of long-range percolation converges to that of the critical parameter on G. The proof contains some basic structure theory of vertex transitive graphs, renormalisation techniques, and making use of the long-edges. Joint work with Yago Moreno Alonso

Anja Sturm: On min-max games on trees and beyond
We study a random game in which two players in turn play a fixed number of moves. For each move, there are two possible choices. To each possible outcome of the game we assign a winner in an i.i.d. fashion with a fixed parameter p. In the case where all different game histories lead to different outcomes, a classical result due to Pearl (1980) says that in the limit when the number of moves is large, there is a sharp threshold in the parameter p that separates the regimes in which either player has with high probability a winning strategy.
We are interested in a modification of this game where the outcome is determined by the exact sequence of moves played by the first player (as in a game tree) and by the number of times the second player has played each of the two possible moves. We show that also in this case, there is a sharp threshold in the parameter p that separates the regimes in which either player has with high probability a winning strategy. Since in the modified game, different game histories can lead to the same outcome, the graph associated with the game is no longer a tree which means independence is lost. As a result, the analysis becomes more complicated and open problems remain. This is joint work with Jan Swart (UTIA Prague) and Natalia Cardona Tobon (Universidad Nacional de Colombia).