Random planar maps are simple models for random (discrete) surfaces. Usually two very different point of views are taken to study limits of large maps: the scaling limits, leading to continuum surfaces that describe the macroscopic behaviour, and the local limits, leading to infinite discrete graphs that describe the microscopic behaviour. In this talk we will stand at a mesoscopic scale, and study limits of maps conditioned to have a fixed large number of vertices, edges, and faces at the same time, and the limit will be continuum objects (trees in fact) glued along an infinite discrete graph structure.

In this talk, we introduce a misere tree searching game, where players take turns to guess vertices in a tree with a secret ‘poisoned’ vertex. After each turn, the guessed vertex is removed from the tree and the game continues on the component containing the poisoned vertex, and as soon as a player guesses the poisoned vertex, they lose. We describe the solution when the game is played on a path graph, both between two optimal players and between a player who makes their decisions uniformly at random and an opponent who plays to exploit this. We show that, with two perfect players, the solution involves different guessing strategies depending on the value of n modulo 4. We then show that, with a random and an exploitative player, the probability that the exploitative player wins approaches a constant (approximately 0.599) as n increases, and that the vertices one away from the leaves of the path are always optimal guesses for them. We also solve the game played on a star graph, and briefly discuss the possibility for extending the analysis to more general trees.