Voronoi tessellations in the CRT and continuum random maps of finite excess

Given a large graph G and k agents on this graph, we consider the Voronoi tessellation induced by the graph distance. Each agent gets control of the portion of the graph that is closer to itself than to any other agent. We study the limit law of the vector Vor := (V1/n, V2/n, ..., Vk/n), whose i’th coordinate records the fraction of vertices of G controlled by the i’th agent, as n tends to infinity. We show that if G is a uniform random tree, and the agents are placed uniformly at random, the limit law of Vor is uniform on the (k − 1)dimensional simplex. In particular, when k = 2, the two agents each get a uniform random fraction of the territory. In fact, we prove the result directly on the Brownian continuum random tree (CRT), and we also prove the same result for a “higher genus” analogue of the CRT that we call the continuum random unicellular map, indexed by a genus parameter g ≥ 0. As a key step of independent interest, we study the case when G is a random planar embedded graph with a finite number of faces. The main idea of the proof is to show that Vor has the same distribution as another partition of mass Int := (I1/n, I2/n, ..., Ik/n) where Ij is the contour length separating the i-th agent from the next one in clockwise order around the graph.