Titolo: The Ant in the Labyrinth
Abstract: In 1976, Pierre-Gilles de Gennes posed the problem of the “ant in the labyrinth” to understand diffusion, including heat flow, in disordered media: how does a random walk on a percolation cluster behave over long times and large distances? This question has inspired decades of research in mathematics and physics and was an important motivation for the development of analysis on fractals. The problem is especially subtle at criticality since the large clusters have a fractal structure. Kesten showed in 1986 that in this setting in two dimensions the random walk is subdiffusive, meaning that it spreads out more slowly than an ordinary random walk, but identifying the scaling limit remained a long-standing open problem. In this talk, I will trace the mathematical history of the problem and then present recent work establishing the scaling limit of random walk on two-dimensional critical percolation clusters. This is based on joint work with Valeria Ambrosio, Irina Dankovic, Maarten Markering, and Yizheng Yuan.