Monochromatic Exponents for 2D Percolation, and Related Questions

Description

Bernoulli percolation is a classical model for random media, which amounts to populate the vertices of a lattice in an independent fashion. For some parameterp, each vertex contains a particle with probabilityp, and it is left vacant with probability 1 − p. As one lets the parameter vary, a phase transition can be observed, i.e. a drastic change of behavior, at some critical threshold pc: forp<pc, particles form only finite (and tiny) connected components, while forp>pc, a giant (infinite) connected component of particles arises. Percolation theory is widely used in applications, to understand spatial correlations in various systems. We are interested in its critical regime, and more generally in the so-called near-critical regime, which amounts to describing the whole phase transition. More precisely, we want to improve our understanding of a set of exceptional events known as the monochromatic arm events, which play a key role to describe the phase transition. These events require the existence of a given number of disjoint connections, all made of occupied vertices, through annular regions. The case of one single connection has been well understood since the seminal works of Lawler, Schramm, Smirnov and Werner around 2000, yielding the exponent (rac{5}{48}). We recently obtained an exact formula for the backbone exponent, which is the particular case ofj= 2 disjoint connections. Somewhat surprisingly, this exponent has a transcendental value, in contrast to all previously known exponents for 2D percolation. Accurate numerical estimates had been obtained for this exponent, but finding its exact value had remained an open question since at least the 1970s, both in mathematics and also in theoretical physics. We hope to build on our recent work in that case to improve our understanding of other monochromatic exponents, for a larger numberj≥ 3 of connections. We also want to explore potential consequences on the structure of the discrete model, since our proof involves in particular Schramm-Loewner Evolution bubbles. In addition, external perimeters of connected components play a central role. Finally, inequalities were proved between these exponents and the so-called polychromatic exponents, which are now very well understood. Hence, we want to consider possible refinements and extensions of these relations.