Random growth models: Shape and convergence rate

Abstract

Random growth models are fundamental objects in modern probability theory, have given rise to new mathematics, and have numerous applications, including tumor growth and fluid flow in porous media. In this article, we introduce some of the typical models and the basic analytical questions and properties, like existence of asymptotic shapes, fluctuations of infection times, and relations to particle systems. We then specialize to models built on percolation (first-passage percolation and last-passage percolation) and give a self-contained treatment of the shape theorem, the subadditive ergodic theorem, and conjectured and proven properties of asymptotic shapes. We finish by discussing the rate of convergence to the limit shape, along with definitions of scaling exponents and a sketch of the proof of the KPZ scaling relation.2010 Mathematics Subject Classification. 60K35, 60K37, 82B43. This work was done with support from an NSF CAREER grant.

Fluctuations in first-passage percolation

“…Under our assumptions, the leading order behavior of T n is linear. This follows from the subadditive ergodic theorem, which is presented in [8]. Here we just record the result for future reference.…”

“…Another important prediction from physics, the universal scaling relation, links the order of the fluctuations χ to another exponent ξ, the transversal fluctuation exponent. Discussing this here would take us too far afield, so we refer to [8] for a statement of the scaling relation and discussion of its derivation under suitable assumptions.…”

Busemann functions, geodesics, and the competition interface for directed last-passage percolation

“…However, it turns out that one can prove a stochastic version of Fekete's subadditive lemma and apply this subadditive ergodic theorem to´G 0,nξ to obtain the limit (5.1). A version of this theorem is given in [18] as Theorem 3.2. A bit more work is needed for the general case ξ P r0, 8q 2 and even more work is needed to get more uniform control and prove (5.3).…”

“…A bit more work is needed for the general case ξ P r0, 8q 2 and even more work is needed to get more uniform control and prove (5.3). The details are omitted as they are similar to the ones in the proof of the standard first-passage percolation shape theorem, given in [18]. See also Proposition 2.1(i) of [45].…”

“…This raises the next layer of questions regarding the size and statistics of the fluctuations of the random objects from their deterministic limits. See Section 1 of [18] for more details and precise formulations.…”