Stationary cocycles and Busemann functions for the corner growth model

Georgiou, Nicos; Rassoul-Agha, Firas; Seppäläinen, Timo

doi:10.1007/s00440-016-0729-x

Cited by 55 publications

(78 citation statements)

References 61 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…has P-probability at most C exp{−c(r/k) 3/2 }, provided that the hypotheses t −2/3 1,2 |u i − x| c(nt 1,2 ) 1/9 and r k are satisfied. Thus, (19) implies Proposition 6.3 (1).…”

Section: Bouquet Constructionmentioning

confidence: 69%

Exponents governing the rarity of disjoint polymers in Brownian last passage percolation

Hammond

2019

Proc. London Math. Soc.

View full text Add to dashboard Cite

In last passage percolation models lying in the KPZ universality class, long maximizing paths have a typical deviation from the linear interpolation of their endpoints governed by the two‐thirds power of the interpolating distance. This two‐thirds power dictates a choice of scaled coordinates, in which these maximizers, now called polymers, cross unit distances with unit‐order fluctuations. In this article, we consider Brownian last passage percolation in these scaled coordinates, and prove that the probability of the presence of k disjoint polymers crossing a unit‐order region while beginning and ending within a short distance ε of each other is bounded above by ε(k2−1)/2+o(1). This result, which we conjecture to be sharp, yields understanding of the uniform nature of the coalescence structure of polymers, and plays a foundational role in Hammond (Forum Math. Pi 7 (2019) e2, 69) in proving comparison on unit‐order scales to Brownian motion for polymer weight profiles from general initial data. The present paper also contains an on‐scale articulation of the two‐thirds power law for polymer geometry: polymers fluctuate by ε2/3 on short scales ε.

show abstract

“…has P-probability at most C exp{−c(r/k) 3/2 }, provided that the hypotheses t −2/3 1,2 |u i − x| c(nt 1,2 ) 1/9 and r k are satisfied. Thus, (19) implies Proposition 6.3 (1).…”

Section: Bouquet Constructionmentioning

confidence: 69%

Exponents governing the rarity of disjoint polymers in Brownian last passage percolation

Hammond

2019

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…In 2014, a theory of infinite geodesics for last-passage percolation (LPP) with general weights was given by Georgiou-Rassoul-Agha-Seppäläinen [14,15] that parallels the one developed by Damron-Hanson in [11] for FPP. Using directedness of paths, they were able to go further than in [11], proving uniqueness of infinite geodesics and absence of bigeodesics in deterministic directions.…”

Section: Geodesics In Related Modelsmentioning

confidence: 99%

Bigeodesics in First-Passage Percolation

Damron

Hanson

2016

Commun. Math. Phys.

View full text Add to dashboard Cite

In first-passage percolation, we place i.i.d. continuous weights at the edges of Z 2 and consider the weighted graph metric. A distance-minimizing path between points x and y is called a geodesic, and a bigeodesic is a doubly-infinite path whose segments are geodesics. It is a famous conjecture that almost surely, there are no bigeodesics. In the '90s, Licea-Newman showed that, under a curvature assumption on the "asymptotic shape," all infinite geodesics have an asymptotic direction, and there is a full measure set D ⊂ [0, 2π) such that for any θ ∈ D, there are no bigeodesics with one end directed in direction θ. In this paper, we show that there are no bigeodesics with one end directed in any deterministic direction, assuming the shape boundary is differentiable. This rules out existence of ground state pairs for the related disordered ferromagnet whose interface has a deterministic direction. Furthermore, it resolves the BenjaminiKalai-Schramm "midpoint problem" [5, p. 1976] under the extra assumption that the limit shape boundary is differentiable.

show abstract

“…See [12,14,21,22,30] for more on geodesics in first-passage percolation. See [16,17] for more on the connection between the minimizers of similar variational formulas, Busemann functions, and geodesics.…”

Section: Busemann Functions and Geodesicsmentioning

confidence: 99%

Variational Formula for the Time Constant of First‐Passage Percolation

Krishnan

2016

Comm Pure Appl Math

View full text Add to dashboard Cite

We consider first‐passage percolation with positive, stationary‐ergodic weights on the square lattice ℤd. Let T(x) be the first‐passage time from the origin to a point x in ℤd. The convergence of the scaled first‐passage time T([nx])/n to the time constant as n → ∞ can be viewed as a problem of homogenization for a discrete Hamilton‐Jacobi‐Bellman (HJB) equation. We derive an exact variational formula for the time constant and construct an explicit iteration that produces a minimizer of the variational formula (under a symmetry assumption). We explicitly identify when the iteration produces correctors.© 2016 Wiley Periodicals, Inc.

show abstract

Stationary cocycles and Busemann functions for the corner growth model

Cited by 55 publications

References 61 publications

Exponents governing the rarity of disjoint polymers in Brownian last passage percolation

Exponents governing the rarity of disjoint polymers in Brownian last passage percolation

Bigeodesics in First-Passage Percolation

Variational Formula for the Time Constant of First‐Passage Percolation

Contact Info

Product

Resources

About