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Stationary cocycles and Busemann functions for the corner growth model


We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as boundary conditions for stationary last-passage percolation, solve variational formulas that characterize limit shapes, and yield existence of Busemann functions in directions where the shape has some regularity. In a sequel to this paper the cocycles are used to prove results about semi-infinite geodesics and the competition interface.

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The authors thank Yuri Bakhtin and Michael Damron for useful discussions and two anonymous referees for valuable comments. N. Georgiou was partially supported by a Wylie postdoctoral fellowship at the University of Utah and the Strategic Development Fund (SDF) at the University of Sussex. F. Rassoul-Agha and N. Georgiou were partially supported by National Science Foundation grant DMS-0747758. F. Rassoul-Agha was partially supported by National Science Foundation grant DMS-1407574 and by Simons Foundation grant 306576. T. Seppäläinen was partially supported by National Science Foundation grants DMS-1306777 and DMS-1602486, by Simons Foundation grant 338287, and by the Wisconsin Alumni Research Foundation.

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Correspondence to Timo Seppäläinen.

Appendix: Ergodic theorem for cocycles

Appendix: Ergodic theorem for cocycles

Cocycles satisfy a uniform ergodic theorem. The following is a special case of Theorem 9.3 of [28]. Note that a one-sided bound suffices for a hypothesis. Recall Definition 2.1 for the space \(\mathscr {K}_0\) of centered cocycles.

Theorem 7.8

Assume \(\mathbb {P}\) is ergodic under the transformations \(\{T_{e_i}:i\in \{1,2\}\}\). Let \(F\in \mathscr {K}_0\). Assume there exists a function V such that for \(\mathbb {P}\)-a.e. \(\omega \)

$$\begin{aligned} \varlimsup _{\varepsilon \searrow 0}\;\varlimsup _{n\rightarrow \infty } \;\max _{x: \vert x\vert _1\le n}\;\frac{1}{n} \sum _{0\le k\le \varepsilon n} \vert V(T_{x+ke_i}\omega )\vert =0\qquad \text {for }i\in \{1,2\} \end{aligned}$$

and \(\max _{i\in \{1,2\}} F(\omega ,0,e_i)\le V(\omega )\). Then

$$\begin{aligned} \lim _{n\rightarrow \infty }\;\max _{\begin{array}{c} x=z_1+\cdots +z_n\\ z_{1,n}\in \{e_1, e_2\}^n \end{array}} \;\frac{\vert F(\omega ,0,x)\vert }{n}=0 \qquad \text {for } \mathbb {P}\text {-a.e. } \omega .\end{aligned}$$

If the process \(\{V(T_x\omega ):x\in \mathbb {Z}^2\}\) is i.i.d., then a sufficient condition for (7.23) is \(\mathbb {E}(\vert V\vert ^p)<\infty \) for some \(p>2\) [50, Lemma A.4].

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Georgiou, N., Rassoul-Agha, F. & Seppäläinen, T. Stationary cocycles and Busemann functions for the corner growth model. Probab. Theory Relat. Fields 169, 177–222 (2017).

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  • Busemann function
  • Cocycle
  • Competition interface
  • Directed percolation
  • Geodesic
  • Last-passage percolation
  • Percolation cone
  • Queueing fixed point
  • Variational formula

Mathematics Subject Classification

  • 60K35
  • 65K37