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Busemann functions and infinite geodesics in two-dimensional first-passage percolation. (English) Zbl 1293.82014
The first-passage percolation problem is studied on $$\mathbb{Z}^2$$, addressing questions related to the existence and coalescence of infinite geodesics. The problem in turn is related to the structure of the set of ground states of the Edwards-Anderson model, see [L.-P. Arguin and M. Damron, Ann. Inst. Henri Poincaré, Probab. Stat. 50, No. 1, 28–62 (2014; Zbl 1292.82044)]. The departure point are papers by C. Newman and J. Wehr containing a rigorous study of infinite geodesics (and that of ground states of disordered ferromagnetic spin models), under a number of strong assumptions. The authors set a framework for working with distributional limits of Buseman functions and use them to prove some of Newman’s results under minimal assumptions. A purely directional condition is introduced which replaces Newman’s global curvature condition and implies the existence of directional geodesics. Interestingly, the existence of infinite geodesics which are directed in sectors can be proved without the latter condition. An analysis of distributional limits of geodesic graphs allows to prove the nonexistence of infinite backward paths. This is related to the almost-sure coalescence and a conjectured nonexistence of certain types of bigeodesics.

##### MSC:
 82B43 Percolation 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses) 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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