Geodesics and recurrence of random walks in disordered systems (Q1860587)

There are two parts, both concerned with random environment on the edges of \(\mathbb Z^d\). Criteria, especially the order of moment conditions, are precised and/or illustrated through many counter-examples. The first part is about the number of geodesics in first-passage percolation model in \(\mathbb Z^2\). The result of \textit{J. Wehr} [J. Stat. Phys. 87, No. 1-2, 439-447 (1997; Zbl 0937.82020)] for this number to be either \(0\) or \(+\infty\) is improved. A counter-example is constructed where a unique geodesic filling the space is prescribed. The second part is about transience/recurrence criteria for the associated reversible random walk, referring to results of Peres in [\textit{N. Berger}, Commun. Math. Phys. 226, No. 3, 531-558 (2002; Zbl 0991.82017)]. The transience criterion in \(\mathbb Z^d\), \(d \geq 3\), is improved.