First pair coalescence among three geodesics

Abstract

In directed last-passage percolation with i.i.d. exponential vertex weights, the coalescence time $\mathrm{Coal}(a,b)$ of two semi-infinite geodesics started from vertices $a$ and $b$ in a fixed direction has a heavy tail with (KPZ) exponent $2/3$, and hence infinite expectation. We study instead the first coalescence among triples of semi-infinite geodesics in first- and last-passage percolation models (FPP/LPP) on ${\mathbb{Z}}^2$ that arise from stationary compatible families, such as Busemann geodesics. For directed FPP and LPP with arbitrary weights, we prove that $$\mathbb{E}[\min \{\mathrm{Coal}(a,a-e_1),\mathrm{Coal}(a,a-e_2)\}]<\infty .$$ We also consider FPP and LPP models in which antidiagonal motion is allowed, and prove that for every integer $k>0$, \[\mathbb{E}\bigl[\min\{\operatorname{Coal}(a,a-k(e_1-e_2)),\operatorname{Coal}(a,a+k(e_1-e_2))\}\bigr]<\infty.\] These results support the heuristic prediction that adjacent coalescence times are negatively associated.