First pair coalescence among three geodesics

Source abstract

In directed last-passage percolation with i.i.d. exponential vertex weights, the coalescence time $\operatorname{Coal}(a,b)$ 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 $$\mathbb{E}\bigl[\min\{\operatorname{Coal}(a,a-e_1),\operatorname{Coal}(a,a-e_2)\}\bigr] 0$, $$\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.