No bigeodesics in planar first-passage percolation under a minimum-of-four second-moment condition

Theorem 1.1: if the edge-weight law on $[0,\infty)$ has no atoms and the minimum of four independent copies has finite second moment, then almost surely there is no bigeodesic, all paths excluded simultaneously on one probability-one event, with no assumption of differentiability, strict convexity or curvature of the limit shape. The law may be singular continuous, have bounded support or infinite mean. It does not cover laws failing the moment condition, higher dimensions, or directed models.

Theorem 1.1: if the edge-weight law on $[0,\infty)$ has no atoms and the minimum of four independent copies has finite second moment, then almost surely there is no bigeodesic, all paths excluded simultaneously on one probability-one event, with no assumption of differentiability, strict convexity or curvature of the limit shape. The law may be singular continuous, have bounded support or infinite mean. It does not cover laws failing the moment condition, higher dimensions, or directed models.