Exceptional times with random-direction bigeodesics

Determine whether there exist exceptional dynamical times at which the discrete dynamical Brownian last passage percolation model admits a non-trivial bigeodesic whose asymptotic direction is random rather than fixed in advance.

Background

The paper proves that, for every fixed deterministic non-axial direction, almost surely no dynamical time admits a non-trivial bigeodesic in that direction. It also proves that the set of all exceptional times admitting a non-trivial bigeodesic has almost surely Hausdorff dimension zero.

These results do not settle whether the exceptional-time set is empty when the direction is allowed to depend on the random environment or on the exceptional time. The authors explicitly identify this existence question as unresolved and refer to the corresponding question in the cited companion work.

References

Whether non-trivial bigeodesics actually occur at exceptional times remains an open question.

— A non-trivial dynamics on the directed landscape  (2609.35124 - Bhatia, 28 Sep 2026) in Section 1.1, “Dynamics and noise sensitivity,” final paragraph

Whether such times actually exist when the direction is allowed to be random still remains an open question; see Question 2 and the discussion following Conjecture 4 therein for the corresponding existence question and its predicted behaviour in dynamical LPP.

— Geodesic traces in dynamical Brownian last passage percolation  (2609.35103 - Bhatia, 28 Sep 2026) in Section 1, Introduction