Determine the joint law of directed-LQG Busemann processes

Determine the joint law, across different interface times t, of the stable Lévy-process Busemann functions associated with the conjectural directed LQG metrics at gamma = sqrt(4/3) and theta = pi/2.

Background

The paper proves one-time scaling limits for the discrete Busemann function, obtaining stable Lévy processes for longest and shortest directed paths. The conjectural continuum analog involves a family of Busemann functions indexed by the interface time t. The authors explicitly leave the coupling or joint distribution of this family unresolved.

References

It is an interesting open question to determine the joint law of such processes for different values of $t$; see Problem~\ref{prob-busemann-joint}.

— Directed distances in bipolar-oriented triangulations: exact exponents and scaling limits  (2510.26123 - Borga et al., 30 Oct 2025) in Problem 1, Section 2.3 and discussion immediately preceding it