Characterize the transition and shock near the half-space diagonal

Characterize the scaling-limit transition for the point-to-point half-space exponential last-passage process with diagonal rate \(\alpha<\tfrac12\), fixed endpoint \((N,\gamma N)\), and start points varying on the \(\mathcal{O}(N^{-1/3})\) scale near the diagonal; in particular, establish whether the process converges to an Airy\(_2\) process away from the diagonal and develops a shock in the \(\mathcal{O}(N^{-1/3})\) transition window.

Background

Remark 1.14 considers the constant-rate regime $\alpha<\tfrac12$ with $\beta=\gamma=\alpha^2/(1-\alpha)^2$. By a symmetry relation, the relevant point-to-point last-passage process can be viewed either by varying the endpoint near $(N,\gamma N)$ or by varying the start point near the diagonal while keeping the endpoint fixed.

Proposition 1.9 shows that, for start points separated from the diagonal on the relevant scale, geodesics avoid the diagonal with probability tending to one. This suggests Airy$_2$ behavior away from the diagonal, whereas the value at the diagonal has a different limiting one-point law, namely the square of the GOE Tracy–Widom distribution. The authors therefore leave unresolved the precise transition between these regimes and the nature of the anticipated shock on the $\mathcal{O}(N^{-1/3})$ scale.