Near-existence of bigeodesics in dynamical exponential last passage percolation

TL;DR

This work studies the dynamical exponential last passage percolation (LPP) on the planar lattice and shows that nontrivial bigeodesics occur at times with subpolynomial probability. By coupling a dynamical Russo-Margulis framework with precise control of geodesic overlaps and covariances of passage times, the authors establish an Omega(1/log n) lower bound on the probability that, for length-$n$ geodesics, a geodesic passes through the origin at some time t in [0,1]. They outline a second-moment method centered on a carefully defined overlap statistic X_n and develop sharp covariance estimates to bound the second moment in terms of the first. The results suggest that exceptional times are rare but inherently present, and they conjecture the Hausdorff dimension of the exceptional set is zero, highlighting a delicate balance between geodesic coalescence and dynamical perturbations in KPZ‑class geometry.

Abstract

It is believed that, under very general conditions, bi-infinite geodesics (or bigeodesics) do not exist for planar first and last passage percolation (LPP) models. However, if one endows the model with a natural dynamics, thereby gradually perturbing the geometry, then it is plausible that there could exist a non-trivial set $\mathscr{T}$ of exceptional times at which such bigeodesics exist. For dynamical exponential LPP, we show that $\mathscr{T}$ is "very close" to being non-trivial; namely, we obtain an $Ω( 1/\log n)$ lower bound on the probability that there exists a random time $t\in [0,1]$ at which a non-trivial geodesic of length $n$ passes through the origin at its midpoint; note that if the above probability were $Ω(1)$, then it would imply the non-triviality of $\mathscr{T}$. We conjecture that, even if $\mathscr{T}\neq \emptyset$, it a.s. has Hausdorff dimension exactly zero.

Figures (6)

• Figure 1: Statement of Theorem \ref{['thm:4']}: There is at least a $C(\log n)^{-1}$ probability of there existing a $t\in [0,1]$ for which there is a geodesic $\Gamma_p^{q,t}$ between two points $p,q$ on the linear length segments $\ell_{-n,\varepsilon}^\theta, \ell_{n,\varepsilon}^\theta$ which additionally satisfies $\mathbf{0}\in \Gamma_p^{q,t}$.
• Figure 2: We mark $n^{2/3}$-equispaced points $u_j,v_k$ on the $2\varepsilon n$ length line segments $\ell_{-n,\varepsilon}^\theta,\ell_{n,\varepsilon}^\theta$ respectively. Subsequently, we consider the geodesics $\Gamma_{u_j}^{v_{-j},t}$ for all $t\in [0,1]$ and track whether $\mathbf{0}\in \Gamma_{u_j}^{v_{-j},t}$ occurs. For the case $\theta=0$ and when $\varepsilon n^{1/3}=1$, this figure shows a snapshot at a particular $t\in [0,1]$ at which $\mathbf{0}\in \Gamma_{u_0}^{v_0,t}\cap \Gamma_{u_{-1}}^{v_1,t}$ but $\mathbf{0}\notin \Gamma_{u_1}^{v_{-1},t}$. Thus, this value of $t$, contributes to precisely two of the three integrals appearing in the sum in the definition of $X_n$ (see \ref{['eq:522']}). Further, note that in the figure, the overlap set $\Gamma_{u_0}^{v_0,t}\cap \Gamma_{u_{-1}}^{v_1,t}$ consists of exactly $8$ vertices.
• Figure 3: For static exponential LPP and for large $|j_1-j_2|$, by the results from BBB23, we expect the geodesics $\Gamma_{u_{j_1}}^{v_{-j_1}}, \Gamma_{u_{j_2}}^{v_{-j_2}}$ to overlap for an $O(n/|j_1-j_2|^{3})$ contiguous stretch located at a random location in a larger region of length $\Theta(n/|j_1-j_2|)$ about $\{z: \phi(z)=0\}$. Thus, by a KPZ scaling heuristic, we expect $\mathrm{Cov}(T_{u_{j_1}}^{v_{-j_1}}, T_{u_{j_2}}^{v_{-j_2}})$ to originate entirely from the overlap, and thus be $O((n/|j_1-j_2|^3)^{2/3})= O(n^{2/3}/|j_1-j_2|^2)$. In contrast, for dynamical LPP, the overlap set $|\Gamma_{u_{j_1}}^{v_{-j_1},0}\cap \Gamma_{u_{j_2}}^{v_{-j_2},t}|$ is no longer necessarily contiguous if $t\neq 0$. However, we still expect that $\Gamma_{u_{j_1}}^{v_{-j_1},0}\cap \Gamma_{u_{j_2}}^{v_{-j_2},t}$ is located within a $\Theta(n/|j_1-j_2|)$ stretch around $\{z: \phi(z)=0\}$, that is, between the dashed lines above.
• Figure 4: The separation between consecutive points $p_{i,\Delta}$ and $p_{i+1,\Delta}$ is roughly $\Delta^{-2}n^{2/3}$. The region $L_i$ is to the left of the line connecting $p_{i,\Delta}$ and $v_{k+\Delta/2}$ while the region $R_i$ is to the right of it. Here, the region to the left of the brown line is $L_0$ while the region to the right of the orange line is $R_3$. The blue path here is the geodesic $\Gamma_{u_j}^{v_k}$ while the cyan path is one which attains $T_{u_j}^{v_k}\lvert_{L_0}$. Lemma \ref{['lem:97']} shows stretched exponential tails for $T_{u_j}^{v_k}-T_{u_j}^{v_k}\lvert_{L_0}$ at the scale $\Delta^{-1} n^{1/3}$.
• Figure 5: We consider the point $\tilde{z}_{\alpha}$ where the geodesic $\Gamma_{u_j}^{v_k}$ intersects the line $\ell_\alpha$. While the geodesic $\Gamma_{u_j}^{v_k}$ here does not lie in the region $L_0$, we do have $\Gamma_{u_j}^{v_k}\cap \{\phi(z)\geq -n + \lceil \sqrt{\alpha} \Delta^{-3} n\rceil \}\subseteq L_0$. We now consider a path $\gamma$ (cyan) from $u_j$ to $\tilde{z}_{\alpha}$ lying within $L_0\cup \{u_j\}$ and attaining $T_{u_j}^{\tilde{z}_\alpha}\lvert_{L_0}$ and concatenate $\gamma$ with $\Gamma_{\tilde{z}_{\alpha}}^{v_k}$ to obtain a path from $u_j$ to $v_k$ whose length is within $\alpha \Delta^{-1} n^{1/3}$ of the passage time $T_{u_j}^{v_k}$.

Theorems & Definitions (47)

• Theorem 3
• Conjecture 4
• Proposition 5: BHS22, BBS20
• Proposition 6: Kes80
• Proposition 7: GPS10
• Proposition 8: GH24
• Proposition 9
• Proposition 10
• Lemma 11
• proof