Geodesic switches and exceptional times in dynamical Brownian last passage percolation

TL;DR

The paper introduces and analyzes a dynamical BLPP model in which vertex weights are resampled in time, and it studies how the dynamic geodesic structure evolves. By quantifying coarse-grained geodesic switches and relating them to hitsets, the authors obtain a sharp $n^{5/3+o(1)}(t-s)$ bound on the expected number of switches, revealing a KPZ-typical temporal complexity. Leveraging this switch control, they prove fractal bounds: the exceptional times set $oldsymbol{igmathscr T}$ for bi-geodesics has Hausdorff dimension at most $1/2$, and for any fixed direction $ heta$, the corresponding set $oldsymbol{igmathscr T}^ heta$ has dimension $0$, supporting the broader conjecture that such exceptional times are sparse. The analysis combines BLPP line ensembles with Brownian Gibbs technology, a Brownian comparison framework, Poisson-sprinkling techniques to access on-scale geodesics, and a twin-peaks routing framework to control near-geodesic excursions across scales, contributing to the understanding of noise-sensitivity and universality in the KPZ class.

Abstract

We consider Brownian last passage percolation evolving dynamically via a discrete resampling procedure. Using $Γ_{(0,0)}^{(n,n),r}$ to denote a geodesic from $(0,0)$ to $(n,n)$ at time $r$, we prove that the expected total number of coarse-grained changes (or "switches") accumulated by $Γ_{(0,0)}^{(n,n),r}$ away from its endpoints during a time interval $[s,t]$ is at most $n^{5/3+o(1)}(t-s)$; we expect the exponent $5/3$ to be tight. Using the above estimate, we establish that the set $\mathscr{T}$ of exceptional times at which a non-trivial bi-infinite geodesic exists a.s. has Hausdorff dimension at most $1/2$. Further, for any fixed direction $θ$, we show that the set $\mathscr{T}^θ\subseteq \mathscr{T}$ of times at which a non-trivial bi-infinite geodesic directed along $θ$ exists a.s. has Hausdorff dimension equal to $0$.

Figures (12)

• Figure 1: Given the Brownian motions $W_n^t$ associated to the dynamical BLPP, we can consider the processes $X_{i,n}^t(x)= W_n^t(x+i)-W_n(x)$ for $x\in [0,1]$. The BLPP dynamics is defined by associated an independent exponential clock to each such $(i,n)\in \mathbb{Z}^2$ and then freshly resampling $X_{i,n}^t$ at any $t$ at which the clock corresponding to $(i,n)$ rings.
• Figure 2: Here, in the setting of dynamical BLPP, we look at the geodesic between $\mathbf{0}=(0,0)$ and $\mathbf{6}=(6,6)$ as time is varied. In the given figure, it happens to be the case that $\mathcal{T}_{\mathbf{0}}^{\mathbf{6},[0,1]}=\{s_1,s_2,s_3\}$ for some $s_1<s_2<s_3\in (0,1)$. Here, $\mathrm{Coarse}(\Gamma_{\mathbf{0}}^{\mathbf{6},s_1^-})=\mathrm{Coarse}(\Gamma_{\mathbf{0}}^{\mathbf{6},0})\subseteq \mathcal{M}_{\mathbf{0}}^{\mathbf{6}}$ is equal to the set $\{(-1,0),(0,0), (0,1), (1,1), (1,2), (2,2), (2,3), (3,3), (3,4), (3,5), (4,5), (5,5), (5,6), (6,6)\}$, and thus $|\mathrm{HitSet}_{\mathbf{0}}^{\mathbf{6},\{0\}}(\mathbb{R}^2)|=14$. At time $s_1$, the geodesic changes from $\Gamma_{\mathbf{0}}^{\mathbf{6},s_1^-}$ to $\Gamma_{\mathbf{0}}^{\mathbf{6},s_1}$ and the set $\Gamma_{\mathbf{0}}^{\mathbf{6},s_1}\setminus \Gamma_{\mathbf{0}}^{\mathbf{6},s_1^-}$ is shown in cyan. Note that $\mathrm{Coarse}(\Gamma_{\mathbf{0}}^{\mathbf{6},s_1})\setminus \mathrm{Coarse}(\Gamma_{\mathbf{0}}^{\mathbf{6},s_1^-})= \{(4,3), (4,4)\}$-- this is marked by red squares. Thereafter, at time $s_2$, the geodesic happens to change back to the original blue path $\Gamma_{\mathbf{0}}^{\mathbf{6},s_1^-}$, and the set $\mathrm{Coarse}(\Gamma_{\mathbf{0}}^{\mathbf{6},s_2})\setminus \mathrm{Coarse}(\Gamma_{\mathbf{0}}^{\mathbf{6},s_2^-})$ consists of two elements and is marked by red hollow squares. Finally, at time $s_3$, there is another change in the geodesic and the set $\mathrm{Coarse}(\Gamma_{\mathbf{0}}^{\mathbf{6},s_3})\setminus \mathrm{Coarse}(\Gamma_{\mathbf{0}}^{\mathbf{6},s_3^-})$ is a singleton and is marked by a red cross. Here, $\mathrm{Switch}_{\mathbf{0}}^{\mathbf{6},[0,1]}(\mathbb{R}^2)= 2+2+1=5$ but $|\mathrm{HitSet}_{\mathbf{0}}^{\mathbf{6},[0,1]}(\mathbb{R}^2)|=14+2+1=17$. Note that $17=|\mathrm{HitSet}_{\mathbf{0}}^{\mathbf{6},[0,1]}(\mathbb{R}^2)|\leq |\mathrm{HitSet}_{\mathbf{0}}^{\mathbf{6},\{0\}}(\mathbb{R}^2)|+ \mathrm{Switch}_{\mathbf{0}}^{\mathbf{6},[0,1]}(\mathbb{R}^2)=19$, and the difference $19-17=2$ is explained by the intervals $\{4\}_{[3,4]}, \{5\}_{[3,4]}$ corresponding to the hollow red squares being revisited by the geodesic at time $s_2$.
• Figure 3: Here, the weight of a point $\mathfrak{p}$ at a horizontal distance $k$ from the geodesic $\Gamma_{-\mathbf{n}}^{\mathbf{n}}$ has been resampled. For the geodesic to undergo a change, that is, in order to have $\Gamma_{-\mathbf{n}}^{\mathbf{n}}\neq \Gamma_{-\mathbf{n}}^{\mathbf{n},+}$, it is extremely likely that a "twin-peaks" event has to occur on the anti-diagonal line passing through $\mathfrak{p}$. That is, we must have $|T_{-\mathbf{n}}^{\mathbf{n}}-\mathcal{Z}_{-\mathbf{n}}^{\mathbf{n}}(\mathfrak{p})|=O(1)$; by heuristic calculations for a random walk conditioned to be positive, we expect the above probability to be $\Theta(k^{-3/2})$. Now, if we indeed have $\Gamma_{-\mathbf{n}}^{\mathbf{n}}\neq \Gamma_{-\mathbf{n}}^{\mathbf{n},+}$, then by the KPZ 1:2:3 scaling, we expect $|\Gamma_{-\mathbf{n}}^{\mathbf{n},+}\setminus \Gamma_{-\mathbf{n}}^{\mathbf{n}}|=\Theta(k^{3/2})$. As a result, we should have $\mathbb{E} |\Gamma_{-\mathbf{n}}^{\mathbf{n},+}\setminus \Gamma_{-\mathbf{n}}^{\mathbf{n}}|= \Theta( k^{-3/2}\times k^{3/2})=\Theta(1)$, which we note does not depend on $k$.
• Figure 4: Here, we have $(p_1,q_1)\in \mathrm{Basin}_n^{\mathrm{ELPP}}(\Gamma_{-\mathbf{n}}^{\mathbf{n}})$ since the geodesic $\Gamma_{p_1}^{q_1}$ only disagrees with $\Gamma_{-\mathbf{n}}^{\mathbf{n}}$ outside the region between the dashed lines. This should be contrasted with the pair $(p_2,q_2)$ which does not belong to the set $\mathrm{Basin}_n^{\mathrm{ELPP}}(\Gamma_{-\mathbf{n}}^{\mathbf{n}})$. With some work, Proposition \ref{['prop:45']} can be used to show that $\mathrm{Basin}_n^{\mathrm{ELPP}}(\Gamma_{-\mathbf{n}}^{\mathbf{n}})\geq \varepsilon^2 n^{10/3}$ with stretched exponentially high probability in $\varepsilon^{-1}$.
• Figure 5: Here, the points $(p_1,q_1), (p_2,q_2), (p_3,q_3)$ all belong to the Poisson process $\mathcal{Q}_n^{\mathrm{ELPP}}$, and we depict geodesics for $T^0$ between the points $u\in \ell_{-n}$ and $u+2\mathbf{n}\in \ell_{n}$. Here, the high probability event from \ref{['eq:568']} occurs and thus the portion of all such geodesics $\Gamma_u^{u+2\mathbf{n},0}$ in between the dotted lines is entirely covered by the union $\Gamma_{p_1}^{q_1,0}\cup \Gamma_{p_2}^{q_2,0}\cup\Gamma_{p_3}^{q_3,0}$.

Theorems & Definitions (143)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 5: LR10,DV21+
• Lemma 6
• Proposition 7
• Proposition 8: GH20
• Proposition 9: BBBK25
• Lemma 10