Geodesic switches and exceptional times in dynamical Brownian last passage percolation (2510.27589v1)

Abstract: We consider Brownian last passage percolation evolving dynamically via a discrete resampling procedure. Using $\Gamma_{(0,0)}^{(n,n),r}$ 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 $\Gamma_{(0,0)}^{(n,n),r}$ 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 $\theta$, we show that the set $\mathscr{T}^\theta\subseteq \mathscr{T}$ of times at which a non-trivial bi-infinite geodesic directed along $\theta$ exists a.s. has Hausdorff dimension equal to $0$.

• The paper establishes rigorous upper bounds on geodesic switching, showing that the expected switches scale as O(n^(5/3 + o(1)) (t-s)).
• The study identifies exceptional times with a Hausdorff dimension bounded by 1/2, highlighting their sparse occurrence in the stochastic model.
• Utilizing advanced Brownian resampling and geometric techniques, the research provides a framework to analyze temporal dynamics in random media.

Geodesic Switches and Exceptional Times in Dynamical Brownian Last Passage Percolation

Abstract

The paper investigates the dynamic behavior of Brownian last passage percolation (BLPP) through a novel lens by considering geodesic switches and the occurrence of exceptional times when bi-infinite geodesics exist. The authors provide rigorous upper bounds on the number of such switches and establish dimensional constraints on the set of exceptional times.

Introduction

Dynamical LPP models, incorporating stochastic resampling of weights, provide a rich framework for exploring temporal evolution in random media. This paper extends this exploration to Brownian environments, where the discrete and continuous evolution of geodesics offers insights into the interplay between randomness and structure.

Geodesic Switches

Concept and Boundaries

In the dynamical BLPP model, a geodesic switch is defined as a transition where a geodesic changes due to weight resampling at a specific time. A key result of this paper is the upper bound on the expected number of geodesic switches over a given time interval, denoted as $[s, t]$. The authors establish that this expectation scales as $O(n^{5/3 + o(1)}(t-s))$, suggesting a nuanced balance between spatial and temporal fluctuations in the passage process.

Proof Strategies

The proof utilizes a combination of Brownian resampling estimates and advanced geometric considerations of the path structure within BLPP. By understanding path overlaps and the role of noise sensitivity, the authors rigorously derive these bounds, offering deeper insights into the dynamic nature of percolation paths.

Exceptional Times and Dimensions

Definition and Upper Bounds

Exceptional times are critical instants where non-trivial bi-infinite geodesics can exist. The authors construct frameworks to measure the Hausdorff dimension of these times, revealing that they are constrained to a dimension of at most 1/2. This implies a sparse occurrence of such exceptional times in the stochastic LPP environment.

Methodological Approach

The approach combines probabilistic arguments with geometric analysis to argue the rarity of these exceptional occurrences. The paper leverages insights from the broader KPZ universality class, proposing that this dimension constraint arises from fundamental stochastic geometrical properties.

Implications and Future Directions

The results have significant implications for the understanding of temporal processes in random media. By subsuming noise-induced changes into quantifiable metrics, the research opens avenues for future exploration of temporal dynamics in other stochastic frameworks. The paper anticipates extensions to different types of noise models and potential applications in modeling real-world phenomena such as material resistance and network communications.

Conclusion

This paper provides a pioneering exploration of geodesic switching and exceptional time dimensions within the BLPP framework. By characterizing these phenomena through rigorous bounds, it lays the groundwork for continued research into stochastic temporal processes and their applications in mathematical physics and beyond.