2024/11/12 by Shirshendu Ganguly, Victor Ginsburg, Ganguly, Shirshendu +3
Mathematics · Neuroscience · Physics and Astronomy · #Biology #Complex Network Analysis Techniques #Computer science #Neuroscience #Opinion Dynamics and Social Influence #Percolation (cognitive psychology) #Physics #Statistical physics #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.2411.08018
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2024/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
Last passage percolation (LPP) is a model of a directed metric and a zero-temperature polymer where the main observable is a directed path evolving in a random environment accruing as energy the sum of the random weights along itself. When the environment has light tails and a fast decay of correlation, the fluctuations of LPP are predicted to be explained by the Kardar-Parisi-Zhang (KPZ) universality theory. However, the KPZ theory is not expected to apply for many natural environments, particularly "critical" ones exhibiting a hierarchical structure often leading to logarithmic correlations. In this article, we initiate a novel study of LPP in such hierarchical environments by investigating two particularly interesting examples. The first is an i.i.d. environment but with a power-law distribution with an inverse quadratic tail decay which is conjectured to be the critical point for the validity of the KPZ scaling relation. The second is the Branching Random Walk which is a hierarchical approximation of the two-dimensional Gaussian Free Field. The second example may be viewed as a high-temperature (weak coupling) directed version of Liouville Quantum Gravity, which is a model of random geometry driven by the exponential of a logarithmically correlated field. Due to the underlying fractal structure, LPP in such environments is expected to exhibit logarithmic correction terms with novel critical exponents. While discussions about such critical models appear in the physics literature, precise predictions about exponents seem to be missing. Developing a framework based on multi-scale analysis, we obtain bounds on such exponents and prove almost optimal concentration results in all dimensions for both models. As a byproduct of our analysis we answer a long-standing question of Martin on necessary and sufficient conditions for the linear growth of the LPP energy in i.i.d. environments.