Bounds on the distance exponent for higher-dimensional Liouville first passage percolation

2025/04/12 by Andres A. Contreras Hip, Zijie Zhuang, Hip, Andres A. Contreras +1
Mathematics · Computer Science · #Stochastic processes and statistical mechanics #Random Matrices and Applications #Bayesian Methods and Mixture Models

paper · pdf · doi:10.48550/arxiv.2504.09141

Abstract

For ξ≥ 0 and d ≥ 3, the higher-dimensional Liouville first passage percolation (LFPP) is a random metric on εℤ^d obtained by reweighting each vertex by e^ξh_ε(x), where h_ε(x) is a continuous mollification of the whole-space log-correlated Gaussian field. This metric generalizes the two-dimensional LFPP, which is related to Liouville quantum gravity. We derive several estimates for the set-to-set distance exponent of this metric, including upper and lower bounds and bounds on its derivative with respect to ξ. In the subcritical region for ξ, we derive estimates for the fractal dimension and show that it is continuous and strictly increasing with respect to ξ. In particular, our result is an important step towards proving a technical assumption made in previous work by the first author and Gwynne. These are also the first bounds on the distance exponent for LFPP in higher dimensions.

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