2019/03/22 by Ewain Gwynne, Gwynne, Ewain, Joshua Pfeffer +1 · 1 citation
Mathematics · #Stochastic processes and statistical mechanics #Markov Chains and Monte Carlo Methods #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1903.09561
For \ξ \≥ 0, Liouville first passage percolation (LFPP) is the random\nmetric on \ε mathbb Z2 obtained by weighting each vertex by\n\ε e\ξ h_\ε(z), where h_\ε(z) is the average\nof the whole-plane Gaussian free field h over the circle \∂\nB_\ε(z). Ding and Gwynne (2018) showed that for \γ \∈ (0,2),\nLFPP with parameter \ξ = \γ/d_\γ is related to \γ-Liouville\nquantum gravity (LQG), where d_\γ is the \γ-LQG dimension exponent.\nFor \ξ > 2/d2, LFPP is instead expected to be related to LQG with central\ncharge greater than 1.\n We prove several estimates for LFPP distances for general \ξ\≥ 0. For\n\ξ\≤ 2/d2, this leads to new bounds for d_\γ which improve on the\nbest previously known upper (resp. lower) bounds for d_\γ in the case\nwhen \γ > \√(8/3) (resp. \γ \∈ (0.4981, \√(8/3))). These\nbounds are consistent with the Watabiki (1993) prediction for d_\γ.\nHowever, for \ξ > 1/\√ 3 (or equivalently for LQG with central charge\nlarger than 17) our bounds are inconsistent with the analytic continuation of\nWatabiki's prediction to the \ξ >2/d2 regime. We also obtain an upper bound\nfor the Euclidean dimension of LFPP geodesics.\n