Tightness of supercritical Liouville first passage percolation
Jian Ding
University of Pennsylvania, Philadelphia, USAEwain Gwynne
University of Chicago, USA
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Abstract
Liouville first passage percolation (LFPP) with parameter is the family of random distance functions on the plane obtained by integrating along paths, where for is a smooth mollification of the planar Gaussian free field. Previous work by Ding–Dubédat–Dunlap–Falconet and Gwynne–Miller has shown that there is a critical value such that for , LFPP converges under appropriate re-scaling to a random metric on the plane which induces the same topology as the Euclidean metric (the so-called -Liouville quantum gravity metric for ).
We show that for all , the LFPP metrics are tight with respect to the topology on lower semicontinuous functions. For , every possible subsequential limit is a metric on the plane which does not induce the Euclidean topology: rather, there is an uncountable, dense, Lebesgue measure-zero set of points such that for every . We expect that these subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in .
Cite this article
Jian Ding, Ewain Gwynne, Tightness of supercritical Liouville first passage percolation. J. Eur. Math. Soc. 25 (2023), no. 10, pp. 3833–3911
DOI 10.4171/JEMS/1273