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Tightness of Liouville first passage percolation for \(\gamma \in (0,2)\)

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Abstract

We study Liouville first passage percolation metrics associated to a Gaussian free field \(h\) mollified by the two-dimensional heat kernel \(p_{t}\) in the bulk, and related star-scale invariant metrics. For \(\gamma \in (0,2)\) and \(\xi = \frac{\gamma }{d_{\gamma }}\), where \(d_{\gamma }\) is the Liouville quantum gravity dimension defined in Ding and Gwynne (Commun. Math. Phys. 374:1877–1934, 2020), we show that renormalized metrics \((\lambda _{t}^{-1} e^{ \xi p_{t} * h} ds)_{t \in (0,1)}\) are tight with respect to the uniform topology. We also show that subsequential limits are bi-Hölder with respect to the Euclidean metric, obtain tail estimates for side-to-side distances, and derive error bounds for the normalizing constants \(\lambda _{t}\).

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Author information

Author notes

  1. Alexander Dunlap

    Present address: Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

Authors and Affiliations

  1. Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Jian Ding

  2. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Julien Dubédat & Hugo Falconet

  3. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Alexander Dunlap

Authors
  1. Jian Ding
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  2. Julien Dubédat
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  3. Alexander Dunlap
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  4. Hugo Falconet
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Additional information

J. Ding was partially supported by NSF grant DMS-1757479 and an Alfred Sloan fellowship.

J. Dubédat was Partially supported by NSF grant DMS-1512853.

A. Dunlap was partially supported by an NSF Graduate Research Fellowship.

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Ding, J., Dubédat, J., Dunlap, A. et al. Tightness of Liouville first passage percolation for \(\gamma \in (0,2)\). Publ.math.IHES 132, 353–403 (2020). https://doi.org/10.1007/s10240-020-00121-1

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  • DOI: https://doi.org/10.1007/s10240-020-00121-1

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