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A mating-of-trees approach for graph distances in random planar maps

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Peer-reviewed

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Authors

Gwynne, E 
Holden, N 
Sun, X 

Abstract

jats:titleAbstract</jats:title>jats:pWe introduce a general technique for proving estimates for certain random planar maps which belong to the jats:inline-formulajats:alternativesjats:tex-math$$\gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miγ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-Liouville quantum gravity (LQG) universality class for jats:inline-formulajats:alternativesjats:tex-math$$\gamma \in (0,2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo∈</mml:mo> mml:mo(</mml:mo> mml:mn0</mml:mn> mml:mo,</mml:mo> mml:mn2</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; jats:inline-formulajats:alternativesjats:tex-math$$\gamma =\sqrt{8/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo=</mml:mo> mml:msqrt mml:mrow mml:mn8</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>); and planar maps weighted by the number of different spanning trees (jats:inline-formulajats:alternativesjats:tex-math$$\gamma =\sqrt{2}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo=</mml:mo> mml:msqrt mml:mn2</mml:mn> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>), bipolar orientations (jats:inline-formulajats:alternativesjats:tex-math$$\gamma =\sqrt{4/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo=</mml:mo> mml:msqrt mml:mrow mml:mn4</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>), or Schnyder woods (jats:inline-formulajats:alternativesjats:tex-math$$\gamma =1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo=</mml:mo> mml:mn1</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of jats:inline-formulajats:alternativesjats:tex-math$$\gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miγ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map jats:italicM</jats:italic> to a jats:italicmated-CRT map</jats:italic>—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk for jats:italicM</jats:italic> and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in jats:italicM</jats:italic> from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when jats:inline-formulajats:alternativesjats:tex-math$$\gamma =\sqrt{8/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo=</mml:mo> mml:msqrt mml:mrow mml:mn8</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, we instead deduce estimates for the jats:inline-formulajats:alternativesjats:tex-math$$\sqrt{8/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msqrt mml:mrow mml:mn8</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula>-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.</jats:p>

Description

Funder: Norwegian Research Council


Funder: Simons Foundation; doi: http://dx.doi.org/10.13039/100000893

Keywords

60D05 Geometric probability, 60G50 Sums of independent random variables, random walks

Journal Title

Probability Theory and Related Fields

Conference Name

Journal ISSN

0178-8051
1432-2064

Volume Title

177

Publisher

Springer Science and Business Media LLC
Sponsorship
National Science Foundation (DMS1209044)