jats:titleAbstract</jats:title>jats:pWe prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most jats:inline-formulajats:alternativesjats:tex-math$$n^{1/4 + o_n(1)}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:min</mml:mi> mml:mrow mml:mn1</mml:mn> mml:mo/</mml:mo> mml:mn4</mml:mn> mml:mo+</mml:mo> mml:msub mml:mio</mml:mi> mml:min</mml:mi> </mml:msub> mml:mrow mml:mo(</mml:mo> mml:mn1</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> in jats:italicn</jats:italic> units of time. Together with the complementary lower bound proven by Gwynne and Miller (2017) this shows that the typical graph distance displacement of the walk after jats:italicn</jats:italic> steps is jats:inline-formulajats:alternativesjats:tex-math$$n^{1/4 + o_n(1)}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:min</mml:mi> mml:mrow mml:mn1</mml:mn> mml:mo/</mml:mo> mml:mn4</mml:mn> mml:mo+</mml:mo> mml:msub mml:mio</mml:mi> mml:min</mml:mi> </mml:msub> mml:mrow mml:mo(</mml:mo> mml:mn1</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>, as conjectured by Benjamini and Curien (Geom Funct Anal 2(2):501–531, 2013. <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" ext-link-type="uri" xlink:href="http://arxiv.org/abs/1202.5454">arXiv:1202.5454</jats:ext-link>). More generally, we show that the simple random walks on a certain family of random planar maps in 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>—including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps—typically travels graph distance jats:inline-formulajats:alternativesjats:tex-math$$n^{1/d_\gamma + o_n(1)}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:min</mml:mi> mml:mrow mml:mn1</mml:mn> mml:mo/</mml:mo> mml:msub mml:mid</mml:mi> mml:miγ</mml:mi> </mml:msub> mml:mo+</mml:mo> mml:msub mml:mio</mml:mi> mml:min</mml:mi> </mml:msub> mml:mrow mml:mo(</mml:mo> mml:mn1</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> in jats:italicn</jats:italic> units of time, where jats:inline-formulajats:alternativesjats:tex-math$$d_\gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:mid</mml:mi> mml:miγ</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on 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> by Ding and Gwynne (Commun Math Phys 374:1877–1934, 2018. <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" ext-link-type="uri" xlink:href="http://arxiv.org/abs/1807.01072">arXiv:1807.01072</jats:ext-link>). Since jats:inline-formulajats:alternativesjats:tex-math$$d_\gamma > 2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msub mml:mid</mml:mi> mml:miγ</mml:mi> </mml:msub> mml:mo></mml:mo> mml:mn2</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, this shows that the simple random walk on each of these maps is subdiffusive. Our proofs are based on an embedding of the random planar maps under consideration into jats:inline-formulajats:alternativesjats:tex-math$${\mathbb {C}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miC</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> wherein graph distance balls can be compared to Euclidean balls modulo subpolynomial errors. This embedding arises from a coupling of the given random planar map with a mated-CRT map together with the relationship of the latter map to SLE-decorated LQG. </jats:p>