Random walk on random planar maps: spectral dimension, resistance, and displacement
dc.contributor.author  Gwynne, Ewain  en 
dc.contributor.author  Miller, Jason  en 
dc.date.accessioned  20200827T23:30:38Z  
dc.date.available  20200827T23:30:38Z  
dc.date.issued  202105  en 
dc.identifier.issn  00911798  
dc.identifier.uri  https://www.repository.cam.ac.uk/handle/1810/309713  
dc.description.abstract  We study simple random walk on the class of random planar maps which can be encoded by a twodimensional random walk with i.i.d.\ increments or a twodimensional Brownian motion via a ``matingoftrees" type bijection. This class includes the uniform infinite planar triangulation (UIPT), the infinitevolume limits of random planar maps weighted by the number of spanning trees, bipolar orientations, or Schnyder woods they admit, and the $\gamma$matedCRT map for $\gamma \in (0,2)$. For each of these maps, we obtain an upper bound for the Green's function on the diagonal, an upper bound for the effective resistance to the boundary of a metric ball, an upper bound for the return probability of the random walk to its starting point after $n$ steps, and a lower bound for the graphdistance displacement of the random walk, all of which are sharp up to polylogarithmic factors. When combined with work of Lee (2017), our bound for the return probability shows that the spectral dimension of each of these random planar maps is a.s.\ equal to 2, i.e., the (quenched) probability that the simple random walk returns to its starting point after $2n$ steps is $n^{1+o_n(1)}$. Our results also show that the amount of time that it takes a random walk to exit a metric ball is at least its volume (up to a polylogarithmic factor). In the special case of the UIPT, this implies that random walk typically travels at least $n^{1/4  o_n(1)}$ units of graph distance in $n$ units of time. The matching upper bound for the displacement is proven by Gwynne and Hutchcroft (2018). These two works together resolve a conjecture of Benjamini and Curien (2013) in the UIPT case. Our proofs are based on estimates for the matedCRT map (which come from its relationship to SLEdecorated Liouville quantum gravity) and a strong coupling of the matedCRT map with the other random planar map models.  
dc.publisher  Institute of Mathematical Statistics  
dc.rights  All rights reserved  
dc.title  Random walk on random planar maps: spectral dimension, resistance, and displacement  en 
dc.type  Article  
prism.publicationDate  2021  en 
prism.publicationName  Annals of Probability  en 
dc.identifier.doi  10.17863/CAM.56807  
dcterms.dateAccepted  20200825  en 
rioxxterms.versionofrecord  10.1214/20AOP1471  en 
rioxxterms.version  AM  
rioxxterms.licenseref.uri  http://www.rioxx.net/licenses/allrightsreserved  en 
rioxxterms.licenseref.startdate  202105  en 
rioxxterms.type  Journal Article/Review  en 
cam.issuedOnline  20210407  en 
cam.orpheus.counter  44  * 
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