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The Liouville and the intersection properties are equivalent for planar graphs

   
@article{ECP1913,
	author = {Itai Benjamini and Nicolas Curien and Agelos Georgakopoulos},
	title = {The Liouville and the intersection properties are equivalent for planar graphs},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {17},
	year = {2012},
	keywords = {Random walks ; Planar graphs ; Liouville property ; intersection},
	abstract = {It is shown that if a planar graph admits no non-constant bounded harmonic function then the trajectories of two independent simple random walks  intersect almost surely.},
	pages = {no. 42, 1-5},
	issn = {1083-589X},
	doi = {10.1214/ECP.v17-1913},    
        url = {http://ecp.ejpecp.org/article/view/1913}}