𝐿₁-distortion of Wasserstein metrics: A tale of two dimensions

By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid { 0 , 1 , … , n } 2 \{0,1,\dots , n\}^2 has L 1 L_1 -distortion bounded below by a constant multiple of log ⁡ n \sqrt {\log n} . We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if { G n } n = 1 ∞ \{G_n\}_{n=1}^\infty is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number δ ∈ [ 2 , ∞ ) \delta \in [2,\infty ) , then the 1-Wasserstein metric over G n G_n has L 1 L_1 -distortion bounded below by a constant multiple of ( log ⁡ | G n | ) 1 δ (\log |G_n|)^{\frac {1}{\delta }} . We proceed to compute these dimensions for ⊘ \oslash -powers of certain graphs. In particular, we get that the sequence of diamond graphs { D n } n = 1 ∞ \{\mathsf {D}_n\}_{n=1}^\infty has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over D n \mathsf {D}_n has L 1 L_1 -distortion bounded below by a constant multiple of log ⁡ | D n | \sqrt {\log | \mathsf {D}_n|} . This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of L 1 L_1 -embeddable graphs whose sequence of 1-Wasserstein metrics is not L 1 L_1 -embeddable.