Transportation cost spaces and stochastic trees

Abstract: We study transportation cost spaces over finite metric spaces, also known as Lipschitz free spaces. Our work is motivated by a core problem posed by S. Dilworth, D. Kutzarova and M. Ostrovskii, namely, find a condition on a metric space $M$ equivalent to the Banach-Mazur distance between the transportation cost space over $M$ and $\ell_1^N$ of the corresponding dimension, which we call the $\ell_1^N$-distortion of $M$. In this regard, some examples have been studied like the $N\times N$ grid by Naor and Schechtman (2007) and the Laakso and diamond graphs by Dilworth, Kutzarova and Ostrovskii (2020), later studied by Baudier, Gartland and Schlumprecht (2023). We present here three main results. Firstly, we give a partial solution to this problem relating to the tree-like structure of the metric space. For that purpose, we develop a new technique that could potentially lead to a complete solution of the problem and utilize it to find an asymptotically tight upper bound of the $\ell_1^N$-distortion of the Laakso graphs, fully solving an open problem raised by Dilworth, Kutzarova and Ostrovskii. Finally, we apply our technique to prove that finite hyperbolic approximations of doubling metric spaces have uniformly bounded $\ell_1^N$-distortion.