Finite-dimensional distance laws of the Brownian map

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(\mathcal M,d,\mu)$ be a Brownian map with its volume measure normalized to a probability measure. This random measured compact metric space is obtained from scaling limits of uniform finite planar maps equipped with graph distance multiplied by the conventional $n^{-1/4}$ factor and uniform vertex probability measure. Conditional on $(\mathcal M,d,\mu)$, sample $X_1,\ldots,X_p$ independently from $\mu$, and form $\mathsf D_p=(d(X_i,X_j))_{1\le i,j\le p}$.

Statement recorded in the supplied source.

For every integer $p\ge4$, determine the joint probability distribution of all entries of $\mathsf D_p$. Any choice of overall Brownian-map distance normalization should be stated explicitly in a resulting formula. [S2]

Short English statement

Find the joint distance law for four or more independent volume-sampled points of the Brownian map.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000080 (AMR-099-0080). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Random Planar Metrics; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 3.1, extension (2), p. 5. https://arquivo.pt/noFrame/replay/20201231041544id_/http://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf.