Talagrand’s question on periodic two-dimensional random matching

Let $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ be the flat unit torus, let $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ be independent samples of $n$ independent points uniformly distributed on $\mathbb{T}^2$ , and let $d_{\mathbb{T}^2}$ denote the torus distance. For every finite $q\ge 1$ , determine the exact leading asymptotics, including the leading constant, of

C_{n,q}=\mathbb{E}\left[\min_{\sigma\in S_n}\frac{1}{n}\sum_{i=1}^n d_{\mathbb{T}^2}(X_i,Y_{\sigma(i)})^q\right]

as $n\to\infty$ . In particular, determine the exact asymptotics in the endpoint case $q=1$ for periodic matching on the flat two-dimensional torus.

References

Primary source

arXiv

Additional references

Exact Asymptotics for the 2D Euclidean Random Matching Problem — arXiv

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

An unrefereed preprint claims to determine the exact leading constant for periodic two-dimensional random matching at every finite power cost, including the unresolved endpoint.

Talagrand’s question concerns the leading asymptotics of random matching on the flat two-dimensional torus for finite power costs. Earlier work established important quadratic-case asymptotics, but not the full finite-power statement.

Known results

For quadratic cost, self-matching on the flat torus is known up to an additive double-logarithmic error, with leading term $\frac{\log n}{4\pi n}$ .
For quadratic bipartite matching, Ambrosio, Goldman, and Trevisan (2021) proved the logarithmic asymptotic for bounded two-dimensional domains with positive Hölder-continuous densities.
Earlier quadratic results include the compact-manifold bipartite constant $\frac{1}{2\pi}$ and unequal-cardinality extensions.

September 8, 2026 claimed resolution

A preprint titled Exact Asymptotics for the 2D Euclidean Random Matching Problem claims the leading constant for the flat-torus problem for every finite power, including $q=1$ , which would settle Talagrand’s question and unify the finite-power cases. The claim is unrefereed and has not been independently verified in the retrieved sources.

Current status (as of September 2026): the all-finite-power claim, including $q=1$ , is reported but unverified; earlier quadratic cases remain established.

Sources

No solutions have been posted yet.