Direct construction of the permuton- and graphon-valued limiting diffusion

Construct the limiting permuton- and graphon-valued Feller diffusion directly on the limiting space, rather than obtaining it through semi-discrete approximations of finite permutation- and graph-valued up-down chains.

Background

The paper constructs scaling limits of duplication–deletion chains on finite permutations and finite graphs, obtaining Feller diffusions on the spaces of permutons and graphons. It also provides semi-discrete approximations whose state space is already the limiting space, but these approximations are not a direct construction of the target diffusion itself.

The authors explicitly identify the direct construction of the limiting process as unresolved. Such a construction would provide an intrinsic description of the diffusion on permutons or graphons, independent of the finite combinatorial approximations.