Random coloured complexes

In the 1970’s <a>M. Pezzana<span class="abstract"> Sulla struttura topologica delle variety compatte. Atti Sem. Mat. Fis. Univ. Modena, 23:269-277, 1974.</span></a> discovered a way of coding a piecewise-linear (PL) manifold by a properly edge-coloured graph. The idea goes as follows: choose a triangulation \(K\) of this given manifold \(M\). Consider then its first barycentric subdivision \(K_1\). It is properly colourable, namely its vertices can be equipped with colours chosen among \(\dim M +1\) colours such that two adjacent vertices bear different colours. Hence the name coloured triangulations. Dually, the \(1\)-skeleton of \(K_{1}^{*}\) is a properly edge-colourable graph. It turns out that the colouring of the graph is sufficient to reconstruct \(M\) completely. The discovery of M. Pezzana allows to bring combinatorial and graph theoretical methods into PL topology. This correspondence between PL manifolds and coloured graphs has been further developed by <a>M. Ferri, C. Gagliardi and their group<span class="abstract"> A graph-theoretical representation of PL-manifolds - A survey on cristallizations. Aequationes Mathematicae, 31:121-141, 1986.</span></a> (1986). It has also been independently rediscovered by A. Vince (1983), S. Lins and A. Mandel (1985), and, to a certain extent, by R. Gurau (2011) in the context of Group Field Theory.

Inspired by the wonderful results about Brownian (\(2D\)) surfaces, I am working at developing models and tools for the definition of higher dimensionnal random metric spaces. To this aim, I consider various classes of coloured graphs equipped with different probability measures. For example, let \(C_n\) be a family of coloured graphs with \(2n\) vertices, endowed with a probability measure. Having in mind a possible continuum limit, I am, as a first step, interested in the following simple questions:

  • At large \(n\), how many vertices does the random dual complexe have?
  • At large \(n\), what is the typical distance between two uniform vertices?

In a recent article, Ariane Carrance (PhD student of G. Miermont and myself) proved that

  • if \(C_n\) is the set of bipartite \((D+1)\)-regular properly edge-coloured graphs equipped with a uniform measure, then the dual complex has almost surely only \(D+1\) vertices.
  • If \(C_n\) is the set of so-called uniform quartic melonic \((D+1)\)-coloured graphs, then the number of vertices of the dual complex is linear in \(n\) but the distance between two uniform vertices is almost surely two.

We are currently working to find random discrete spaces having a chance to have a non trivial continuum limit.