The random assignment problem concerns finding the minimum cost assignment or matching in a complete bipartite graph with edge weights being i.i.d. with some distribution, say exponential(1)â€¦ (More)

We prove that the local weak convergence of a sequence of graphs is enough to guarantee the convergence of their normalized matching numbers. The limiting quantity is described by a local recursionâ€¦ (More)

Every graph eigenvalue is in particular a totally real algebraic integer, i.e. a zero of some real-rooted monic polynomial with integer coefficients. Conversely, the fact that every such numberâ€¦ (More)

Reduced `-cohomology in degree 1 (for short "LpR1") is a useful quasiisometry invariant of graphs [of bounded valency] whose definition is relatively simple. On a graph, there is a natural gradientâ€¦ (More)

We establish existence and uniqueness of the solution to the cavity equation for the random assignment problem in pseudo-dimension d > 1, as conjectured by Aldous and Bandyopadhyay (Annals of Appliedâ€¦ (More)

In the so-called sparse regime where the numbers of edges and vertices tend to infinity in a comparable way, the asymptotic behavior of many graph invariants is expected to depend only upon localâ€¦ (More)

Proceedings of the National Academy of Sciencesâ€¦

2015

We introduce a minimal theory of glass formation based on the ideas of molecular crowding and resultant string-like cooperative rearrangement, and address the effects of free interfaces. In the bulkâ€¦ (More)

We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime where the degree is kept fixed while the number of verticesâ€¦ (More)