  Consider a simple locally finite hypergraph on a countable vertex set, where
each edge represents one unit of load which should be distributed among the
vertices defining the edge. An allocation of load is called balanced if load
cannot be moved from a vertex to another that is carrying less load. We analyze
the properties of balanced allocations of load. We extend the concept of
balancedness from finite hypergraphs to their local weak limits in the sense of
Benjamini and Schramm (2001) and Aldous and Steele (2004). To do this, we
define a notion of unimodularity for hypergraphs which could be considered an
extension of unimodularity in graphs. We give a variational formula for the
balanced load distribution and, in particular, we characterize it in the
special case of unimodular hypergraph Galton Watson processes. Moreover, we
prove the convergence of the maximum load under some conditions. Our work is an
extension to hypergraphs of Anantharam and Salez (2016), which considered load
balancing in graphs, and is aimed at more comprehensively resolving conjectures
of Hajek (1990).
