Component Spectrum of Large Sparse Uniformly Random Magical Squares

In this paper, we shall try to deduce asymptotic behaviour of component spectrum of random $n \times n$ magical squares with line sum $r \in \mathbb{N}$, which can also be identified as $r$-regular bipartite graphs on $2n$ vertices, chosen uniformly from the set of all possible such squares as the dimension $n$ grows large keeping $r$ fixed. We shall focus on limits (after appropriate centering and scaling) of various statistic depending upon the component structure, e.g., number of small components, size of the smallest and largest components, total number of components etc. We shall observe that for the case $r=2$, this analysis falls into the domain of Logarithmic combinatorial structures, although we shall present a new approach for this case relying only on the asymptotic results for random permutations which also helps us to demonstrate an importance sampling algorithm to estimate parameters defined in terms of uniform distribution on magical squares. The case $r \geq 3$ although does not fall in the domain of the Logarithmic combinatorial structures, we shall establish that its component structure is rather trivial, using techniques based on a power series approach.