  A long genomic segment inherited by a pair of individuals from a single,
recent common ancestor is said to be identical-by-descent (IBD). Shared IBD
segments have numerous applications in genetics, from demographic inference to
phasing, imputation, pedigree reconstruction, and disease mapping. Here, we
provide a theoretical analysis of IBD sharing under Markovian approximations of
the coalescent with recombination. We describe a general framework for the IBD
process along the chromosome under the Markovian models (SMC/SMC'), as well as
introduce and justify a new model, which we term the renewal approximation,
under which lengths of successive segments are independent. Then, considering
the infinite-chromosome limit of the IBD process, we recover previous results
(for SMC) and derive new results (for SMC') for the mean number of shared
segments longer than a cutoff and the fraction of the chromosome found in such
segments. We then use renewal theory to derive an expression (in Laplace space)
for the distribution of the number of shared segments and demonstrate
implications for demographic inference. We also compute (again, in Laplace
space) the distribution of the fraction of the chromosome in shared segments,
from which we obtain explicit expressions for the first two moments. Finally,
we generalize all results to populations with a variable effective size.
