  We suggest a new random model for links based on meander diagrams and graphs.
We then prove that trivial links appear with vanishing probability in this
model, no link $L$ is obtained with probability 1, and there is a lower bound
for the number of non-isotopic knots obtained for a fixed number of crossings.
A random meander diagram is obtained through matching pairs of parentheses, a
well-studied problem in combinatorics. Hence tools from combinatorics can be
used to investigate properties of random links in this model, and, moreover, of
the respective 3-manifolds that are link complements in 3-sphere. We use this
for exploring geometric properties of a link complement. Specifically, we give
expected twist number of a link diagram and use it to bound expected hyperbolic
and simplicial volume of random links. The tools from combinatorics that we use
include Catalan and Narayana numbers, and Zeilberger's algorithm.
