Random Walks on the Hamiltonian Diffeomorphism Group

What is a random Hamiltonian diffeomorphism? In this talk, we propose an answer by constructing a family of probability measures on the group Ham(M) of Hamiltonian diffeomorphisms of a closed symplectic manifold M. We outline the construction, which is based on Gaussian processes. We then use these measures to obtain irreducible random walks on Ham(M). These random walks can be used to probe the large-scale geometry of Ham(M) with respect to the Hofer metric. On complex projective space

𝐶𝑃𝑛, we will see that the expected Hofer distance of the random walk from the identity grows at least as fast as the square root of the number of steps. By contrast, when the random walk is restricted to a commutative subgroup, the expected Hofer distance cannot grow faster than the square root of the number of steps. We conclude by showing that these bounds agree for the random walk restricted to the subgroup of Hamiltonian diffeomorphisms that admit 𝑇𝑛-invariant Hamiltonian functions.