arXiv subjects

Adrian Dawid

Publications and source records attributed to Adrian Dawid.

3 recordsLinked to original sources

arXiv2026

Random Hamiltonians II: A central limit theorem and the Hofer geometry of random walks

This paper investigates the global geometry of the group of Hamiltonian diffeomorphisms $\operatorname{Ham}(M,\omega)$ using random walks. On a large class of symplectic manifolds, we show that the expected Hofer norm of such a random walk grows at least as fast as the square root of the number of steps. Furthermore, we show that if the random walk is restricted to an abelian subgroup, then the growth rate is also bounded from above by the square root of the number of steps. We also provide some numerical evidence suggesting that this upper bound fails away from commutative subgroups. This provides, for subgroups of the group of Hamiltonian diffeomorphisms, a probabilistic version of the flatness observed in commutative finite-dimensional Lie groups. En route, we show that the class of probability measures introduced in the prequel is a class of Borel measures with respect to the $C^\infty$-topology on $\operatorname{Ham}(M,\omega)$, show the measurability of the stable commutator length, and show a central limit theorem for Hofer-Lipschitz quasimorphisms on $\operatorname{Ham}(M,\omega)$.

math.SG↗

arXiv2025

Random Hamiltonians I: Probability measures and random walks on the Hamiltonian diffeomorphism group

We construct a family of probability measures on the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,\omega)$. We show that these measures are Borel measures with respect to the topology induced by the Hofer metric. Further, we show that these measures turn any Hofer-Lipschitz function into a random variable with finite expectation. These measures have (for suitable choices of parameters) several desirable properties, such as full support on $\text{Ham}(M,\omega)$, explicit estimates of the measure of Hofer-balls, and certain controls under the action of the group. We also define a family of probability measures on the space of autonomous Hamiltonian diffeomorphisms. These measures have similar properties and give rise to a random walk on the group $\text{Ham}(M,\omega)$. Finally, we show that under certain limits this construction gives rise to probability measures on the space of Hamiltonian homeomorphisms and on the metric completion of $\text{Ham}(M,\omega)$ with respect to the Hofer metric and the spectral metric.

math.SG↗

arXiv2024

Hofer geometry of $A_3$-configurations

Let $L_0,L_1,L_2 \subset M$ be exact Lagrangian spheres in a Liouville domain $M$ with $2c_1(M)=0$. If $L_0,L_1,L_2$ form an $A_3$-configuration, we show that $\mathscr{L}(L_0)$ and $\mathscr{L}(L_2)$ endowed with the Hofer metric contain quasi-isometric embeddings of $(\mathbb{R}^\infty, \|\cdot\|_\infty)$, i.e. infinite-dimensional quasi-flats. A corollary of the proof presented here establishes that $\text{Ham}_c(M)$ itself contains an infinite-dimensional quasi-flat. We also show that for a Dehn twist $\tau: M \to M$ along $L_1$ the boundary depth of $CF(\tau^{2\ell}(L_0), L')$ is unbounded in $L' \in \mathscr{L}(L_2)$ for any $\ell \in \mathbb{N}_0$.

math.SG↗