La recherche

Numerical experiments of dynamo action designed to understand the generation of Earth's magnetic field produce different regime branches identified within bifurcation diagrams. Notable are distinct branches where the resultant magnetic field is either weak or strong. Weak‐field solutions are identified by the prominent role of viscosity (and/or inertia) on the motion, whereas the magnetic field has a leading‐order effect on the flow in strong‐field solutions. We demonstrate the persistence of the strong‐field branch, preserving the expected force balance of Earth's core, and provide scaling laws governing its onset as parameters move toward values appropriate for the Geodynamo. We introduce a new output parameter, based on dynamically important parts of rotational and magnetic forces, that captures expected values of strong‐field solutions throughout input parameter space. This new measure of the field strength and our bounds on scaling laws can guide future studies in locating strong‐field dynamos in parameter space.

In this thesis, we investigate combinatorial, geometric, and probabilistic properties of wreath products and other group extensions. The work is divided into the following two parts. [1] Non-extendable geodesics in Cayley graphs. We study the property of having unbounded depth in Cayley graphs of wreath products. That is, whether there exist elements at arbitrarily large distance from other elements of larger word length. We prove that for any finite group A and any finitely generated group B, the wreath product A ≀ B admits a standard generating set with unbounded depth. If B is abelian, then the above is true for every standard generating set. This generalizes the case B = ℤ, due to Cleary and Taback. When B = H ∗ K for two finite groups H and K, we characterize which standard generators of A ≀ B have unbounded depth in terms of a geometrical constant related to the Cayley graphs of H and K. [2] Random walks and Poisson boundaries of groups. First, we study random walks on the lampshuffler group FSym(H) ⋊ H, where H is a finitely generated group and FSym(H) is the group of finitary permutations of H. We show that for any step distribution µ with a finite first moment that induces a transient random walk on H, the permutation coordinate of the random walk almost surely stabilizes pointwise to a limit function. Our main result states that for H = ℤ, the Poisson boundary of the random walk (FSym(ℤ)⋊ℤ, μ) is equal to the space of limit functions endowed with the hitting measure. Our result provides new examples of completely described non-trivial Poisson boundaries of elementary amenable groups. Next, in collaboration with Joshua Frisch, we completely describe the Poisson boundary of the wreath product A ≀ B of countable groups A and B, for all probability measures µ with finite entropy and such that the lamp configurations stabilize almost surely along sample paths. If in addition the projection of µ to B is Liouville, we prove that the Poisson boundary of (A ≀ B, µ) coincides with the space of limit lamp configurations, endowed with the corresponding hitting measure. This improves earlier results by Lyons-Peres and, in particular, we answer an open question asked by Kaimanovich and Lyons-Peres for B = ℤᵈ, d ≥ 3, and measures µ with a finite first moment.