Law of large numbers for the drift of the two-dimensional wreath product

We prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $$(2+\epsilon )$$ ( 2 + ϵ ) -moment. This result is in contrast with classical examples of abelian groups, where the displacement after n steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised n-step displacement admits a density whose support is $$[0,\infty )$$ [ 0 , ∞ ) . We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

What’s inside a hairy black hole in massive gravity?

In the context of massive gravity theories, we study holographic flows driven by a relevant scalar operator and interpolating between a UV 3-dimensional CFT and a trans-IR Kasner universe. For a large class of scalar potentials, the Cauchy horizon never forms in presence of a non-trivial scalar hair, although, in absence of it, the black hole solution has an inner horizon due to the finite graviton mass. We show that the instability of the Cauchy horizon triggered by the scalar field is associated to a rapid collapse of the Einstein-Rosen bridge. The corresponding flows run smoothly through the event horizon and at late times end in a spacelike singularity at which the asymptotic geometry takes a general Kasner form dominated by the scalar hair kinetic term. Interestingly, we discover deviations from the simple Kasner universe whenever the potential terms become larger than the kinetic one. Finally, we study the effects of the scalar deformation and the graviton mass on the Kasner singularity exponents and show the relationship between the Kasner exponents and the entanglement and butterfly velocities probing the black hole dynamics. Differently from the holographic superconductor case, we can prove explicitly that Josephson oscillations in the interior of the BH are absent.

Evolution of the Iberian Massif as deduced from its crustal thickness and geometry of a mid-crustal (Conrad) discontinuity

Normal incidence seismic data provide the best images of the crust and lithosphere. When properly designed and continuous, these sections greatly contribute to understanding the geometry of orogens and, along with surface geology, unraveling their evolution. In this paper we present the most complete transect, to date, of the Iberian Massif, the westernmost exposure of the European Variscides. Despite the heterogeneity of the dataset, acquired during the last 30 years, the images resulting from reprocessing the data with a homogeneous workflow allow us to clearly define the crustal thickness and its internal architecture. The Iberian Massif crust, formed by the amalgamation of continental pieces belonging to Gondwana and Laurussia (Avalonian margin), is well structured in the upper and lower crust. A conspicuous mid-crustal discontinuity is clearly defined by the top of the reflective lower crust and by the asymptotic geometry of reflections that merge into it, suggesting that it has often acted as a detachment. The geometry and position of this discontinuity can give us insights into the evolution of the orogen (i.e., of the magnitude of compression and the effects and extent of later-Variscan gravitational collapse). Moreover, the limited thickness of the lower crust below, in central and northwestern Iberia, might have constrained the response of the Iberian microplate to Alpine shortening. Here, this discontinuity, featuring a Vp (P-wave velocity) increase, is observed as an orogen-scale boundary with characteristics compatible with those of the globally debated Conrad discontinuity.

Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

This paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of $$\delta $$ δ -hyperbolicity.

Evolution of the Iberian Massif as deduced from its crustal thickness and geometry of a mid-crustal (Conrad) discontinuity

Normal incidence seismic data provide the best images of the crust and lithosphere. When properly designed and continuous, these sections greatly contribute to understanding the geometry of orogens and, together with surface geology, to unravel their evolution. In this paper we present an almost complete transect of the Iberian Massif, the westernmost exposure of the European Variscides. Despite the heterogeneity of the dataset, acquired during the last 30 years, the images resulting from reprocessing with a homogeneous workflow allow us to clearly define the crustal thickness and its internal architecture. The Iberian Massif crust, formed by the amalgamation of continental pieces belonging to Gondwana and Laurussia (Avalonian margin) is well structured in upper and lower crust. A conspicuous mid-crustal discontinuity is clearly defined by the top of the reflective lower crust and by the asymptotic geometry of reflections that merge into it, suggesting that it has often acted as a detachment. The geometry and position of this discontinuity can give us insights on the evolution of the orogen, i.e. of the effects and extent of the late Variscan gravitational collapse. Also, its position and the limited thickness of the lower crust in central and NW Iberia constraints the response of the Iberian microplate to Alpine shortening. This discontinuity is here observed as an orogeny-scale feature with characteristics compatible with those of the worldwide, Conrad discontinuity.

Energy-minimizing maps from manifolds with nonnegative Ricci curvature

The energy of any [Formula: see text] representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only if the original map is totally geodesic. This conclusion also holds under the weaker assumption that the domain is finitely covered by a diffeomorphic product, and its universal covering space splits isometrically as a product with a flat factor, in a commutative diagram that follows from the Cheeger–Gromoll splitting theorem.

Zariski closures of images of algebraic subsets under the uniformization map on finite-volume quotients of the complex unit ball

We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball $\mathbb{B}^{n}$ using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ and the associated uniformization map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$. Given an algebraic subset $S\,\subset \,\mathbb{B}^{n}$ and writing $Z$ for the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ (which is equipped with a canonical quasi-projective structure), in some precise sense we realize $Z$ as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component $\widetilde{Z}$ of $\unicode[STIX]{x1D70B}^{-1}(Z)$ as $\widetilde{Z}$ exits the boundary $\unicode[STIX]{x2202}\mathbb{B}^{n}$ by exploiting the strict pseudoconvexity of $\mathbb{B}^{n}$, culminating in the proof that $\widetilde{Z}\,\subset \,\mathbb{B}^{n}$ is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of $\text{ Aut}(\unicode[STIX]{x1D6FA})$ for (possibly reducible) bounded symmetric domains $\unicode[STIX]{x1D6FA}$.