arXiv subjects

Emilio Corso

Publications and source records attributed to Emilio Corso.

8 recordsLinked to original sources

arXiv2026

Large deviations for random walks generated by countable groups: a unified approach

We present a unified treatment of various large deviation principles in the context of random walks induced by actions of countable groups. The fundamental notion undergirding the global perspective is that of weakly almost additive random walks, inspired by earlier works on the metric geometry of group actions as well as by recent developments in the theory of nonadditive thermodynamic formalism. We treat large deviations of the renormalized vector-valued distance function from the origin for groups acting simultaneously, by isometries and in a weakly almost additive fashion, on an arbitrary number of metric spaces. The foremost class of examples is provided by group actions with contracting elements, which encompasses, among others, actions on Gromov-hyperbolic spaces and mapping class group actions on Teichmüller spaces. We further establish large deviation principles for quasimorphisms on countable groups, and for random walks generated by compositions of Euclidean isometries; also, we discuss extensions and streamlining of known large deviation results for singular values of random matrix products. Finally, in all the contemplated cases, we investigate the analytic properties of the rate function, leveraging its expression as the Fenchel-Legendre transform of a pressure function for certain nonadditive sequences of potentials on shift spaces.

math.PR↗

arXiv2026

The moduli space of quasimorphisms on subshifts and its pressure geometry

We introduce a metric geometry on the space of positive, linearly growing quasimorphisms on a mixing subshift of finite type. Two such quasimorphisms are identified when their homogenizations agree up to multiplicative scaling. The resulting moduli space carries a natural symmetrised Thurston-type metric, defined from the extremal ratios of their periodic orbit lengths. We prove that this is a genuine metric and determine several global features of the resulting space: it is unbounded, separable, incomplete, and has geodesics that are canonically generated by thermodynamic pressure functions. We then describe the endpoints of these geodesics and develop a boundary theory for our moduli space. We apply this theory to metric structures on right-angled Artin and Coxeter groups. In particular we construct isometric embeddings of geometric moduli spaces into our quasimorphism moduli spaces. As a consequence, Manhattan curves for metrics on these groups are $C^1$, with equality cases characterised by rough similarity of the underlying metrics. This provides insights into Manhattan curves in settings where neither classical thermodynamic formalism for Hölder potentials nor Patterson-Sullivan theory apply. The same construction recovers the corresponding results for hyperbolic groups and places these examples within a common dynamical-geometric framework.

math.DS↗

arXiv2025

Unboundedness of shapes of unit lattices in totally real cubic fields

The question of the distribution of shapes of unit lattices in number fields, pioneered by Margulis and Gromov, has lately attracted considerable interest, not least because of the lack of available results. Here we prove that the set of shapes of orders of totally real cubic fields is unbounded in the modular surface.

math.NT↗

arXiv2024

Dynamical self-similarity, $L^{q}$-dimensions and Furstenberg slicing in $\mathbb{R}^d$

We extend a theorem of the second author on the $L^q$-dimensions of dynamically driven self-similar measures from the real line to arbitrary dimension. Our approach provides a novel, simpler proof even in the one-dimensional case. As consequences, we show that, under mild separation conditions, the $L^q$-dimensions of homogeneous self-similar measures in $\mathbb{R}^d$ take the expected values, and we derive higher rank slicing theorems in the spirit of Furstenberg's slicing conjecture.

math.CA↗

arXiv2024

The distribution of dilating sets: a journey from Euclidean to hyperbolic geometry

We survey the distributional properties of progressively dilating sets under projection by covering maps, focusing on manifolds of constant sectional curvature. In the Euclidean case, we review previously known results and formulate some generalizations, derived as a direct byproduct of recent developments on the problem of Fourier decay of finite measures. In the hyperbolic setting, we consider a natural upgrade of the problem to unit tangent bundles; confining ourselves to compact hyperbolic surfaces, we discuss an extension of our recent result with Ravotti on expanding circle arcs, establishing a precise asymptotic expansion for averages along expanding translates of homogeneous curves.

math.DS↗

arXiv2022

Large hyperbolic circles

We consider circles of common centre and increasing radius on a compact hyperbolic surface and, more generally, on its unit tangent bundle. We establish a precise asymptotics for their rate of equidistribution. Our result holds for translates of any circle arc by arbitrary elements of $\text{SL}_2(\mathbb{R})$. Our proof relies on a spectral method pioneered by Ratner and subsequently developed by Burger in the study of geodesic and horocycle flows. We further derive statistical limit theorems, with compactly supported limiting distribution, for appropriately rescaled circle averages of sufficient regular observables. Finally, we discuss applications to the classical circle problem in the hyperbolic plane, following the approach of Duke-Rudnick-Sarnak and Eskin-McMullen.

math.DS↗

arXiv2021

Large deviations for irreducible random walks on relatively hyperbolic groups

We show existence of the weak large deviation principle, with a convex rate function, for the renormalized distance from the starting point of irreducible random walks on relatively hyperbolic groups. Under the assumption of finiteness of exponential moments, the full large deviation principle holds, and the rate function governing it can be expressed as the Fenchel-Legendre transform of the limiting logarithmic moment generating function of the sequence of renormalized distances.

math.PR↗

arXiv2021

Large deviations for random walks on free products of finitely generated groups

We prove existence of the large deviation principle, with a proper convex rate function, for the distribution of the renormalized distance from the origin of a random walk on a free product of finitely generated groups. As a consequence, we derive the same principle for nearest-neighbour random walks on regular trees.

math.PR↗