We study a self-attractive random walk such that each trajectory of length is penalised by a factor proportional to , where is the set of sites visited by the walk. We show that the range of such a walk is close to a solid Euclidean ball of radius approximately , for some explicit constant . This proves a conjecture of Bolthausen  who obtained this result in the case .
We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank . More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective products of resonant and co-resonant states converge weakly to the Liouville measure. We prove this result by establishing an explicit quantum-classical correspondence between eigenspaces of the scalar Laplacian and the resonant states of the first band of Ruelle resonances, which also leads to a new description of Patterson–Sullivan distributions.
Typical weighted random simplices , , in a Poisson–Delaunay tessellation in are considered, where the weight is given by the st power of the volume. As special cases this includes the typical () and the usual volume-weighted () Poisson–Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of satisfies a central limit theorem in high dimensions, that is, as . In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight to depend on the dimension as well. A number of special cases are discussed separately. For fixed also mod- convergence and the large deviations behaviour of the logarithmic volume of are investigated.
A Lagrangian subspace of a weak symplectic vector space is called split Lagrangian if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for to have a closed complement, which can then be moved to become isotropic. The purpose of this note is to develop the theory of compositions and reductions of split canonical relations for symplectic vector spaces. We give conditions on a coisotropic subspace of a weak symplectic space which imply that the induced canonical relation from to is split, and, from these, we find sufficient conditions for split canonical relations to compose well. We prove that the canonical relations arising in the Poisson sigma model from the Lagrangian field theoretical approach are split, giving a description of symplectic groupoids integrating Poisson manifolds in terms of split canonical relations.
We show that the tessellation of a compact, negatively curved surface induced by a long random geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation – for instance, the fraction of triangles – approach those of the limiting Poisson line process.
For each prime , we study the eigenvalues of a 3-regular graph on roughly vertices constructed from the Markoff surface. We show they asymptotically follow the Kesten–McKay law, which also describes the eigenvalues of a random regular graph. The proof is based on the method of moments and takes advantage of a natural group action on the Markoff surface.
We propose a general strategy to derive Lieb–Thirring inequalities for scale-covariant quantum many-body systems. As an application, we obtain a generalization of the Lieb–Thirring inequality to wave functions vanishing on the diagonal set of the configuration space, without any statistical assumption on the particles.
We define a partition and a -rotation (-action defined by rotations) on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel–Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition and a -rotation on whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that is a Markov partition for the -rotation on . We prove in both cases that the toral -rotation is the maximal equicontinuous factor of the minimal subshifts and that the set of fiber cardinalities of the factor map is . The two minimal subshifts are uniquely ergodic and are isomorphic as measure-preserving dynamical systems to the toral -rotations. It provides a construction of these Wang shifts as model sets of 4-to-2 cut and project schemes. A do-it-yourself puzzle is available in the appendix to illustrate the results.
Motivated by the problem of designing inference-friendly Bayesian nonparametric models in probabilistic programming languages, we introduce a general class of partially exchangeable random arrays which generalizes the notion of hierarchical exchangeability introduced in Austin and Panchenko (2014). We say that our partially exchangeable arrays are DAG-exchangeable since their partially exchangeable structure is governed by a collection of Directed Acyclic Graphs. More specifically, such a random array is indexed by for some DAG , and its exchangeability structure is governed by the edge set . We prove a representation theorem for such arrays which generalizes the Aldous-Hoover and Austin–Panchenko representation theorems.