Natural Probability Measure Research Articles

Generalized unistochastic matrices

We study a class of bistochastic matrices generalizing unistochastic matrices. Given a complex bipartite unitary operator, we construct a bistochastic matrix having as entries the normalized squared Frobenius norm of the blocks. We show that the closure of the set of generalized unistochastic matrices is the whole Birkhoff polytope. We characterize the points on the edges of the Birkhoff polytope that belong to a given level of our family of sets, proving that the different (non-convex) levels have a rich inclusion structure. We also study the corresponding generalization of orthostochastic matrices. Finally, we introduce and study the natural probability measures induced on our sets by the Haar measure of the unitary group. These probability measures interpolate between the natural measure on the set of unistochastic matrices and the Dirac measure supported on the van der Waerden matrix.

Exponential rarefaction of maximal real algebraic hypersurfaces

Given an ample real Hermitian holomorphic line bundle L over a real algebraic variety X , the space of real holomorphic sections of L^{\otimes d} inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section s of L^{\otimes d} defines a maximal hypersurface tends to 0 exponentially fast as d goes to infinity. This extends to any dimension a result of Gayet and Welschinger (2011) valid for maximal real algebraic curves inside a real algebraic surface. The starting point is a low degree approximation property which relates the topology of the real vanishing locus of a real holomorphic section of L^{\otimes d} with the topology of the real vanishing locus a real holomorphic section of L^{\otimes d'} for a sufficiently smaller d'<d . Such a statement is inspired by the recent work of Diatta and Lerario (2022).

A note on dyadic approximation in Cantor’s set

We consider the convergence theory for dyadic approximation in the middle-third Cantor set, K, for approximation functions of the form ψτ(n)=n−τ (τ⩾0). In particular, we show that for values of τ beyond a certain threshold we have that almost no point in K is dyadically ψτ-well approximable with respect to the natural probability measure on K. This refines a previous result in this direction obtained by the first, third, and fourth named authors.

The multinomial tiling model

Given a graph G and collection of subgraphs T (called tiles), we consider covering G with copies of tiles in T so that each vertex v∈G is covered with a predetermined multiplicity. The multinomial tiling model is a natural probability measure on such configurations (it is the uniform measure on standard tilings of the corresponding “blow-up” of G). In the limit of large multiplicities, we compute the asymptotic growth rate of the number of multinomial tilings. We show that the individual tile densities tend to a Gaussian field defined by an associated discrete Laplacian. We also find an exact discrete Coulomb gas limit when we vary the multiplicities. For tilings of Zd with translates of a single tile and a small density of defects, we study a crystallization phenomenon when the defect density tends to zero, and give examples of naturally occurring quasicrystals in this framework.

(Non)-escape of mass and equidistribution for horospherical actions on trees

Let G be a large group acting on a biregular tree T and Gamma le G a geometrically finite lattice. In an earlier work, the authors classified orbit closures of the action of the horospherical subgroups on G/Gamma . In this article we show that there is no escape of mass and use this to prove that, in fact, dense orbits equidistribute to the Haar measure on G/Gamma . On the other hand, we show that new dynamical phenomena for horospherical actions appear on quotients by non-geometrically finite lattices: we give examples of non-geometrically finite lattices where an escape of mass phenomenon occurs and where the orbital averages along a Følner sequence do not converge. In the last part, as a by-product of our methods, we show that projections to Gamma backslash T of the uniform distributions on large spheres in the tree T converge to a natural probability measure on Gamma backslash T. Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term.

RANDOM REAL BRANCHED COVERINGS OF THE PROJECTIVE LINE

AbstractIn this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C} \mathbb {P}^1,\textit{conj} )$ . We prove that the space of degree d real branched coverings having “many” real branched points (for example, more than $\sqrt {d}^{1+\alpha }$ , for any $\alpha>0$ ) has exponentially small measure. In particular, maximal real branched coverings – that is, real branched coverings such that all the branched points are real – are exponentially rare.

Equidistribution of families of expanding horospheres on moduli spaces of hyperbolic surfaces

Given a simple closed curve $$\gamma $$ on a connected, oriented, closed surface S of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on S having a simple closed geodesic of length L of the same topological type as $$\gamma $$ equidistributes with respect to a natural probability measure as $$L \rightarrow \infty $$ . We prove several generalizations of Mirzakhani’s result and discuss some of the technical aspects ommited in her original work. The dynamics of the earthquake flow play a fundamental role in the arguments in this paper.

Random ideal hyperbolic quadrilaterals, the cross ratio distribution and punctured tori

Earlier work introduced a geometrically natural probability measure on the group of all M\"obius transformations of the hyperbolic plane so as to be able to study "random" groups of M\"obius transformations, and in particular random two-generator groups. Here we extend these results to consider random punctured tori. These Riemann surfaces have finite hyperbolic area $2\pi$ and fundamental group the free group of rank 2. They can be obtained by pairing (identifying) the opposite sides of an ideal hyperbolic quadrilateral. There is a natural distribution on ideal quadrilateral given by the cross ratio of their vertices. We identify this distribution and then calculate the distributions of various geometric quantities associated with random punctured tori such as the base of the geodesic length spectrum and the conformal modulus, along with more subtle things such as the distribution of the distance in Teichm\"uller space to the central "square" punctured torus.

Random Kleinian Groups, II Two Parabolic Generators

In earlier work, we introduced geometrically natural probability measures on the group of all Mobius transformations in order to study "random" groups of Mobius transformations, random surfaces, and in particular random two-generator groups, that is, groups where the generators are selected randomly, with a view to estimating the likelihood that such groups are discrete and then to make calculations of the expectation of their associated parameters, geometry, and topology. In this paper, we continue that study and identify the precise probability that a Fuchsian group generated by two parabolic Mobius transformations is discrete, and give estimates for the case of Kleinian groups generated by a pair of random parabolic elements which we support with a computational investigation of the Riley slice as identified by Bowditch's condition, and establish rigorous bounds.

Exact Hausdorff and Packing measure of certain Cantor sets, not necessarily self-similar or homogeneous

We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous. The calculation is based on the local behavior of the natural probability measure supported on the sets.

Open and closed random walks with fixed edgelengths in

In this paper, we consider fixed edgelength n-step random walks in . We give an explicit construction for the closest closed equilateral random walk to almost any open equilateral random walk based on the geometric median, providing a natural map from open polygons to closed polygons of the same edgelength. Using this, we first prove that a natural reconfiguration distance to closure converges in distribution to a Nakagami random variable as . We then strengthen this to an explicit probabilistic bound on the distance to closure for a random n-gon in any dimension with any collection of fixed edgelengths wi. Numerical evidence supports the conjecture that our closure map pushes forward the natural probability measure on open polygons to something very close to the natural probability measure on closed polygons; if this is so, we can draw some conclusions about the frequency of local knots in closed polygons of fixed edgelength.

On the minimum of a positive definite quadratic form over non-zero lattice points. Theory and applications

Let Σd++ be the set of positive definite matrices with determinant 1 in dimension d⩾2. Identifying any two SLd(Z)-congruent elements in Σd++ gives rise to the space of reduced quadratic forms of determinant one, which in turn can be identified with the locally symmetric space Xd:=SLd(Z)\\SLd(R)/SOd(R). Equip the latter space with its natural probability measure coming from a Haar measure on SLd(R). In 1998, Kleinbock and Margulis established sharp estimates for the probability that an element of Xd takes a value less than a given real number δ>0 over the non-zero lattice points Zd\\{0}.In this article, these estimates are extended to a large class of probability measures arising either from the spectral or the Cholesky decomposition of an element of Σd++. The sharpness of the bounds thus obtained are also established (up to multiplicative constants) for a subclass of these measures.Although of an independent interest, this theory is partly developed here with a view towards application to Information Theory. More precisely, after providing a concise introduction to this topic fitted to our needs, we lay the theoretical foundations of the study of some manifolds frequently appearing in the theory of Signal Processing. This is then applied to the recently introduced Integer-Forcing Receiver Architecture channel whose importance stems from its expected high performance. Here, we give sharp estimates for the probabilistic distribution of the so-called Effective Signal-to-Noise Ratio, which is an essential quantity in the evaluation of the performance of this model.

Partition-free families of sets

For an n-element set X and a real number 0<p<1 there is a very natural probability measure wp on the power set 2X. The measure of a family F⊂2X is defined as the sum of the measure of its members, and for F⊂X one sets w(F)=p|F|⋅(1−p)n−|F|. A family F⊂2X is called partition-free if it has no pairwise disjoint members whose union is X. Denoting the maximum of wp(F) over all partition-free families F⊂2X by m(n,p) we prove the rather surprising fact that while mn,1k=1−1k for all integers k≥2, m(n,p)→1 as n→∞ for all other values of p. This problem was raised recently by Fan Wei.

A natural probability measure derived from Stern’s diatomic sequence

Stern's diatomic sequence with its intrinsic repetition and refinement structure between consecutive powers of $2$ gives rise to a rather natural probability measure on the unit interval. We construct this measure and show that it is purely singular continuous, with a strictly increasing, Holder continuous distribution function. Moreover, we relate this function with the solution of the dilation equation for Stern's diatomic sequence.

English

Acute triangles are defined by having all angles less than π/2, and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension n ≥ 3, acuteness is defined by demanding that all dihedral angles between (n−1)-dimensional faces are smaller than π/2. However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property of acuteness is logically independent of the property of containing the circumcenter when the dimension is greater than two. In this article, we show that the latter property is also quite rare in higher dimensions. In a natural probability measure on the set of n-dimensional simplices, we show that the probability that a uniformly random n-simplex contains its circumcenter is 1/2 n .

Random curves on surfaces induced from the Laplacian determinant

We define natural probability measures on finite multicurves (finite collections of pairwise disjoint simple closed curves) on curved surfaces. These measures arise as universal scaling limits of probability measures on cycle-rooted spanning forests (CRSFs) on graphs embedded on a surface with a Riemannian metric, in the limit as the mesh size tends to zero. These in turn are defined from the Laplacian determinant and depend on the choice of a unitary connection on the surface. Wilson’s algorithm for generating spanning trees on a graph generalizes to a cycle-popping algorithm for generating CRSFs for a general family of weights on the cycles. We use this to sample the above measures. The sampling algorithm, which relates these measures to the loop-erased random walk, is also used to prove tightness of the sequence of measures, a key step in the proof of their convergence. We set the framework for the study of these probability measures and their scaling limits and state some of their properties.

Small Scale Equidistribution of Random Eigenbases

We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e. eigenbases) on a compact manifold M. Assume that the group of isometries acts transitively on M and the multiplicity of eigenfrequency tends to infinity at least logarithmically. We prove that, with respect to the natural probability measure on the space of eigenbases, almost surely a random eigenbasis is equidistributed at small scales; furthermore, the scales depend on the growth rate of multiplicity. In particular, this implies that almost surely random eigenbases on the n-dimensional sphere (n>=2) and the n-dimensional tori (n>=5) are equidistributed at polynomial scales.

A FRAMEWORK FOR INTERNEURAL DYNAMICS IN A CEREBRAL CORTEX CENTER

We model the innervation dynamics of interneurons in a cerebral cortex center A between the time of initial sensory input and acquisition of a sustained steady state. The model assumes that interneurons in A are heavily interconnected allowing synchronization. This invites modeling the dynamics by means of a discrete time map. The model takes into account the influence of excitatory and inhibitory cells and reflects the architecture of synapses along the axons. The acquisition of a sustained chaotic state is characterized by means of a natural invariant probability measure. The time to attain this probability measure can be estimated.

Ergodicity criteria for non-expanding transformations of 2-adic spheres

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $\langle f;\mathbf S_{2^-r}(a)\rangle$ on 2-adic spheres $\mathbf S_{2^-r}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$.

Injections de Sobolev probabilistes et applications

In this article, we give probabilistic versions of Sobolev embeddings on any Riemannian manifold $(M,g)$. More precisely, we prove that for natural probability measures on $L^2(M)$, almost every function belong to all spaces $L^p(M)$, $p<+\infty$. We then give applications to the study of the growth of the $L^p$ norms of spherical harmonics on spheres $\mathbb{S}^d$: we prove (again for natural probability measures) that almost every Hilbert base of $L^2( \mathbb{S}^d)$ made of spherical harmonics has all its elements uniformly bounded in all $L^p(\mathbb{S}^d), p<+\infty$ spaces. We also prove similar results on tori $\mathbb{T}^d$. We give then an application to the study of the decay rate of damped wave equations in a frame-work where the geometric control property on Bardos-Lebeau-Rauch is not satisfied. Assuming that it is violated for a measure $0$ set of trajectories, we prove that there exists almost surely a rate. Finally, we conclude with an application to the study of the $H^1$-supercritical wave equation, for which we prove that for almost all initial data, the weak solutions are strong and unique, locally in time.