Lehmer points and visible points on affine varieties over finite fields

Let $V$ be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on $V$ is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on $V$ are obtained, and the distribution of visible points into different congruence classes is investigated.Comment: 16 pages, accepted for publication in Math. Proc. Cambridge Philos. Soc. Theorem 2 of this paper is improved from the published versio

Generic properties of subgroups of free groups and finite presentations

Asymptotic properties of finitely generated subgroups of free groups, and of finite group presentations, can be considered in several fashions, depending on the way these objects are represented and on the distribution assumed on these representations: here we assume that they are represented by tuples of reduced words (generators of a subgroup) or of cyclically reduced words (relators). Classical models consider fixed size tuples of words (e.g. the few-generator model) or exponential size tuples (e.g. Gromov's density model), and they usually consider that equal length words are equally likely. We generalize both the few-generator and the density models with probabilistic schemes that also allow variability in the size of tuples and non-uniform distributions on words of a given length.Our first results rely on a relatively mild prefix-heaviness hypothesis on the distributions, which states essentially that the probability of a word decreases exponentially fast as its length grows. Under this hypothesis, we generalize several classical results: exponentially generically a randomly chosen tuple is a basis of the subgroup it generates, this subgroup is malnormal and the tuple satisfies a small cancellation property, even for exponential size tuples. In the special case of the uniform distribution on words of a given length, we give a phase transition theorem for the central tree property, a combinatorial property closely linked to the fact that a tuple freely generates a subgroup. We then further refine our results when the distribution is specified by a Markovian scheme, and in particular we give a phase transition theorem which generalizes the classical results on the densities up to which a tuple of cyclically reduced words chosen uniformly at random exponentially generically satisfies a small cancellation property, and beyond which it presents a trivial group

Symmetries of statistics on lattice paths between two boundaries

We prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that lie between two fixed boundaries T and B (which are themselves lattice paths), the statistics `number of E steps shared with B' and `number of E steps shared with T' have a symmetric joint distribution. To do so, we give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps S=(0,-1) at prescribed x-coordinates. We also show that a similar equidistribution result for path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. We extend the two theorems to k-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, to watermelon configurations, to pattern-avoiding permutations, and to the generalized Tamari lattice. Finally, we prove a conjecture of Nicol\'as about the distribution of degrees of k consecutive vertices in k-triangulations of a convex n-gon. To achieve this goal, we provide a new statistic-preserving bijection between certain k-tuples of non-crossing paths and k-flagged semistandard Young tableaux, which is based on local moves reminiscent of jeu de taquin.Comment: Small typos corrected, and journal reference and grant info adde

Polynomials with prescribed bad primes

We tabulate polynomials in Z[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and infinity. We explain how these sets of polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification. Finally we discuss the similar but technically more complicated tabulation problem corresponding to removing the auxiliary conditions.Comment: 26 pages, 3 figure

Statistics for traces of cyclic trigonal curves over finite fields

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over a field of q elements as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of the Frobenius equals the sum of q+1 independent random variables taking the value 0 with probability 2/(q+2) and 1, e^{(2pi i)/3}, e^{(4pi i)/3} each with probability q/(3(q+2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.Comment: 30 pages, added statement and sketch of proof in Section 7 for generalization of results to p-fold covers of the projective line, the final version of this article will be published in International Mathematics Research Notice

Integration with respect to the Haar measure on unitary, orthogonal and symplectic group

We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson-Zuber type

Biased statistics for traces of cyclic p-fold covers over finite fields

In this paper, we discuss in more detail some of the results on the statistics of the trace of the Frobenius endomorphism associated to cyclic p-fold covers of the projective line that were presented in [1]. We also show new findings regarding statistics associated to such curves where we fix the number of zeros in some of the factors of the equation in the affine model

Fluctuations for analytic test functions in the Single Ring Theorem

We consider a non-Hermitian random matrix $A$ whose distribution is invariant under the left and right actions of the unitary group. The so-called Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni, states that the empirical eigenvalue distribution of $A$ converges to a limit measure supported by a ring $S$. In this text, we establish the convergence in distribution of random variables of the type $Tr (f(A)M)$ where $f$ is analytic on $S$ and the Frobenius norm of $M$ has order $\sqrt{N}$. As corollaries, we obtain central limit theorems for linear spectral statistics of $A$ (for analytic test functions) and for finite rank projections of $f(A)$ (like matrix entries). As an application, we locate outliers in multiplicative perturbations of $A$.Comment: 29 pages, 1 figure. In Version v2, we slightly modified the assumptions, in order to fix a problem un the control of the tails (see Assumption 2.3). In v3, some minors typos were corrected. In v4, some explanations were added in the introduction and some typos were corrected. To appear in Indiana Univ. Math.