Related papers: On the generalized circle problem for a random lat…
We describe a procedure for determining the generalised scaling functions $f_n(g)$ at all the values of the coupling constant. These functions describe the high spin contribution to the anomalous dimension of large twist operators (in the…
In this article we report a preliminary investigation of the large $N$ limit of a generalized one-matrix model which represents an $O(n)$ symmetric model on a random lattice. The model on a regular lattice is known to be critical only for…
For a random walk $S_n$ on $\mathbb{R}^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n = n^{-1} \sum_{i=1}^n S_i$. For lattice distributions we give conditions for a local limit theorem to hold. We prove…
The central limit theorem for convex bodies says that with high probability the marginal of an isotropic log-concave distribution along a random direction is close to a Gaussian, with the quantitative difference determined asymptotically by…
The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_k(n)^2$, where $P_k(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of…
We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a…
For $\Gamma$ a cocompact or cofinite Fuchsian group, we study the lattice point problem on the Riemann surface $\Gamma\backslash\mathbb{H}$. The main asymptotic for the counting of the orbit $\Gamma z$ inside a circle of radius $r$ centered…
We study certain cases of convoluted Fourier coefficients of $GL_n$-automorphic functions. We establish identities that express them in terms of Fourier coefficients related to unipotent orbits. The most general case that is studied is…
A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function $f$ over $\mathbb{F}_2^{n}$ has the same absolute value, namely $|\hat{f}(\alpha)|=1/2^k$ for every $\alpha$ in the…
Let $[a,b]\subset\mathbb{R}$ be a non empty and non singleton closed interval and $P=\{a=x_0<\cdots<x_n=b\}$ is a partition of it. Then $f:I\to\mathbb{R}$ is said to be a function of $r$-bounded variation, if the expression…
In this paper we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centered at the origin is the only minimizer of such a…
We consider the probability distributions of values in the complex plane attained by Fourier sums of the form \sum_{j=1}^n a_j exp(-2\pi i j nu) /sqrt{n} when the frequency nu is drawn uniformly at random from an interval of length 1. If…
We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}\alpha(n)$ where $\alpha$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved…
In this paper, we prove that the Euclidean distance between two independent random vectors uniformly distributed on $l_p^n$-balls $(1 \leq p \leq \infty)$ or on its boundary satisfies a central limit theorem as $n$ tends to $\infty$. Also,…
We prove that the Newton quotient of the average R(t) of a lipschitzian function (with non vanishing variation) with respect to the SRB measure on a transversal family f_t of piecewise expanding unimodal maps, after an appropriated…
Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_n(\beta)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the…
The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the…
This paper concerns the so-called diffusion in the curl of the 2d Gaussian free field, and its generalization to higher dimensions $n \geq 2$, building on the scale-by-scale homogenization approach developed recently by Chatzigeorgiou,…
In [1], the authors consider a random walk $(Z_{n,1},\ldots,Z_{n,K+1})\in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. A functional central limit theorem for the first…
The main result of this paper is a functional limit theorem for the sine-process. In particular, we study the limit distribution, in the space of trajectories, for the number of particles in a growing interval. The sine-process has the…