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Eero Saksman

Publications and source records attributed to Eero Saksman.

At least 37 records · Page 2Linked to original sources
arXiv2020

Homogenization of iterated singular integrals with applications to random quasiconformal maps

We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let $(F_j)_{j \geq 1}$ be a sequence of normalized homeomorphic solutions to the planar Beltrami equation $\overline{\partial} F_j (z)=μ_j(z,ω) \partial F_j(z),$ where the random dilatation satisfies $|μ_j|\leq k<1$ and has locally periodic statistics, for example of the type $$μ_j (z,ω)=ϕ(z)\sum_{n\in \mathbf{Z}^2}g(2^j z-n,X_{n}(ω)), $$ where $g(z,ω)$ decays rapidly in $z$, the random variables $X_{n}$ are i.i.d., and $ϕ\in C^\infty_0$. We establish the almost sure and local uniform convergence as $j\to\infty$ of the maps $F_j$ to a deterministic quasiconformal limit $F_\infty$. This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let $T_1,\ldots , T_{m}$ be translation and dilation invariant singular integrals on ${\bf R}^d, $ and consider a $d$-dimensional version of $μ_j$, e.g., as defined above or within a more general setting. We then prove that there is a deterministic function $f$ such that almost surely as $j\to\infty$, $$ μ_j T_{m}μ_j\ldots T_1μ_j\to f \quad \textrm{weakly in } L^p,\quad 1 < p < \infty\ . $$

math.CV↗
arXiv2020

Local times and sample path properties of the Rosenblatt process

Let $Z = (Z_t)_{t \geq 0}$ be the Rosenblatt process with Hurst index $H \in (1/2, 1)$. We prove joint continuity for the local time of $Z$, and establish Hölder conditions for the local time. These results are then used to study the irregularity of the sample paths of $Z$. Based on analogy with similar known results in the case of fractional Brownian motion, we believe our results are sharp. A main ingredient of our proof is a rather delicate spectral analysis of arbitrary linear combinations of integral operators, which arise from the representation of the Rosenblatt process as an element in the second chaos.

math.PR↗
arXiv2020

Coefficient estimates for $H^p$ spaces with $0<p<1$

Let $C(k,p)$ denote the smallest real number such that the estimate $|a_k|\leq C(k,p)\|f\|_{H^p}$ holds for every $f(z)=\sum_{n\geq0}a_n z^n$ in the $H^p$ space of the unit disc. We compute $C(2,p)$ for $0<p<1$ and $C(3,2/3)$, and identify the functions attaining equality in the estimate.

math.FA↗
arXiv2019

On the regularity of complex multiplicative chaos

Denote by $μ_β="\exp(βX)"$ the Gaussian multiplicative chaos which is defined using a log-correlated Gaussian field $X$ on a domain $U\subset\mathbb{R}^d$. The case $β\in\mathbb{R}$ has been studied quite intensively, and then $μ_β$ is a random measure on $U$. It is known that $μ_β$ can also be defined for complex values $β$ lying in certain subdomain of $\mathbb{C}$, and then the realizations of $μ_β$ are random generalized functions on $U$. In this note we complement the results of Junnila et al. (where the case of purely imaginary $β$ was considered) by studying the Besov-regularity of $μ_β$ and the finiteness of moments for general complex values of $β$.

math.PR↗
arXiv2019

Linear space properties of $H^p$ spaces of Dirichlet series

We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of $\mathcal{H}^p$. More specifically, we study linear functionals generated by fractional primitives of the Riemann zeta function; our estimates rely on certain Hardy--Littlewood inequalities and display an interesting phenomenon, called contractive symmetry between $\mathcal{H}^p$ and $\mathcal{H}^{4/p}$, contrasting the usual $L^p$ duality. We next deduce general coefficient estimates, based on an interplay between the multiplicative structure of $\mathcal{H}^p$ and certain new one variable bounds. Finally, we deduce general estimates for the norm of the partial sum operator $\sum_{n=1}^\infty a_n n^{-s}\mapsto \sum_{n=1}^N a_n n^{-s}$ on $\mathcal{H}^p$ with $0< p \le 1$, supplementing a classical result of Helson for the range $1<p<\infty$. The results for the coefficient estimates and for the partial sum operator exhibit the traditional schism between the ranges $1\le p \le \infty$ and $0<p<1$.

math.FA↗
arXiv2019

On the convergence of Hamiltonian Monte Carlo

This paper discusses the irreducibility and geometric ergodicity of the Hamiltonian Monte Carlo (HMC) algorithm. We consider cases where the number of steps of the symplectic integrator is either fixed or random. Under mild conditions on the potential $\F$ associated with target distribution $π$, we first show that the Markov kernel associated to the HMC algorithm is irreducible and recurrent. Under more stringent conditions, we then establish that the Markov kernel is Harris recurrent. Finally, we provide verifiable conditions on $\F$ under which the HMC sampler is geometrically ergodic.

stat.CO↗
arXiv2019

Decompositions of log-correlated fields with applications

In this article we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos. We prove two decomposition theorems. The first one is a global one and says that if the difference between the covariance kernels of two Gaussian fields, taking values in some Sobolev space, has suitable Sobolev regularity, then these fields differ by a Hölder continuous Gaussian process. Our second decomposition theorem is more specialized and is in the setting of Gaussian fields whose covariance kernel has a logarithmic singularity on the diagonal -- or log-correlated Gaussian fields. The theorem states that any log-correlated Gaussian field $X$ can be decomposed locally into a sum of a Hölder continuous function and an independent almost $\star$-scale invariant field (a special class of stationary log-correlated fields with 'cone-like' white noise representations). This decomposition holds whenever the term $g$ in the covariance kernel $C_X(x,y)=\log(1/|x-y|)+g(x,y)$ has locally $H^{d+\varepsilon}$ Sobolev smoothness. We use these decompositions to extend several results that have been known basically only for $\star$-scale invariant fields to general log-correlated fields. These include the existence of critical multiplicative chaos, analytic continuation of the subcritical chaos in the so-called inverse temperature parameter $β$, as well as generalised Onsager-type covariance inequalities which play a role in the study of imaginary multiplicative chaos.

math.PR↗
arXiv2018

Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model

In this article we study imaginary Gaussian multiplicative chaos -- namely a family of random generalized functions which can formally be written as $e^{i X(x)}$, where $X$ is a log-correlated real-valued Gaussian field on $\mathbb{R}^d$, i.e. it has a logarithmic singularity on the diagonal of its covariance. We study basic analytic properties of these random generalized functions, such as what spaces of distributions do these objects live in, along with their basic stochastic properties, such as moment and tail estimates. After this, we discuss connections between imaginary multiplicative chaos and the critical planar Ising model, namely that the scaling limit of the spin field of the so called critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, i.e. the real part of an imaginary multiplicative chaos distribution. Moreover, if one adds a magnetic perturbation to the XOR-Ising model, then the scaling limit of the spin field can be expressed in terms of the cosine of the sine-Gordon field, which can also be viewed as the real part of an imaginary multiplicative chaos distribution. The first sections of the article have been written in the style of a review, and we hope that the text will also serve as an introduction to imaginary chaos for an uninitiated reader.

math.PR↗
arXiv2018

On certain sums over ordinates of zeta-zeros II

Let $γ$ denote the imaginary parts of complex zeros $ρ= β+iγ$ of $ζ(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{γ> 0}γ^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critical line $\int_1^T|G(1/2+it)|^2\,dt$ are revisited. This paper is a continuation of work begun by the second author in 2001.

math.NT↗
arXiv2018

Multiplicative chaos and the characteristic polynomial of the CUE: the $L^1$-phase

In this note we prove that suitable positive powers of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix converge in law, as the size of the matrix tends to infinity, to a Gaussian multiplicative chaos measure once correctly normalized. We prove this in the whole $L^1$- or subcritical phase of the chaos measure.

math.PR↗
arXiv2018

Pseudomoments of the Riemann zeta function

The $2$kth pseudomoments of the Riemann zeta function $ζ(s)$ are, following Conrey and Gamburd, the $2k$th integral moments of the partial sums of $ζ(s)$ on the critical line. For fixed $k>1/2$, these moments are known to grow like $(\log N)^{k^2}$, where $N$ is the length of the partial sum, but the true order of magnitude remains unknown when $k\le 1/2$. We deduce new Hardy--Littlewood inequalities and apply one of them to improve on an earlier asymptotic estimate when $k\to\infty$. In the case $k<1/2$, we consider pseudomoments of $ζ^α(s)$ for $α>1$ and the question of whether the lower bound $(\log N)^{k^2α^2}$ known from earlier work yields the true growth rate. Using ideas from recent work of Harper, Nikeghbali, and Radziwi{łł} and some probabilistic estimates of Harper, we obtain the somewhat unexpected result that these pseudomements are bounded below by $\log N$ to a power larger than $k^2α^2$ when $k<1/e$ and $N$ is sufficiently large.

math.FA↗
arXiv2018

The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line

We prove that if $ω$ is uniformly distributed on $[0,1]$, then as $T\to\infty$, $t\mapsto ζ(iωT+it+1/2)$ converges to a non-trivial random generalized function, which in turn is identified as a product of a very well behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between number theory and the theory of multiplicative chaos -- the latter is known to be connected to many other areas of mathematics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let $δ_T$ approach zero slowly enough as $T\to\infty$, then $t\mapsto ζ(1/2+iδ_T t+iωT)$ is asymptotically a product of a divergent scalar quantity suggested by Selberg's central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This essentially says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale.

math.PR↗
arXiv2017

Fatou and brother Riesz theorems in the infinite-dimensional polydisc

We study the boundary behavior of functions in the Hardy spaces on the infinite dimensional polydisk. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz-Zygmund type theorems for radial convergence. As a consequence one obtains easy new proofs of the brothers F. and M. Riesz theorems in infinite dimension. Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.

math.CV↗
arXiv2017

Structural rigidity of generalised Volterra operators on $H^p$

We show that the non-compact generalised analytic Volterra operators $T_g$, where $g \in \mathit{BMOA}$, have the following structural rigidity property on the Hardy spaces $H^p$ for $1 \le p < \infty$ and $p \neq 2$: if $T_g$ is bounded below on an infinite-dimensional subspace $M \subset H^p$, then $M$ contains a subspace linearly isomorphic to $\ell^p$. This implies in particular that any Volterra operator $T_g\colon H^p \to H^p$ is $\ell^2$-singular for $p \neq 2$.

math.FA↗
arXiv2016

Hypercyclicity Properties of Commutator Maps

We investigate the hypercyclic properties of commutator maps acting on separable ideals of operators. As the main result we prove the commutator map induced by scalar multiples of the backward shift operator fails to be hypercyclic on the space of compact operators on $\ell^2$. We also establish some necessary conditions which identify large classes of operators that do not induce hypercyclic commutator maps.

math.FA↗
arXiv2016

Triebel-Lizorkin spaces on metric spaces via hyperbolic fillings

We give a new characterization of (homogeneous) Triebel-Lizorkin spaces $\dot{\mathcal F}^{s}_{p,q}(Z)$ in the smoothness range $0 < s < 1$ for a fairly general class of metric measure spaces $Z$. The characterization uses Gromov hyperbolic fillings of $Z$. This gives a short proof of the quasisymmetric invariance of these spaces in case $Z$ is $Q$-Ahlfors regular and $sp = Q > 1$. We also obtain first results on complex interpolation for these spaces in the framework of doubling metric measure spaces.

math.CA↗