Conference Agenda

A sequence of real numbers is said to have Poissonian pair correlations (PPC) if $\lim_{N\to\infty}\frac{1}{N}\#\left\{1\leq i,j\leq N\middle | i\neq j, \Vert x_i-x_j\Vert\leq \frac{s}{N}\right\}=2s$ for every $s>0$. Because PPC characterizes equidistribution on a local scale, it has been a topic of large interest within the community in recent years. However, the focus has so far been primarily on deterministic sequences, and whenever random variables are involved, on iid sequences. While it is known that iid sequences of uniformly distributed random variables generically have Poissonian pair correlations, the case of dependent random variables is more complicated. This talk takes a closer look at the case of dependent random variables, which to the best of the author's knowledge has not yet been discussed within the literature yet. More specifically, two types of sequences on the torus are investigated: For sequences of jittered samples, the PPC property depends on how the finite jittered sample is extended to infinite sequences. Second, for random walks on the torus where results by Schatte for the convergence of sums on the torus can be used to show generic PPC.

The construction of uniform spacings is well-established. Consider $n-1$ independent points uniformly distributed over the unit interval. Uniform $1$-spacings correspond to the distances between consecutive points, while $k$-spacings are defined as the distances between points separated by exactly $k-1$ other points. There is an extensive body of work on the exact and asymptotic properties of uniform $1$-spacings, and to a lesser extent, $k$-spacings, as well as on their numerous statistical applications.

We propose an apparently novel approach, referred to as the local Poisson approximation for $k$-spacings. This method provides a detailed understanding of their asymptotic behavior on small scales across the entire range of possible values. This framework not only immediately yields existing limit theorems for the minimum and maximum $k$-spacings, which were previously proved using more complex techniques such as the Chen-Stein method, but also extends the analysis to encompass the asymptotic properties of all $k$-spacings, regardless of their length.

Let $X_1,\ldots, X_n \in \mathbb{R}^d$ be a sequence of iid random vectors, where $n\ll d$. A fundamental problem in high-dimensional statistics concerns normal approximations and convergence properties of the maximum statistic $$M_n=\max_{1\leq k\leq d} \frac{1}{\sqrt{n}}\sum_{i=1}^n X_{i,k},$$ whose study was initiated in seminal works by by Chernozhukov, Chetverikov and Kato. A next step in understanding the asymptotic properties of $M_n$ and accompanying quantile approximations is the development of Edgeworth-type expansions and corresponding bootstrap methods. A very recent result in this direction was established by Koike, developing an Edgeworth expansion for $\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i$ based on Stein kernels, subject to some regularity conditions. In our project, we view the problem through the lens of Poisson-approximations to directly construct an Edgeworth expansion for $M_n$. Our main assumptions are a Cram\'{e}r-type condition for all pairs of components of the $X_i$ and a notion of weak dependence across the dimension. Inverting the Edgeworth expansions, we obtain a Cornish-Fisher-type expansion for the quantiles of $M_n$, which is also second-order accurate. Furthermore, we extend our results to studentized case, i.e. to the statistic $\max_{1\leq k\leq d} T_{n,k}$, where $T_{n,k}$ is Student-t statistic of the $k$-th components of the $X_i$.

Let $(A_v)_{v\in \mathcal{T}}$ be the balanced Gaussian Branching Random Walk on a $d$-ary tree $\mathcal{T}$ and let $M^A$ be the multiplicative chaos with parameter $\gamma \in (0, \sqrt{2\log d})$ constructed from $A$.

In this work we establish the precise first order asymptotics of %the the logarithm of negative exponential moment of $M^A$, i.e.\ we prove that for $t_k = \lambda p_\gamma^k$ with $\lambda>0$ and $p_\gamma$ an explicit constant depending only on $\gamma$, we have as $k \to \infty$, \begin{equation*} -\frac{1}{d^k} \log \mathbb{E}[e^{-\lambda p_\gamma^k M^A } ] \to h(\lambda), \end{equation*} where $h\colon (0,\infty)\to \mathbb{R}$ is a non-explicit positive continuous function.

This result allows us to study the law of $A$ tilted by $e^{-t_k M^A}$ for particular values of $\lambda$, with $k\to \infty$. In this setting we prove that the normalized $L^1$ norm of $A$ in generation $k-a$ is bounded and converges to $0$ when first $k\to \infty$ and then $a\to 0$.

As an application we prove that in this setting, under the tilt $e^{-t_k M^A}$ and with $k\to \infty$, the Branching Random Walk $A$ exhibits a weak decay of correlation, which is not present in the non-tilted model.

Our methods also apply to the usual Branching Random Walk $(S_v)_{v\in \mathcal{T}}$ and with $M^A$ replaced by $\frac{1}{2}(M^+ + M^- )$, where $M^+$ and $M^-$ are the multiplicative chaoses with parameter $\gamma \in (0, \sqrt{2\log d})$ constructed from $S$ and $-S$. In that case we prove that, as $k\to \infty$, \begin{equation*} -\frac{1}{d^k} \log \mathbb{E}[e^{- \frac{\lambda p_\gamma^k}{2}( M^+ + M^-) }] \to \tilde h(\lambda), \end{equation*} where $\tilde h\colon (0,\infty)\to \mathbb{R}$ is again a non-explicit positive continuous function.

Our models are motivated by Euclidean field theory and can be seen as hierarchical versions of the massless Liouville and the sinh-Gordon field theory in infinite volume. From this perspective our analysis sheds new light on the existence and the decay or correlations in these models, which are among the major open questions in this area.