arXiv subjects

Duncan Dauvergne

Publications and source records attributed to Duncan Dauvergne.

27 records · Page 2Linked to original sources

arXiv2022

Non-uniqueness times for the maximizer of the KPZ fixed point

Let $\mathfrak h_t$ be the KPZ fixed point started from any initial condition that guarantees $\mathfrak h_t$ has a maximum at every time $t$ almost surely. For any fixed $t$, almost surely $\max \mathfrak h_t$ is uniquely attained. However, there are exceptional times $t \in (0, \infty)$ when $\max \mathfrak h_t$ is achieved at multiple points. Let $\mathcal T_k \subset (0, \infty)$ denote the set of times when $\max \mathfrak h_t$ is achieved at exactly $k$ points. We show that almost surely $\mathcal T_2$ has Hausdorff dimension $2/3$ and is dense, $\mathcal T_3$ has Hausdorff dimension $1/3$ and is dense, $\mathcal T_4$ has Hausdorff dimension $0$, and there are no times when $\max \mathfrak h_t$ is achieved at $5$ or more points. This resolves two conjectures of Corwin, Hammond, Hegde, and Matetski.

math.PR↗

arXiv2021

Last passage isometries for the directed landscape

Consider the restriction of the directed landscape $\mathcal L(x, s; y, t)$ to a set of the form $\{x_1, \dots, x_k\} \times \{s_0\} \times \mathbb R \times \{t_0\}$. We show that on any such set, the directed landscape is given by a last passage problem across $k$ locally Brownian functions. The $k$ functions in this last passage isometry are built from certain marginals of the extended directed landscape. As applications of this construction, we show that the Airy difference profile is locally absolutely continuous with respect to Brownian local time, that the KPZ fixed point started from two narrow wedges has a Brownian-Bessel decomposition around its cusp point, and that the directed landscape is a function of its geodesic shapes.

math.PR↗

arXiv2021

A necessary and sufficient condition for convergence of the zeros of random polynomials

Consider random polynomials of the form $G_n = \sum_{i=0}^n ξ_i p_i$, where the $ξ_i$ are i.i.d.\ non-degenerate complex random variables, and $\{p_i\}$ is a sequence of orthonormal polynomials with respect to a regular measure $τ$ supported on a compact set $K$. We show that the zero measure of $G_n$ converges weakly almost surely to the equilibrium measure of $K$ if and only if $\mathbb{E} \log(1 + |ξ_0|) < \infty$. This generalizes the corresponding result of Ibragimov and Zaporozhets in the case when $p_i(z) = z^i$. We also show that the zero measure of $G_n$ converges weakly in probability to the equilibrium measure of $K$ if and only if $\mathbb{P} (|ξ_0| > e^n) = o(n^{-1})$. Our proofs rely on results from small ball probability and exploit the structure of general orthogonal polynomials. Our methods also work for sequences of asymptotically minimal polynomials in $L^p(τ)$, where $p \in (0, \infty]$. In particular, sequences of $L^p$-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class.

math.PR↗

arXiv2021

The Archimedean limit of random sorting networks

A sorting network (also known as a reduced decomposition of the reverse permutation), is a shortest path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of the symmetric group $S_n$ generated by adjacent transpositions. We prove that in a uniform random $n$-element sorting network $σ^n$, all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-$t$ permutation matrix measures of $σ^n$. As a corollary of these results, we show that if $S_n$ is embedded into $\mathbb{R}^n$ via the map $τ\mapsto (τ(1), τ(2), \dots τ(n))$, then with high probability, the path $σ^n$ is close to a great circle on a particular $(n-2)$-dimensional sphere in $\mathbb{R}^n$. These results prove conjectures of Angel, Holroyd, Romik, and Virag.

math.PR↗

arXiv2021

RSK in last passage percolation: a unified approach

We present a version of the RSK correspondence based on the Pitman transform and geometric considerations. This version unifies ordinary RSK, dual RSK and continuous RSK. We show that this version is both a bijection and an isometry, two crucial properties for taking limits of last passage percolation models. We use the bijective property to give a non-computational proof that dual RSK maps Bernoulli walks to nonintersecting Bernoulli walks.

math.PR↗

arXiv2021

Bulk properties of the Airy line ensemble

The Airy line ensemble is a central object in random matrix theory and last passage percolation defined by a determinantal formula. The goal of this paper is to provide a set of tools which allow for precise probabilistic analysis of the Airy line ensemble. The two main theorems are a representation in terms of independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines. Along the way, we give tail bounds and moduli of continuity for nonintersecting Brownian ensembles, and a quick proof of tightness for Dyson's Brownian motion converging to the Airy line ensemble.

math.PR↗

arXiv2019

The Local Limit of Random Sorting Networks

A sorting network is a geodesic path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of $S_n$ generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space-time locations of transpositions in a neighbourhood of $an$ for $a\in[0,1]$ as $n\to\infty$. Here time is scaled by a factor of $1/n$ and space is not scaled. The limit is a swap process $U$ on $\mathbb{Z}$. We show that $U$ is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on $a$ is through time scaling by a factor of $\sqrt{a(1-a)}$. To establish the existence of $U$, we find a local limit for staircase-shaped Young tableaux. These Young tableaux are related to sorting networks through a bijection of Edelman and Greene.

math.PR↗

arXiv2018

Circular support in random sorting networks

A sorting network is a shortest path from $12 \cdots n$ to $n \cdots 2 1$ in the Cayley graph of the symmetric group generated by adjacent transpositions. For a uniform random sorting network, we prove that in the global limit, particle trajectories are supported on $π$-Lipschitz paths. We show that the weak limit of the permutation matrix of a random sorting network at any fixed time is supported within a particular ellipse. This is conjectured to be an optimal bound on the support. We also show that in the global limit, trajectories of particles that start within distance $ε$ of the edge are within $\sqrt{2ε}$ of a sine curve in uniform norm.

math.PR↗

arXiv2018

Asymptotic zero distribution of random orthogonal polynomials

We consider random polynomials of the form $H_n(z)=\sum_{j=0}^nξ_jq_j(z)$ where the $\{ξ_j\}$ are i.i.d non-degenerate complex random variables, and the $\{q_j(z)\}$ are orthonormal polynomials with respect to a compactly supported measure $τ$ satisfying the Bernstein-Markov property on a regular compact set $K \subset \mathbb{C}$. We show that if $\mathbb{P}(|ξ_0|>e^{|z|})=o(|z|^{-1})$, then the normalized counting measure of the zeros of $H_n$ converges weakly in probability to the equilibrium measure of $K.$ This is the best possible result, in the sense that the roots of $G_n(z)=\sum_{j=0}^nξ_jz^j$ fail to converge in probability to the appropriate equilibrium measure when the above condition on the $ξ_j$ is not satisfied. In addition, we give a multivariable version of this result. We also consider random polynomials of the form $\sum_{k=0}^nξ_kf_{n,k}z^k$, where the coefficients $f_{n,k}$ are complex constants satisfying certain conditions, and the random variables $\{ξ_k\}$ satisfy $\mathbb{E} \log(1 + |ξ_0|) < \infty$. In this case, we establish almost sure convergence of the normalized counting measure of the zeros to an appropriate limiting measure. Again, this is the best possible result in the same sense as above.

math.PR↗