arXiv subjects

Duncan Dauvergne

Publications and source records attributed to Duncan Dauvergne.

At least 19 recordsLinked to original sources

arXiv2026

The directed landscape from Brownian motion

We construct an almost sure bijection that recovers the directed landscape on the half-plane from a sequence of independent Brownian motions. This map is the natural scaling limit of the Robinson--Schensted--Knuth (RSK) correspondence. The Brownian motions arise as the marginals of the multi-path stationary horizon associated with the directed landscape. The inverse map is fully explicit and yields a natural coupling in which Brownian last-passage percolation converges in probability to the directed landscape. As an application, we prove that the directed landscape restricted to a strip can be reconstructed from the parabolic Airy line ensemble, resolving a conjecture of the first author and Zhang. Along the way we develop two new versions of RSK in the semi-discrete setting, introduce a general theory of sorting via Pitman operators that generates a faithful action of the biHecke monoid, and establish key identities for the multi-path stationary horizon for both the directed landscape and Brownian last-passage percolation.

math.PR↗

arXiv2026

The directed landscape in half-space

We prove that two half-space models in the KPZ universality class, exponential last-passage percolation and a family of Poisson-avoiding metrics generalizing colored TASEP, converge to a common scaling limit. This scaling limit is the directed landscape in half-space, a random directed metric in the half-plane indexed by a parameter which determines the strength of the boundary interaction. As part of our analysis, we characterize the half-space directed landscape in terms of the half-space KPZ fixed point, and prove convergence of geodesics. We also give an explicit construction of joint stationary measures (or horizons) in half-space for the log-gamma polymer, the KPZ equation, exponential and geometric last passage percolation, and the directed landscape itself.

math.PR↗

arXiv2026

Hidden invariance of last passage percolation and directed polymers

Last passage percolation and directed polymer models on $\mathbb Z^2$ are invariant under translation and certain reflections. When these models have an integrable structure coming from either the RSK correspondence or the geometric RSK correspondence (e.g. geometric last passage percolation or the log-gamma polymer), we show that these basic invariances can be combined with a decoupling property to yield a rich new set of symmetries. Among other results, we prove shift and rearrangement invariance statements for last passage times, geodesic locations, disjointness probabilities, polymer partition functions, and quenched polymer measures. We also use our framework to find `scrambled' versions of the classical RSK correspondence, and to find an RSK correspondence for moon polyominoes. The results extend to limiting models, including the KPZ equation and the Airy sheet.

math.PR↗

arXiv2026

The Airy line ensemble at the rough-smooth boundary

We study the rough-smooth boundary in the two-periodic Aztec diamond, a random domino tiling model exhibiting three types of macroscopic regions. We show that the height function at this boundary converges to an independent sum of an Airy surface and an i.i.d. noise field with fluctuations governed by the full-plane smooth phase. Going further, we prove convergence of a family of Temperleyan backbone paths to the Airy line ensemble. This gives the first convergence result for a family of undirected paths converging to the Airy line ensemble, as well as Airy convergence at a noisy boundary.

math.PR↗

arXiv2026

Geodesic networks and the disjointness gap in the directed landscape

The directed landscape is a random directed metric on the plane that arises as the scaling limit of metric models in the KPZ universality class. For a pair of points p, q, the disjointness gap G(p; q) measures the shortfall when we optimize length over pairs of disjoint paths from p to q versus optimizing over all pairs of paths. Any spatial marginal of G is simply the gap between the top two lines in an Airy line ensemble. In this paper, we show that when the start and end time are fixed, the disjointness gap fully encodes the set of exceptional geodesic networks. The correspondence uses simple features of the disjointness gap, e.g. zeroes, local minima. We give a similar correspondence relating semi-infinite geodesic networks to a Busemann gap function. The proofs are deterministic given a list of soft properties related to the coalescent geometry of the directed landscape.

math.PR↗

arXiv2026

Bulk heights of the KPZ line ensemble

For $t > 0$, let $\{\mathcal{H}^{(t)}_n, n \in \mathbb{N}\}$ be the KPZ line ensemble with parameter $t$, satisfying the homogeneous $\mathbf{H}$-Brownian Gibbs property with $\mathbf{H}(x) =e^x$. We prove quantitative concentration estimates for the $n$th line $\mathcal{H}^{(t)}_n$ which yield the asymptotics $\mathcal{H}^{(t)}_n = n \log n + o(n^{3/4 + ε})$ as $n \to \infty$. A key step in the proof is a general integration by parts formula for $\mathbf{H}$-Brownian Gibbs line ensembles which yields the identity $\mathbb{E} \exp(\mathcal{H}^{(t)}_{n + 1}(x) - \mathcal{H}^{(t)}_n (x)) = n t^{-1}$ for any $n, t, x$.

math.PR↗

arXiv2025

Characterization of the directed landscape from the KPZ fixed point

We show that the directed landscape is the unique coupling of the KPZ fixed point from all initial conditions satisfying three natural properties: independent increments, monotonicity, and shift commutativity. Equivalently, we show that the directed landscape is the unique directed metric on $\mathbb R^2$ with independent increments and KPZ fixed point marginals. This unifies the two central objects in the KPZ universality class. Our main theorem also provides a general framework for proving convergence to the directed landscape given convergence to the KPZ fixed point. We apply this framework to prove landscape convergence in a range of models: exotic couplings of ASEP and TASEP, the random walk and Brownian web distances, and a class of non-integrable asymmetric exclusion processes with the basic coupling that perturb off of TASEP (this final class requires random initial data). All of our convergence theorems are new except for colored TASEP, where we provide a short alternative proof.

math.PR↗

arXiv2024

Three-halves variation of geodesics in the directed landscape

We show that geodesics in the directed landscape have $3/2$-variation and that weight functions along the geodesics have cubic variation. We show that the geodesic and its landscape environment around an interior point has a small-scale limit. This limit is given in terms of the directed landscape with Brownian-Bessel boundary conditions. The environments around different interior points are asymptotically independent. We give tail bounds with optimal exponents for geodesic and weight function increments. As an application of our results, we show that geodesics are not Hölder-$2/3$ and that weight functions are not Hölder-$1/3$, although these objects are known to be Hölder with all lower exponents.

math.PR↗

arXiv2024

Wiener densities for the Airy line ensemble

The parabolic Airy line ensemble $\mathfrak A$ is a central limit object in the KPZ universality class and related areas. On any compact set $K = \{1, \dots, k\} \times [a, a + t]$, the law of the recentered ensemble $\mathfrak A - \mathfrak A(a)$ has a density $X_K$ with respect to the law of $k$ independent Brownian motions. We show that $$ X_K(f) = \exp \left(-\textsf{S}(f) + o(\textsf{S}(f))\right) $$ where $\textsf{S}$ is an explicit, tractable, non-negative function of $f$. We use this formula to show that $X_K$ is bounded above by a $K$-dependent constant, give a sharp estimate on the size of the set where $X_K < ε$ as $ε\to 0$, and prove a large deviation principle for $\mathfrak A$. We also give density estimates that take into account the relative positions of the Airy lines, and prove sharp two-point tail bounds that are stronger than those for Brownian motion. These estimates are a key input in the classification of geodesic networks in the directed landscape. The paper is essentially self-contained, requiring only tail bounds on the Airy point process and the Brownian Gibbs property as inputs.

math.PR↗

arXiv2024

Upper tail large deviations of the directed landscape

Starting from one-point tail bounds, we establish an upper tail large deviation principle for the directed landscape at the metric level. Metrics of finite rate are in one-to-one correspondence with measures supported on a set of countably many paths, and the rate function is given by a certain Kruzhkov entropy of these measures. As an application of our main result, we prove a large deviation principle for the directed geodesic.

math.PR↗

arXiv2023

The 27 geodesic networks in the directed landscape

The directed landscape is a random directed metric on the plane that arises as the scaling limit of classical metric models in the KPZ universality class. Typical pairs of points in the directed landscape are connected by a unique geodesic. However, there are exceptional pairs of points connected by more complicated geodesic networks. We show that up to isomorphism there are exactly $27$ geodesic networks that appear in the directed landscape, and find Hausdorff dimensions in a scaling-adapted metric on $\mathbb R^4_\uparrow$ for the sets of endpoints of each of these networks.

math.PR↗

arXiv2023

Uniform convergence to the Airy line ensemble

We show that classical integrable models of last passage percolation and the related nonintersecting random walks converge uniformly on compact sets to the Airy line ensemble. Our core approach is to show convergence of nonintersecting Bernoulli random walks in all feasible directions in the parameter space. We then use coupling arguments to extend convergence to other models.

math.PR↗

arXiv2022

The scaling limit of the longest increasing subsequence

We provide a framework for proving convergence to the directed landscape, the central object in the Kardar-Parisi-Zhang universality class. For last passage models, we show that compact convergence to the Airy line ensemble implies convergence to the Airy sheet. In i.i.d. environments, we show that Airy sheet convergence implies convergence of distances and geodesics to their counterparts in the directed landscape. Our results imply convergence of classical last passage models and interacting particle systems. Our framework is built on the notion of a directed metric, a generalization of metrics which behaves better under limits. As a consequence of our results, we present a solution to an old problem: the scaled longest increasing subsequence in a uniform permutation converges to the directed geodesic.

math.PR↗

arXiv2022

The directed landscape

The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that last passage geodesics converge to random functions with Holder-2/3- continuous paths. This work completes the construction of the central object in the Kardar-Parisi-Zhang universality class, the directed landscape.

math.PR↗

arXiv2022

The SIR model in a moving population: propagation of infection and herd immunity

In a collection of particles performing independent random walks on $\mathbb Z^d$ we study the spread of an infection with SIR dynamics. Susceptible particles become infected when they meet an infected particle. Infected particles heal and are removed at rate $ν$. We show that when $ν$ is small, with positive probability the infection survives forever and grows linearly. Furthermore, after the infection reaches a region, it quickly passes through and leaves behind a $\textit{herd immunity}$ regime consisting of recovered particles, a small positive density of susceptible particles, and no infected particles. One notable feature of this model is the simultaneously existence of supercritical and subcritical phases on either side of an infection front of $O(1)$ width.

math.PR↗

arXiv2022

Disjoint optimizers and the directed landscape

We study maximal length collections of disjoint paths, or 'disjoint optimizers', in the directed landscape. We show that disjoint optimizers always exist, and that their lengths can be used to construct an extended directed landscape. The extended directed landscape can be built from an independent collection of extended Airy sheets, which we define from the parabolic Airy line ensemble. We show that the extended directed landscape and disjoint optimizers are scaling limits of the corresponding objects in Brownian last passage percolation (LPP). As two consequences of this work, we show that one direction of the Robinson-Schensted-Knuth bijection passes to the KPZ limit, and we find a criterion for geodesic disjointness in the directed landscape that uses only a single parabolic Airy line ensemble. The proofs rely on a new notion of multi-point LPP across the parabolic Airy line ensemble, combinatorial properties of multi-point LPP, and probabilistic resampling ideas.

math.PR↗

arXiv2022

Spread of infections in a heterogeneous moving population

We consider a model where an infection moves through a collection of particles performing independent random walks. In this model, Kesten and Sidoravicius established linear growth of the infected region when infected and susceptible particles move at the same speed. In this paper we establish a linear growth rate when infected and susceptible particles move at different speeds, answering an open problem from their work. Our proof combines an intricate coupling of Poisson processes with a streamlined version of a percolation model of Sidoravicius and Stauffer.

math.PR↗

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↗