arXiv subjects

Christian Serio

Publications and source records attributed to Christian Serio.

8 recordsLinked to original sources

arXiv2026

The pinned half-space Airy line ensemble

Half-space models in the Kardar-Parisi-Zhang (KPZ) universality class exhibit rich boundary phenomena that alter the asymptotic behavior familiar from their full-space counterparts. A distinguishing feature of these systems is the presence of a boundary parameter that governs a transition between subcritical, critical, and supercritical regimes, characterized by different scaling exponents and fluctuation statistics. In this paper we construct the pinned half-space Airy line ensemble $\mathcal{A}^{\mathrm{hs}; \infty}$ on $[0,\infty)$ -- a natural half-space analogue of the Airy line ensemble -- expected to arise as the universal scaling limit of supercritical half-space KPZ models. The ensemble $\mathcal{A}^{\mathrm{hs}; \infty}$ is obtained as the weak limit of the critical half-space Airy line ensembles $\mathcal{A}^{\mathrm{hs}; \varpi}$ introduced in arXiv:2505.01798 as the boundary parameter $\varpi$ tends to infinity. We show that $\mathcal{A}^{\mathrm{hs}; \infty}$ has a Pfaffian point process structure with an explicit correlation kernel and that, after a parabolic shift, it satisfies a one-sided Brownian Gibbs property described by pairwise pinned Brownian motions. Far from the origin, $\mathcal{A}^{\mathrm{hs}; \infty}$ converges to the standard Airy line ensemble, while at the origin its distribution coincides with that of the ordered eigenvalues (with doubled multiplicity) of the stochastic Airy operator with $\beta = 4$.

math.PR↗

arXiv2025

The half-space KPZ line ensemble and its scaling limit

For each $\alpha \in \mathbb{R}$, $t \geq 1$, we show that there exists a unique $\mathbb{N}$-indexed line ensemble of random continuous curves $\mathbb{R}_{\le 0} \to \mathbb{R}$ with the following properties: (1) The top curve is distributed as the time-$t$ Cole--Hopf solution to the half-space KPZ equation with narrow wedge initial condition and Neumann boundary condition with parameter $\alpha$. (2) The line ensemble satisfies a one-sided resampling invariance property, involving softly non-intersecting Brownian motions with an attractive potential between pairs at the boundary. We call this object the half-space KPZ line ensemble. For $\alpha=\mu t^{-1/3}$ with $\mu \in \mathbb{R}$ fixed (critical regime) and for $\alpha>0$ fixed (supercritical regime), we show that the half-space KPZ line ensemble is tight under 1:2:3 KPZ scaling as $t\to\infty$. Moreover, all subsequential limits approximate a parabola and enjoy a one-sided Brownian Gibbs property, described by non-intersecting Brownian motions with pairwise interaction at the boundary. In the critical case this agrees with the half-space Airy line ensemble recently constructed by Dimitrov and Yang. In the supercritical case, we demonstrate a novel structure involving pairwise pinned Brownian motions, one of the main technical contributions of this paper.

math.PR↗

arXiv2025

Scaling limit and tail bounds for a random walk model of SOS level lines

This paper analyzes a random walk model for the level lines appearing in the entropic repulsion phenomena of three-dimensional discrete random interfaces above a hard wall; we are particularly motivated by the low-temperature (2+1)D solid-on-solid (SOS) model, where the emergence of these level lines has been rigorously established. The model we consider is a line ensemble of non-crossing random walk bridges above a wall with geometrically growing area tilts. Our main result, which in particular resolves a question of Caputo, Ioffe, and Wachtel (2019), is an edge 1:2:3 scaling limit for this ensemble as the domain size $N$ diverges, with a growing number of walks (including the number of level lines of the SOS model) and high boundary conditions (covering the maximum upper deviation of the SOS level lines). As a key input, we establish Tracy--Widom-type upper tail bounds for each of the relevant curves in the line ensemble. An ingredient which may be of independent interest is a ballot theorem for random walk bridges under a broader range of boundary values than available in the literature.

math.PR↗

arXiv2024

Convergence to stationary measures for the half-space log-gamma polymer

We consider the point-to-point half-space log-gamma polymer model in the unbound phase. We prove that the free energy increment process on the anti-diagonal path converges to the top marginal of a two-layer Markov chain with an explicit description, which can be interpreted as two random walks conditioned softly never to intersect. This limiting law is a stationary measure for the polymer on the anti-diagonal path. The starting point of our analysis is an embedding of the free energy into the half-space log-gamma line ensemble recently constructed by Barraquand, Corwin, and Das. Given the Gibbsian line ensemble structure, the main contribution of our work lies in developing a route to access and prove convergence to stationary measures via line ensemble techniques. Our argument relies on a description of the limiting behavior of two softly non-intersecting random walk bridges around their starting point, a result established in this paper that may be of independent interest.

math.PR↗

arXiv2023

Scaling limit for line ensembles of random walks with geometric area tilts

We consider line ensembles of non-intersecting random walks constrained by a hard wall, each tilted by the area underneath it with geometrically growing pre-factors $\mathfrak{b}^i$ where $\mathfrak{b}>1$. This is a model for the level lines of the $(2+1)$D SOS model above a hard wall, which itself mimics the low-temperature 3D Ising interface. A similar model with $\mathfrak{b}=1$ and a fixed number of curves was studied by Ioffe, Velenik, and Wachtel (2018), who derived a scaling limit as the time interval $[-N,N]$ tends to infinity. Line ensembles of Brownian bridges with geometric area tilts ($\mathfrak{b}>1$) were studied by Caputo, Ioffe, and Wachtel (2019), and later by Dembo, Lubetzky, and Zeitouni (2022). Their results show that as the time interval and the number of curves $n$ tend to infinity, the top $k$ paths converge to a limiting measure $μ$. In this paper we address the open problem of proving existence of a scaling limit for random walk ensembles with geometric area tilts. We prove that with mild assumptions on the jump distribution, under suitable scaling the top $k$ paths converge to the same measure $μ$ as $N\to\infty$ followed by $n\to\infty$. We do so both in the case of bridges fixed at $\pm N$ and of walks fixed only at $-N$.

math.PR↗

arXiv2023

Uniform convergence of Dyson Ferrari--Spohn diffusions to the Airy line ensemble

We consider the Dyson Ferrari--Spohn diffusion $\mathcal{X}^N = (\mathcal{X}^N_1,\dots,\mathcal{X}^N_N)$, consisting of $N$ non-intersecting Ferrari--Spohn diffusions $\mathcal{X}^N_1 > \cdots > \mathcal{X}^N_N > 0$ on $\mathbb{R}$. This object was introduced by Ioffe, Velenik, and Wachtel (2018) as a scaling limit for line ensembles of $N$ non-intersecting random walks above a hard wall with area tilts, which model certain three-dimensional interfaces in statistical physics. It was shown by Ferrari and Shlosman (2023) that as $N\to\infty$, after a spatial shift of order $N^{2/3}$ and constant rescaling in time, the top curve $\mathcal{X}^N_1$ converges to the $\mathrm{Airy}_2$ process in the sense of finite-dimensional distributions. We extend this result by showing that the full ensemble $\mathcal{X}^N$ converges with the same shift and time scaling to the Airy line ensemble in the topology of uniform convergence on compact sets. In our argument we formulate a Brownian Gibbs property with area tilts for $\mathcal{X}^N$, which we show is equivalent after a global parabolic shift to the usual Brownian Gibbs property introduced by Corwin and Hammond (2014).

math.PR↗

arXiv2023

Tightness of discrete Gibbsian line ensembles

A discrete Gibbsian line ensemble $\mathfrak{L} = (L_1,\dots,L_N)$ consists of $N$ independent random walks on the integers conditioned not to cross one another, i.e., $L_1 \geq \cdots \geq L_N$. In this paper we provide sufficient conditions for convergence of a sequence of suitably scaled discrete Gibbsian line ensembles $f^N = (f_1^N,\dots,f_N^N)$ as the number of curves $N$ tends to infinity. Assuming log-concavity and a KMT-type coupling for the random walk jump distribution, we prove that under mild control of the one-point marginals of the top curves with a global parabolic shift, the full sequence $(f^N)$ is tight in the topology of uniform convergence over compact sets, and moreover any weak subsequential limit possesses the Brownian Gibbs property. If in addition the top curves converge in finite-dimensional distributions to the parabolic $\mathrm{Airy}_2$ process, then a result of arXiv:2002.00684 implies that $(f^N)$ converges to the parabolically shifted Airy line ensemble. These results apply to a broad class of discrete jump distributions, including geometric as well as any log-concave distribution whose support forms a compact integer interval.

math.PR↗

arXiv2021

Tightness of Bernoulli Gibbsian line ensembles

A Bernoulli Gibbsian line ensemble $\mathfrak{L} = (L_1, \dots, L_N)$ is the law of the trajectories of $N-1$ independent Bernoulli random walkers $L_1, \dots, L_{N-1}$ with possibly random initial and terminal locations that are conditioned to never cross each other or a given random up-right path $L_N$ (i.e. $L_1 \geq \cdots \geq L_N$). In this paper we investigate the asymptotic behavior of sequences of Bernoulli Gibbsian line ensembles $\mathfrak{L}^N = (L^N_1, \dots, L^N_N)$ when the number of walkers $N$ tends to infinity. We prove that if one has mild but uniform control of the one-point marginals of the lowest-indexed (or top) curves $L_1^N$ then the sequence $\mathfrak{L}^N$ is tight in the space of line ensembles. Furthermore, we show that if the top curves $L_1^N$ converge in the finite dimensional sense to the parabolic Airy$_2$ process then $\mathfrak{L}^N$ converge to the parabolically shifted Airy line ensemble.

math.PR↗