arXiv subjects

Levi Haunschmid-Sibitz

Publications and source records attributed to Levi Haunschmid-Sibitz.

5 recordsLinked to original sources

arXiv2026

The stochastic six-vertex model speed process

For the stochastic six-vertex model on the quadrant $\mathbb Z_{\geq0}\times\mathbb Z_{\geq0}$ with step initial conditions and a single second-class particle at the origin, we show almost sure convergence of the speed of the second-class particle to a random limit. This allows us to define the stochastic six-vertex model speed process, whose law we show to be ergodic and stationary for the dynamics of the multi-class stochastic six-vertex process. To prove this, we develop precise bounds on the fluctuations of the height function of the stochastic six-vertex model around its limit shape using methods from integrable probability. As part of the proof, we also obtain a novel geometric stochastic domination result that states that a second-class particle to the right of any number of third-class particles will at any fixed time be overtaken by at most a geometric number of third-class particles, which is of independent interest.

math.PR↗

arXiv2026

The censored stochastic six-vertex model and parabolic Kazhdan--Lusztig $R$-polynomials

We introduce a censored version of the stochastic six-vertex model. We show that for parameters $b_1 < b_2$, this model started from the initial condition ${1}_{x>0}$ is stochastically dominated at any time by the blocking measure. This is a partial analog of the censoring inequality for monotone spin systems. In particular, this result allows us to control the behavior of second-class particles. The proof uses parabolic Kazhdan--Lusztig $R$-polynomials, whose appearance is explained using a connection between the stochastic six-vertex model and the Iwahori--Hecke algebras of symmetric groups. Furthermore, we find an intertwining relation for this process using normalized parabolic Kazhdan--Lusztig $R$-polynomials as an intertwining kernel.

math.PR↗

arXiv2024

$(\log t)^\frac{2}{3}$-superdiffusivity for the 2d stochastic Burgers equation

The Stochastic Burgers equation was introduced in [H. van Beijeren, R. Kutner and H. Spohn, Excess noise for driven diffusive systems, PRL, 1985] as a continuous approximation of the fluctuations of the asymmetric simple exclusion process. It is formally given by $$\partial_tη=\frac{1}{2}Δη+ \mathfrak w\cdot\nabla(η^2) + \nabla\cdotξ,$$ where $ξ$ is $d$-dimensional space time white noise and $\mathfrak w$ is a fixed non-zero vector. In the critical dimension $d=2$ at stationarity, we show that this system exhibits superdiffusve behaviour: more specifically, its bulk diffusion coefficient behaves like $(\log t)^\frac23$, in a Tauberian sense, up to $\log\log\log t$ corrections. This confirms a prediction made in the physics literature and complements [G. Cannizzarro, M. Gubinelli, F. Toninelli, Gaussian Fluctuations for the stochastic Burgers equation in dimension $d\geq 2$, CMP, 2024], where the same equation was studied in the weak-coupling regime. Furthermore this model can be seen as a continuous analogue to [H.T. Yau, $(\log t)^\frac{2}{3}$ law of the two dimensional asymmetric simple exclusion process, Annals of Mathematics, 2004].

math.PR↗

arXiv2024

Near-critical dimers and massive SLE

We consider the dimer model on the square and hexagonal lattices with doubly periodic weights. The purpose of this paper is threefold: (a) we establish a rigourous connection with the massive SLE$_2$ constructed by Makarov and Smirnov (and recently revisited by Chelkak and Wan); (b) we show that the convergence takes place in arbitrary bounded domains subject to Temperleyan boundary conditions, and that the scaling limit is universal; and (c) we prove conformal covariance of the scaling limit. For this we introduce an inhomogeneous near-critical dimer model, corresponding to a drift for the underlying random walk which is a smoothly varying vector field or alternatively to an inhomogeneous mass profile. When the vector field derives from a log-convex potential we prove that the corresponding loop-erased random walk has a universal scaling limit. Our techniques rely on an exact discrete Girsanov identity on the triangular lattice which may be of independent interest. We complement our results by stating precise conjectures making connections to a generalised Sine-Gordon model at the free fermion point.

math.PR↗

arXiv2022

$\sqrt{\log t}$-superdiffusivity for a Brownian particle in the curl of the 2d GFF

The present work is devoted to the study of the large time behaviour of a critical Brownian diffusion in two dimensions, whose drift is divergence-free, ergodic and given by the curl of the 2-dimensional Gaussian Free Field. We prove the conjecture, made in [B. Tóth, B. Valkó, J. Stat. Phys., 2012], according to which the diffusion coefficient $D(t)$ diverges as $\sqrt{\log t}$ for $t\to\infty$. Starting from the fundamental work by Alder and Wainwright [B. Alder, T. Wainright, Phys. Rev. Lett. 1967], logarithmically superdiffusive behaviour has been predicted to occur for a wide variety of out-of-equilibrium systems in the critical spatial dimension $d=2$. Examples include the diffusion of a tracer particle in a fluid, self-repelling polymers and random walks, Brownian particles in divergence-free random environments, and, more recently, the 2-dimensional critical Anisotropic KPZ equation. Even if in all of these cases it is expected that $D(t)\sim\sqrt{\log t}$, to the best of the authors' knowledge, this is the first instance in which such precise asymptotics is rigorously established.

math.PR↗