There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar-Parisi-Zhang (KPZ) equation in dimension $d\geq 3$ as the smoothing parameter is switched off, but most of the results did not hold in the full temperature regions where they should. Inspired by martingale techniques coming from the directed polymers literature, we first extend the law of large numbers for SHE obtained in [MSZ16] to the full weak disorder region of the associated polymer model and to more general initial conditions. We further extend the Edwards-Wilkinson regime of the SHE and KPZ equation studied in [GRZ18,MU17,DGRZ20] to the full $L^2$-region, along with multidimensional convergence and general initial conditions for the KPZ equation (and SHE), which were not proven before. To do so, we rely on a martingale CLT combined with a refinement of the local limit theorem for polymers.
We prove, using probabilistic techniques and analysis on the Wiener space, that the large scale fluctuations of the KPZ equation in $d\geq 3$ with a small coupling constant, driven by a white in time and colored in space noise, are given by the Edwards-Wilkinson model. This gives an alternative proof, that avoids perturbation expansions, to the results of Magnen and Unterberger \cite{magnen2017diffusive}.
We study a new model of interactive particle systems which we call the randomly activated cascading exclusion process (RACEP). Particles wake up according to exponential clocks and then take a geometric number of steps. If another particle is encountered during these steps, the first particle goes to sleep at that location and the second is activated and proceeds accordingly. We consider a totally asymmetric version of this model which we refer as Hall-Littlewood-PushTASEP (HL-PushTASEP) on $\mathbb{Z}_{\geq 0}$ lattice where particles only move right and where initially particles are distribu…
We consider the stochastic higher spin six vertex (SHS6V) model introduced in [Corwin-Petrov, 2016] with general integer spin parameters $I, J$. Starting from near stationary initial condition, we prove that the SHS6V model converges to the KPZ equation under weakly asymmetric scaling. This generalizes the result of [Corwin-Ghosal-Shen-Tsai, 2018] from $I = J =1$ to general $I, J$.
We consider the half-line stochastic heat equation (SHE) with Robin boundary parameter $A = -\frac{1}{2}$. Under narrow wedge initial condition, we compute every positive (including non-integer) Lyapunov exponents of the half-line SHE. As a consequence, we prove a large deviation principle for the upper tail of the half-line KPZ equation under Neumann boundary parameter $A = -\frac{1}{2}$ with rate function $Φ_+^{\text{hf}} (s) = \frac{2}{3} s^{\frac{3}{2}}$. This confirms the prediction of [Krajenbrink and Le Doussal 2018] and [Meerson, Vilenkin 2018] for the upper tail exponent of the half-l…
In this paper we mingle the Gaussian free field, the Schramm-Loewner evolution and the KPZ relation in a natural way, shedding new light on all of them. Our principal result shows that the level lines and the SLE$_κ$ flow lines of the Gaussian free field do not satisfy the usual KPZ relation. In order to prove this, we have to make a technical detour: by a careful study of a certain diffusion process, we provide exact estimates of the exponential moments of winding of chordal SLE curves conditioned to pass nearby a fixed point. This extends previous results on winding of SLE curves by Schramm.
The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because the nonlinearity is marginally defined with respect to the roughness of the forcing noise. However, its Cole-Hopf solution, defined as the logarithm of the solution of the linear stochastic heat equation (SHE) with a multiplicative noise, is a mathematically well-defined object. In fact, Hairer [13] has recently proved that the solution of SHE can actually be derived through the Cole-Hopf transform of the solution of the KPZ equation with a suitable renormalization under periodic bou…
We consider the asymptotic behavior of the KPZ fixed point $\{\mathsf H(x,t)\}_{x\in\mathbb R, t>0}$ conditioned on $\mathsf H(0,T)=L$ as $L$ goes to infinity. The main result is a conditional limit theorem for the fluctuations of $\mathsf H$ in the region near the line segment connecting the origin $(0,0)$ and $(0,T)$ for both step and flat initial conditions. The limit random field can be represented as a functional of two independent Brownian bridges, and in addition the limit random field depends also on the initial law of the KPZ fixed point. In particular for temporal fluctuations, the l…
This paper has two main goals. The first is universality of the KPZ equation for fluctuations of dynamic interfaces associated to interacting particle systems in the presence of open boundary. We consider generalizations on the open-ASEP from [Corwin-Shen '16, Parekh '17] but admitting non-simple interactions both at the boundary and within the bulk of the particle system. These variations on open-ASEP are not integrable models, similar to the long-range variations on ASEP considered in [Dembo-Tsai '15, Y '20]. We establish the KPZ equation with the appropriate Robin boundary conditions as sca…
For each $t\geq 1$ we construct an $\mathbf{N}$-indexed ensemble of random continuous curves with three properties: 1. The lowest indexed curve is distributed as the time $t$ Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) stochastic PDE with narrow wedge initial data; 2. The entire ensemble satisfies a resampling invariance which we call the $\mathbf{H}$-Brownian Gibbs property (with $\mathbf{H}(x)=e^{x}$); 3. Increments of the lowest indexed curve, when centered by $-t/24$ and scaled down vertically by $t^{1/3}$ and horizontally by $t^{2/3}$, remain uniformly absolutely continuous (i.e. …
We study the solution $h_\varepsilon$ of the Kardar-Parisi-Zhang (KPZ) equation for $d \geq 3$: $$ \frac{\partial}{\partial t} h_{\varepsilon} = \frac12 Δh_{\varepsilon} + \bigg[\frac12 |\nabla h_\varepsilon |^2 - C_\varepsilon\bigg]+ β\varepsilon^{\frac{d-2}2} ξ_{\varepsilon} $$ with $h_\varepsilon(0,x)=0$. Here $ξ_\varepsilon=ξ\star ϕ_\varepsilon$ is a spatially smoothened (at scale $\varepsilon$) Gaussian space-time white noise and $C_\varepsilon$ is a divergent constant as $\varepsilon\to 0$. When the disorder $β$ is sufficiently small and $\varepsilon\to 0$, $h_\varepsilon(t,x)- \mathfrak…
The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $γ$-Liouville quantum gravity (LQG) surface, $γ\in (0,2)$, decorated with a space-filling form of Schramm's SLE$_κ$, $κ= 16/γ^2 \in (4,\infty)$, $η$ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $η$ which can be defined as a function of $η$ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $η^{-1}(…
The solution of Kardar-Parisi-Zhang equation (KPZ equation) is solved formally via Cole-Hopf transformation $h=\log u$, where $u$ is the solution of multiplicative stochastic heat equation(SHE). In earlier works by Chatterjee and Dunlap, Caravenna, Sun, and Zygouras, and Gu, they consider the solution of two dimensional KPZ equation via the solution $u_\varepsilon$ of SHE with flat initial condition and with noise which is mollified in space on scale in $\varepsilon$ and its strength is weakened as $β_\varepsilon=\hatβ \sqrt{\frac{2π\varepsilon}{-\log \varepsilon}}$, and they prove that when $…
We prove that in the full $L^2$-regime the partition function of the directed polymer model in dimensions $d\geq 3$, if centered, scaled and averaged with respect to a test function $φ\in C_c(\mathbb{R}^d)$, converges in distribution to a Gaussian random variable with explicit variance. Introducing a new idea in this context of a martingale difference representation, we also prove that the log-partition function, which can be viewed as a discretisation of the KPZ equation, exhibits the same fluctuations, when centered and averaged with respect to a test function. Thus, the two models fall with…
This note, dedicated in Harold Widom's memory, is an expanded version of a lecture I gave in fall 2021 at the MSRI program "Universality and Integrability in Random Matrices and Interacting Particle Systems". I will focus on the behavior of the stationary measure for the open KPZ equation, a paradigmatic model for interface growth in contact with boundaries. Much of this will review elements of my joint work with A. Knizel as well as with H. Shen, as well as subsequent works of W. Bryc, A. Kuznetsov, Y. Wang, and J. Wesolowski and of G. Barraquand and P. Le Doussal. The basis for this advance …
We consider the transition probabilities for random walks in $1+1$ dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to th…
We consider the exactly solvable model of exponential directed last passage percolation on $\mathbb{Z}^2$ in the large deviation regime. Conditional on the upper tail large deviation event $\mathcal{U}_δ:=\{T_{n}\geq (4+δ)n\}$ where $T_{n}$ denotes the last passage time from $(1,1)$ to $(n,n)$, we study the geometry of the polymer/geodesic $Γ_{n}$, i.e., the optimal path attaining $T_{n}$. We show that conditioning on $\mathcal{U}_δ$ changes the transversal fluctuation exponent from the characteristic $2/3$ of the KPZ universality class to $1/2$, i.e., conditionally, the smallest strip around …
