Stochastic PDE Limit of the Six Vertex Model

Abstract

We study the stochastic six vertex model and prove that under weak asymmetry scaling (i.e., when the parameter $\Delta \rightarrow 1^+$ so as to zoom into the ferroelectric/disordered phase critical point) its height function fluctuations converge to the solution to the Kardar–Parisi–Zhang (KPZ) equation. We also prove that the one-dimensional family of stochastic Gibbs states for the symmetric six vertex model converge under the same scaling to the stationary solution to the stochastic Burgers equation. Our proofs rely upon the Markov (self) duality of our model. The starting point is an exact microscopic Hopf–Cole transform for the stochastic six vertex model which follows from the model’s known one-particle Markov self-duality. Given this transform, the crucial step is to establish self-averaging for specific quadratic function of the transformed height function. We use the model’s two-particle self-duality to produce explicit expressions (as Bethe ansatz contour integrals) for conditional expectations from which we extract time-decorrelation and hence self-averaging in time. The crux of our Markov duality method is that the entire convergence result reduces to precise estimates on the one-particle and two-particle transition probabilities. Previous to our work, Markov dualities had only been used to prove convergence of particle systems to linear Gaussian SPDEs (e.g. the stochastic heat equation with additive noise).

Appendix A. Quadratic variation in ASEP

In this appendix we expand upon the brief discussion from Sects. 1.3 and 1.4 and explain how our Markov duality method can be applied to ASEP, which is a simpler limit of the stochastic 6V model. We will not carry out the necessary analysis, but rather just point to the main steps.

Recall that ASEP is an interacting particle system on ${\mathbb {Z}}$, where particles inhabit sites index by ${\mathbb {Z}}$ and jump left and right according to continuous-time exponential clocks with rates $\ell > 0$ and $r> 0$ subject to exclusion (jumps to occupied sites are suppressed). We will assume that $\ell +r=1$ and set $\tau := r/\ell$. The ASEP height function $N_{ASEP }(t,x)$ has 1/0 slopes entering occupied/vacant sites as depicted in Fig. 6. For ASEP with near-stationary initial data of density $\rho =\frac{1}{2}$ we define a variant¹⁶ of the Hopf–Cole transform of $N_{ASEP }(t,x)$ by

$$\begin{aligned} Z_{ASEP }(t,x) := \tau ^{N_{ASEP }(t,x)-\frac{1}{2} x} e^{t(1-2\sqrt{\ell r})}, \quad t\in [0,\infty ), \ x\in {\mathbb {Z}}. \end{aligned}$$

This solves the following microscopic SHE:

$$\begin{aligned} d Z_{ASEP }(t,x) = \sqrt{\ell r} \Delta Z_{ASEP }(t,x) dt + dM(t,x), \end{aligned}$$

(A.1)

where $\Delta f(x):= f(x+1)+f(x-1)-2f(x)$ denotes the discrete Laplacian, and, for each $x\in {\mathbb {Z}}$, the process $M(t,x)$, $t\in {\mathbb {R}}_+$, is a martingale.

Under weak asymmetry scaling, i.e., $\tau =\tau _\varepsilon :=e^{-{\sqrt{\varepsilon }}}$ and $(t,x)\mapsto (\varepsilon ^{-2}t,\varepsilon ^{-1}x)$, an informal scaling argument applied to (A.1) indicates that the equation should converge to the continuum SHE. Key to establishing this convergence is the identification of the limiting quadratic variation of $M(t,x)$. Under weak asymmetry scaling, the optional quadratic variation of $M(t,x)$ reads

$$\begin{aligned} d\langle M(t,x), M(t,x') \rangle = \varepsilon {\mathbf {1}}_{{\{x=x'\}}} \Big ( \big ( \tfrac{1}{4} + \varepsilon ^\frac{1}{2} \mathcal {B}_\varepsilon (t,x) \big ) Z_{ASEP }^2(t,x) + {\widetilde{F}}_\varepsilon (t,x) \Big ) dt, \end{aligned}$$

(A.2)

where, following notations in Sect. 7, $\mathcal {B}_\varepsilon (t,x)$ is a generic, uniformly bounded process, and

$$\begin{aligned} {\widetilde{F}}_\varepsilon (t,x) := \varepsilon ^{-\frac{1}{2}}\nabla Z_{ASEP }(t,x) \varepsilon ^{-\frac{1}{2}}\nabla Z_{ASEP }(t,x-1). \end{aligned}$$

(A.3)

Referring to the r.h.s. of (A.2), we see that $\varepsilon ^\frac{1}{2} \mathcal {B}_\varepsilon (t,x)$ is indeed negligible compared to the constant $\frac{1}{4}$ factor. Key to identifying the limiting behavior is to argue that ${\widetilde{F}}(t,x)$ is also negligible. With $\nabla Z_{ASEP }(t,x) = (e^{-{\sqrt{\varepsilon }} \eta (t,x+1)}-1)Z_{ASEP }(t,x)$, we indeed have ${\widetilde{F}}_\varepsilon (t,x) = \mathcal {B}_\varepsilon (t,x) Z_{ASEP }^2(t,x)$, i.e., pointwise bounded up to a multiplicative factor of $Z_{ASEP }^2(t,x)$. On the other hand, it is conceivable that this term ${\widetilde{F}}(t,x)$ does not tend to zero pointwise, i.e., ${\widetilde{F}}(t,x) \not \rightarrow _\text {P} 0$. The crux of the convergence result is to prove that this term converges to zero after time-averaging:

$$\begin{aligned} {\mathbb {E}}\Big [ \Big ( \varepsilon ^{2}\int _0^{\varepsilon ^{-2}T} {\widetilde{F}}_\varepsilon (t,x) dt \Big )^2\Big ] \longrightarrow 0. \end{aligned}$$

(A.4)

This is first achieved in [BG97] by showing the decay as t becomes large of the conditional expectation

$$\begin{aligned} {\mathbb {E}}\big [ {\widetilde{F}}_\varepsilon (t+s,x) \big | {\mathscr {F}}(s) \big ], \end{aligned}$$

where ${\mathscr {F}}$ denotes the canonical filtration of ASEP. Roughly speaking, the estimate starts by using (A.1) to develop a sequence of inequality that bounds the conditional expectation. ‘Closing’ the series of inequality relies crucially on an identity [BG97, (A.6)] for the (semi)-discrete heat kernel. We do not know of a way to generalize this approach from [BG97] to the stochastic 6V model setting.

Here we provide an alternative approach via duality. The Markov duality method also begins with bounding conditional expectations. However, instead of trying to close a sequence of inequalities, this method provides direct access to the conditional expectations. First, the expression ${\widetilde{F}}_\varepsilon (t,x)$ is not convenient for our purpose. Use $\nabla Z_{ASEP }(t,x) = (e^{-{\sqrt{\varepsilon }} (\eta ^+(t,x)-\frac{1}{2}) }-1)Z_{ASEP }(t,x)$ where $\eta ^+(t,x) := \eta (t,x+1)$, and Taylor expand

$$\begin{aligned} \nabla Z_{ASEP }(t,x) ={\sqrt{\varepsilon }} (\tfrac{1}{2} - \eta ^{+}(t,x)) Z_{ASEP }+ \varepsilon \mathcal {B}_\varepsilon (t,x)Z_{ASEP }(t,x) , \end{aligned}$$

(A.5)

where $\mathcal {B}_\varepsilon (t,x)$ stands for a generic uniformly bounded process as in Sect. 7. We can then write ${\widetilde{F}}_\varepsilon (t,x) = F_\varepsilon (t,x) + \varepsilon ^{1/2} \mathcal {B}_\varepsilon (t,x)Z_{ASEP }^2(t,x)$, where

$$\begin{aligned} F_\varepsilon (t,x) = \tfrac{1}{2} Z_{\nabla }(t,x,x-1) + \tfrac{1}{2}Z_{\nabla }(t,x-1,x+1) + {\widetilde{Z}}(t,x-1,x), \end{aligned}$$

(A.6)

where, following the notation in Sect. 7,

$$\begin{aligned} Z_{\nabla }(t,x_1,x_2)&:= (\varepsilon ^{-\frac{1}{2}}\nabla Z_{ASEP }(t,x_1)) Z_{ASEP }(t,x_2),\\ {\widetilde{Z}}(t,x_1,t_2)&:= (\eta ^+Z_{ASEP })(t,x_1) (\eta ^+Z_{ASEP })(t,x_2) - \tfrac{1}{4} Z_{ASEP }(t,x_1) Z_{ASEP }(t,x_2). \end{aligned}$$

To see (A.6), we use (A.5) just as in the proof of Lemma 7.3 (for the stochastic 6V model):

$$\begin{aligned} {\widetilde{F}}_\varepsilon (t,x)&= \big ((\tfrac{1}{2} - \eta ^+) Z_{ASEP }\big )(t,x) \big ((\tfrac{1}{2} - \eta ^+) Z_{ASEP }\big )(t,x-1) + \varepsilon ^{1/2} \mathcal {B}_\varepsilon (t,x)Z_{ASEP }^2(t,x) \\&= \tfrac{1}{2} \big ((\tfrac{1}{2} - \eta ^+) Z_{ASEP }\big )(t,x) Z_{ASEP }(t,x-1)\\&\quad \,\, + \tfrac{1}{2} \big ((\tfrac{1}{2} - \eta ^+) Z_{ASEP }\big )(t,x-1) Z_{ASEP }(t,x)\\&\quad \,\, + {\widetilde{Z}}(t,x-1,x) + \varepsilon ^{1/2} \mathcal {B}_\varepsilon (t,x)Z_{ASEP }^2(t,x) \\&= \text{ r.h.s } \text{ of } {(A.6)} + \varepsilon ^{1/2} \mathcal {B}_\varepsilon (t,x)Z_{ASEP }^2(t,x) . \end{aligned}$$

In the last step, we replace $Z_{ASEP }(t,x)$ with $Z_{ASEP }(t,x\pm 1)$ costing error of order $\varepsilon ^{1/2} \mathcal {B}_\varepsilon (t,x)Z_{ASEP }(t,x)$.

As mentioned in Sect. 1.4, ASEP enjoys self-duality via the functions Q and ${\widetilde{Q}}$ defined therein. Specifically, the $k=2$ duality translates (after tilting and centering) into the following statement, in which we used the notation

$$\begin{aligned} {\mathbf {V}}_{ASEP }\big ((y_1,y_2),(x_1,x_2);t\big ) := e^{2t(1-2\sqrt{\ell r})} \tau ^{-\frac{1}{2}(x_1+x_2-y_1-y_2)} {\mathbb {P}}_\text {ASEP}\big ((y_1,y_2)\rightarrow (x_1,y_2) ;t\big ). \end{aligned}$$

Proposition A.1

For all $x_1<x_2\in {\mathbb {Z}}$ and $t,s\in [0,\infty )$, we have

$$\begin{aligned}&{\mathbb {E}}\Big [ Z_{ASEP }(t+s,x_1) Z_{ASEP }(t+s,x_2) \Big \vert {\mathscr {F}}(s) \Big ]\nonumber \\&\quad = \sum _{y_1<y_2\in {\mathbb {Z}}} {\mathbf {V}}_{ASEP }\big ((y_1,y_2),(x_1,x_2);t\big ) Z_{ASEP }(s,y_1) Z_{ASEP }(s,y_2), \end{aligned}$$

(A.7)

$$\begin{aligned}&{\mathbb {E}}\Big [ (\eta ^{+}Z_{ASEP })(t+s,x_1) (\eta ^{+}Z_{ASEP })(t+s,x_2) \Big \vert {\mathscr {F}}(s) \Big ] \nonumber \\&\quad = \sum _{y_1<y_2\in {\mathbb {Z}}} {\mathbf {V}}_{ASEP }\big ((y_1,y_2),(x_1,x_2);t\big ) \big (\eta ^{+}Z_{ASEP }\big )(s,x_1)\big (\eta ^{+}Z_{ASEP }\big )(s,x_2).\quad \end{aligned}$$

(A.8)

Proposition A.1 provides the necessary ingredients for expressing conditional expectations for the relevant quantities. Specifically, with ${\widetilde{Z}}(t,x-1,x)$ being an linear combination the two observables in (A.7) and in (A.8) at $(x_1,x_2)=(x-1,x)$ we have

$$\begin{aligned} {\mathbb {E}}\Big [{\widetilde{Z}}(t+s,x-1,x)\Big |{\mathscr {F}}(s)\Big ] = \sum _{y_1<y_2\in {\mathbb {Z}}} {\mathbf {V}}_{ASEP }\big ((y_1,y_2),(x-1,x);t\big ) {\widetilde{Z}}(t,y_1,y_2). \end{aligned}$$

(A.9)

Likewise, $Z_{\nabla }(t,x,x-1)$ is the difference of $Z_{ASEP }(t,x+1)Z_{ASEP }(t,x-1)$ and $Z_{ASEP }(t,x)Z_{ASEP }(t,x-1)$. Taking the difference of (A.7) for $(x_1,x_2) = (x+1,x-1)$ and for $(x,x-1)$ gives

$$\begin{aligned}&{\mathbb {E}}\Big [Z_{\nabla }(t+s,x,x-1) \Big |{\mathscr {F}}(s)\Big ] \\&\quad = \sum _{y_1<y_2\in {\mathbb {Z}}} \varepsilon ^{-\frac{1}{2}}\nabla _{x_1}{\mathbf {V}}_{ASEP }\big ((y_1,y_2),(x_1,x_2);t\big ) \big |_{(x_1,x_2)=(x,x-1)} Z_{ASEP }(s,y_1)Z_{ASEP }(s,y_2), \end{aligned}$$

where $\nabla _{x_1}$ denotes the discrete (forward) gradient acting on the variable $x_1$. Similarly,

$$\begin{aligned}&{\mathbb {E}}\Big [Z_{\nabla }(t+s,x-1,x+1)\Big |{\mathscr {F}}(s)\Big ] \\&\quad = \sum _{y_1<y_2\in {\mathbb {Z}}} \varepsilon ^{-\frac{1}{2}}\nabla _{x_1}{\mathbf {V}}_{ASEP }\big ((y_1,y_2),(x_1,x_2); t\big )\big |_{(x_1,x_2)=(x-1,x+1)} Z_{ASEP }(s,y_1)Z_{ASEP }(s,y_2). \end{aligned}$$

From the perspective of duality, roughly speaking, the mechanism of decay in $t\rightarrow \infty$ arises from the discrete gradient $\nabla _{x_1}$. The semigroup ${\mathbf {V}}_{ASEP }$ behaves similar to (two copies of) the heat kernel, so that $\sum _{y_1<y_2}{\mathbf {V}}_{ASEP }\big ((y_1,y_2),(x_1,x_2);t\big ) = \mathcal O(1)$, and each gradient of ${\mathbf {V}}_{ASEP }$ effectively produces a factor of $t^{-1/2}$ for large t. Under the scaling $\varepsilon ^{-2}$ of time, namely $t^{-1/2}\approx \varepsilon ^1$, we expect to trade in $\varepsilon ^{-1/2}\nabla$ for $\varepsilon ^{-1/2}\varepsilon ^{1}=\varepsilon ^{1/2} \rightarrow 0$. In other words, the key heuristic is that the l.h.s of (A.4) behaves as

$$\begin{aligned} {\mathbb {E}}\Big [ \Big ( \varepsilon ^{2}\int _0^{\varepsilon ^{-2}T} {\widetilde{F}}_\varepsilon (t,x) dt \Big )^2\Big ] \approx \varepsilon ^4 \int _0^{\varepsilon ^{-2}T} \!\!\!\! \int _0^{\varepsilon ^{-2}T} \!\!\! \frac{\varepsilon ^{-1/2}}{\sqrt{|t_1-t_2|}}\,dt_1dt_2 \approx \varepsilon ^{\frac{1}{2}} \rightarrow 0.\quad \end{aligned}$$

(A.10)

Note that the identity (A.9) in its current form does not involve gradients of ${\mathbf {V}}_{ASEP }$. This identity can, however, be rewritten via Taylor expansion and summation by parts in a form that exposes the decay in $t\rightarrow \infty$. We do not perform this procedure here, and direct the readers to Lemma 7.5, where the exact same procedure in carried out for the stochastic 6V model. Specifically, the identity (7.28) therein holds with $({\mathbf {V}}_{ASEP },Z_{ASEP },{\mathbb {Z}})$ in place of $({\mathbf {V}}_\varepsilon ,Z,\Xi (s))$, and with $s,t\in [0,\infty )$ instead of ${\mathbb {Z}}_{\ge 0}$.

Given the preceding discussion, the task for bounding conditional expectations boils down to estimating the semigroup ${\mathbf {V}}_{ASEP }$ and its gradients. Thanks to Bethe ansatz, ${\mathbf {V}}_{ASEP }$ permits an explicit, analyzable formula in terms double contour integrals. Under weak asymmetry scaling, we write ${\mathbf {V}}_{ASEP }={\mathbf {V}}_{\varepsilon ,ASEP }$ and the formula reads

$$\begin{aligned} {\mathbf {V}}_{\varepsilon ,ASEP }\big ((y_1,y_2),(x_1,x_2);t\big ) :=&\oint _{\mathcal {C}_r}\oint _{\mathcal {C}_r} \Big ( z_1^{x_1-y_1}z_2^{x_2-y_2} -{\mathfrak {F}}^ASEP _\varepsilon (z_1,z_2) z_1^{x_2-y_1}z_2^{x_1-y_2} \Big )\\&\prod _{i=1}^2 \frac{e^{t{\mathfrak {E}}^ASEP _\varepsilon (z_i)}dz_i}{2\pi {\mathbf {i}}z_i}, \end{aligned}$$

where $\mathcal {C}_r$ is a counter-clockwise oriented, circular contour centered at origin, with a large enough radius r so as to include all poles of the integrand, and

$$\begin{aligned} {\mathfrak {F}}^ASEP _\varepsilon (z_1,z_2) := \frac{1+z_1z_2-(e^{-\frac{1}{2}{\sqrt{\varepsilon }}}+e^{\frac{1}{2}{\sqrt{\varepsilon }}}) z_2}{1+z_1z_2-(e^{-\frac{1}{2}{\sqrt{\varepsilon }}}+e^{\frac{1}{2}{\sqrt{\varepsilon }}}) z_1}, \quad {\mathfrak {E}}^ASEP _\varepsilon (z) := \sqrt{\ell r} \big (z+z^{-1}-2\big ). \end{aligned}$$

This contour integral formula is amenable to steepest decent analysis. Careful analysis jointly in $(x_1,x_2,y_1,y_2,t)$ should produce the relevant estimates on ${\mathbf {V}}_{\varepsilon ,ASEP }$ and its gradient (the result and proof should be analogous to Proposition 6.1). We do not pursue this analysis here.