Current fluctuations of the stationary ASEP and six-vertex model

Abstract

Our results in this paper are two-fold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time T , and show that they are of order T 1/3 along a characteristic line. Upon scaling by T 1/3 , we establish that these fluctuations converge to the long-time height fluctuations of the stationary KPZ equation, that is, to the Baik-Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and Spohn in 2005 (who established the same result for the TASEP), and the work of Balázs and Seppäläinen in 2010 (who established the T 1/3 scaling for the general ASEP).Second, we introduce a class of translation-invariant Gibbs measures that characterizes a oneparameter family of slopes for an arbitrary ferroelectric, symmetric six-vertex model. This family of slopes corresponds to what is known as the conical singularity (or tricritical point) in the free energy profile for the ferroelectric six-vertex model. We consider fluctuations of the height function of this model on a large grid of size T and show that they too are of order T 1/3 along a certain characteristic line; this confirms a prediction of Bukman and Shore from 1995 stating that the ferroelectric six-vertex model should exhibit KPZ growth at the conical singularity.Upon scaling the height fluctuations by T 1/3 , we again recover the Baik-Rains distribution in the large T limit. Recasting this statement in terms of the (asymmetric) stochastic six-vertex model confirms a prediction of Gwa and Spohn from 1992. 11 3. Observables for Models With Double-Sided

Half-Space Stationary Kardar–Parisi–Zhang Equation

“…For a very similar model (TASEP), the spatial correlations were investigated in [ 59 ]. Outside the class of free fermionic models, besides [ 30 , 31 ] that we have already discussed, let us also mention [ 60 ] which proved the one point convergence of ASEP height function towards the Baik–Rains distribution.…”

Tracy-Widom Asymptotics for a River Delta Model

“…Most of them can be realized as more or less direct degenerations of the higher-spin stochastic six-vertex model. This includes particle systems such as exclusion processes (q-TASEP [22,10,33,43] and other models [12,6,36,54]), directed polymers ( [17,21,18,32,40,42]), and the stochastic six-vertex model [3,1,11,19,24].…”

Nonexistence and uniqueness for pure states of ferroelectric six‐vertex models

“…In fact, except for its fourth part concerning frozen phases (whose analysis follows directly from definitions), no aspect of the above phase diagram for ferroelectric six‐vertex pure states had been mathematically proven until recently. To our knowledge, the only result in this direction concerns its second (KPZ) regime and appeared in [1, Appendix A.2], where a pure state $$\mu (s) = \mu _{s, t}$$ of any slope $$(s, t) \in \partial \mathfrak {H}$$ was introduced for the ferroelectric six‐vertex model. Additionally, [1] established both qualitative and quantitative properties for $$\mu (s)$$, which are considerably different from those for pure states of tiling models.…”

“…To our knowledge, the only result in this direction concerns its second (KPZ) regime and appeared in [1, Appendix A.2], where a pure state $$\mu (s) = \mu _{s, t}$$ of any slope $$(s, t) \in \partial \mathfrak {H}$$ was introduced for the ferroelectric six‐vertex model. Additionally, [1] established both qualitative and quantitative properties for $$\mu (s)$$, which are considerably different from those for pure states of tiling models. For instance, it was shown by Kenyon the latter are conformally invariant [23] with Gaussian free field fluctuations [24].…”

“…For instance, it was shown by Kenyon the latter are conformally invariant [23] with Gaussian free field fluctuations [24]. In contrast, the six‐vertex pure states $$\mu (s)$$ are quite anisotropic, exhibiting Baik–Rains fluctuations of exponent $$\frac{1}{3}$$ along a single direction and Gaussian fluctuations of exponent $$\frac{1}{2}$$ elsewhere [1]; this is an indication of the KPZ behavior manifested by these states. Essentially nothing (including existence and uniqueness) remains known about the conjecturally liquid states in the third regime of the phase diagram, where $$(s, t) \notin \overline{\mathfrak {H}}$$.…”