Types of Initial data

Initial data

• curved or step: $X_{0}(i)=-i,i\geq 1$.
• periodic: $X_{0}(i)=2i,i\in \mathbb {Z}$.
• stationary for ASEP: sampling $X_{0}(i)$ from a product Bernouli measure $\nu _{\rho }$ defined as follows by marginals for $n_{i}\in \{0,1\}$

$\nu _{\rho }(X(i_{1})=n_{1},...,X(i_{m})=n_{m})=\prod _{k=1}^{m}(1_{n_{k}=1}\rho (i_{k})+1_{n_{k}=0}(1-\rho (i_{k}))),$

for density function $\rho :\mathbb {Z} \to [0,1]$.

• Half-stationary initial data : we let $X_{0}(i)=i,i\geq 1$ and for $i\leq 0$, we define $X_{0}(i)$ according to a Markov chain $s_{i}$ and parameters $\alpha \in (0,\infty ),q\in (0,1)$

$P(s_{i-1}=s_{i}+1)={\frac {q^{s_{i}}}{\alpha +q^{s_{i}}}}{\text{ and }}P(s_{i-1}=s_{i}-1)={\frac {\alpha }{\alpha +q^{s_{i}}}}.$