## Order of current variance and diffusivity in the asymmetric simple exclusion process.(English)Zbl 1200.60083

The (asymmetric) exclusion process is an interacting particle system, where each point of the integer lattice can be occupied by at most one particle. Each particle, after an independent exponential waiting time makes a coin flip and with probability $$p\in[0,1]$$ attempts to make a jump to the right and with probability $$q=1-p$$ to the left. If the destination is occupied, the jump is suppressed.
For each intensity $$\varrho\in(0,1)$$ of particles, the Bernoulli product measure with success probability $$\varrho$$ is an ergodic invariant measure for the process. Assume that the process is started with this distribution. It is well known that $$v^\varrho=(p-q)(1-2\varrho)$$ is the asymptotic speed at which perturbations travel.
Consider the net flow of particles $$J^v(t)$$ as seen from a traveller at speed $$v$$ up to time $$t$$. That is, $$J^v(t)$$ is the number of particles that were right of $$0$$ at time $$0$$ but are not right of $$vt$$ at time $$t$$ minus the number of particles that were left of $$0$$ at time $$t$$ but are not left of $$vt$$ at time $$t$$. P. A. Ferrari and L. R. G. Fontes [Ann. Probab. 22, No. 2, 820–832 (1994; Zbl 0806.60099)] showed that $$J^v(t)$$ fulfills a central limit theorem with rate $$t^{1/2}$$ and variance $$\sigma^2=\varrho(1-\varrho)|v^\varrho-v|$$. For the critical speed $$v=v^\varrho$$, the variance vanishes and it was conjectured that $$(J^{v^\varrho}(t)-\mathbf{E}[J^{v^\varrho}(t)])/t^{1/3}$$ converges to some nontrivial limit as $$t\to\infty$$.
In the paper under review, the authors show (Corollary 2.3) that $$\mathbf{Var}[J^{v^\varrho}(t)/t^{1/3}]$$ is bounded and bounded away from $$0$$. This is derived from a more general statement (Theorem 2.2) about the position $$Q(t)$$ of a so-called second class particle started at $$0$$. They show that for all $$m\in[1,3)$$ the $$m$$th moments of $$(Q(t)-v^\varrho t)/t^{2/3}$$ are bounded and bounded away from $$0$$.
The proofs are based on a clever coupling of three exclusion processes using the obvious common Poisson clocks.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics 60F99 Limit theorems in probability theory

### Keywords:

exclusion process; nondiffusive limit theorem

Zbl 0806.60099
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### References:

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