Instability of condensation in the zero-range process with random interaction

The zero-range process is a stochastic interacting particle system that is known to exhibit a condensation transition. We present a detailed analysis of this transition in the presence of quenched disorder in the particle interactions. Using rigorous probabilistic arguments, we show that disorder changes the critical exponent in the interaction strength below which a condensation transition may occur. The local critical densities may exhibit large fluctuations, and their distribution shows an interesting crossover from exponential to algebraic behavior.

Figure 1

(Color online) Change of the phase diagram under random perturbations of the jump rates (1). Disorder changes the critical interaction exponent from 1 to $1∕2$ and leads to a critical density that depends on the system size. Inside the red-shaded region, condensation occurs above a nonzero critical density for $b>0$, and for $b<0$ [see (2)] and negative $\sigma$ the critical density is zero.

Figure 2

(Color online) Typical realizations of $\sqrt{n}{\xi }_x\left(n\right)$ (drawn with ${\delta }^2=1$) leave the area enclosed by the dashed parabola only finitely many times, but cross the full line $-\frac{b}{1-\sigma }{n}^{1-\sigma }$ for $\sigma >1∕2$ infinitely often (here $\sigma =0.75$, $b=1$).

Figure 3

(Color online) Background density ${\rho }_{\mathit{bg}}$ Eq. (17), as a function of $N∕L$ for $b=1.2$. Data for fixed disorder with ${\delta }^2=1∕12$ and $L=1024$ show condensation for $\sigma =0.2$ and no condensation for $\sigma =0.8$. Data for ${\delta }^2=0$ are shown by open symbols; dotted lines indicate the critical densities. Data points are averages over 100 MC samples with errors of the size of the symbols.

Figure 4

(Color online) Cumulative tail of the distribution of ${\rho }_c^x$ from $5\times {10}^5$ independent numerical calculations of (14) with $b=1$, where infinite sums have been cut off at $n={10}^6$. The solid line corresponds to a cumulative exponent $-1$. The data exhibit power-law decays and for some parameters also large exponential cutoffs: e.g. $\sim e^{-r∕{10}^5}$ for $\sigma =0.45$, ${\delta }^2=1$ fitted by a dashed line. The expectations are finite but may be very large, $\mathbb{E}\left({\rho }_c^x\right)=1.60$ (△), 2.55 (◆), 1106 (×).