Condensation in zero-range processes with a fast rate

Abstract

We introduce a simple zero-range process with constant rates and one fast rate for a particular occupation number, which diverges with the system size. Surprisingly, this minor modification induces a condensation transition in the thermodynamic limit, where the structure of the condensed phase depends on the scaling of the fast rate. We study this transition and its dependence on system parameters in detail on a rigorous level using size-biased sampling. This approach generalizes to any particle system with stationary product measures, and the techniques used in this paper provide a foundation for a more systematic understanding of condensing models with a non-trivial condensed phase.

Figures (5)

• Figure 1: The density $R_L (\phi )$\ref{['rhol']} (dashed coloured lines) converges pointwise for all $\phi <1$ to $R(\phi )$ (black line) for $L\to\infty$\ref{['rzlim']} (left). The canonical current \ref{['cancurr']} (dotted lines) compares well with the grand-canonical prediction $\Phi_L (\rho )$\ref{['gcancurr']} (dashed lines). For $L\to\infty$ both converge to the same limit $\Phi (\rho )$ (black line), which is the inverse of $R(\phi )$\ref{['rzlim']}. Parameters are $A=2$, $w(n)=1$ for $n=0,1$ and $\theta_L =1/L$.
• Figure 2: Integrated stationary density profiles $S_x (\eta )=\sum_{k=1}^x \eta_x$ for a system with parameters \ref{['simple']} and several values for $\Theta =\theta_L /L$ confirm condensation with bulk density $\rho_c$\ref{['extend']} (indicated by grey lines). We have $A=2$, $L=8192$, $\rho =1$ with $\rho_c =1/2$ (left) and $A=5$, $L=1024$, $\rho =3$ and $4$ with $\rho_c =2$, as well as subcritical $\rho =1$ (right). The size of clusters is increasing with $\Theta$ in accordance with the scale $C_L$\ref{['scale']}.
• Figure 3: (Left) The tail of the size-biased empirical distribution $\overline{F}_{sb} (u):=\frac{1}{N}\sum_{x\in\Lambda} \eta_x \mathbbm{1}\{\eta_x >uC_L\}$ on scale $C_L$\ref{['scale']} for various values of $\Theta$ (coloured lines) compared to the theoretical prediction of Theorem \ref{['thm2']} (dashed black line) with $A=5$, $\rho =3$ or $A=2$, $\rho =1$. (Right) The corresponding tail of the conditioned empirical distribution of cluster sites $\overline{F}_c (u):=\sum_{x\in\Lambda} \mathbbm{1}\{\eta_x >uC_L\} /\sum_{x\in\Lambda} \mathbbm{1}\{\eta_x \geq A\}$ on scale $C_L$\ref{['scale']} shows exponential decay (dashed black line). Other parameters are $L=8192$ with $10$ realizations for $A=2$ and $L=4096$ with $100$ realizations for $A=5$.
• Figure 4: For a ZRP with rates \ref{['eq_rates']}, $A=2$ and $\rho_c =1/2$ in the scaling regime \ref{['lscaling']} with $\theta_N =N^\gamma$ we show the marginal tail distribution of a re-scaled occupation number $Z_x =\eta_x /N$. For $\gamma =2 >1$ (left) the mass condenses on a single lattice site and the distribution shows a plateau at level $1/L$ and a uniform distribution for occupation numbers up to $A$. For $\gamma <1$ (right) the empirical tail distributions of $Z_x$ (coloured lines) fit well with the limiting distributions \ref{['zxlim']} (dashed black lines) for various values of $L$ and $\gamma$, where $N=8192$. In each case $100$ realizations were used.
• Figure 5: For a ZRP with rates \ref{['pdzrp']}, $A=2$ and $\rho_c =1/2$, the recursively computed canonical current \ref{['cancurr']} (coloured dotted lines) and the grand-canonical current \ref{['gcancurr']} (coloured dashed lines) exhibit an overshoot compared to the asymptotic current (full black line) as seen on the left. On the right for $\rho=1$ the size-biased tail distributions $\overline{F}_{sb} (u)$ from recursively computed tails (dotted blue and red lines, analogous to \ref{['cancurr']}) and empirical tails from simulations (green line) compare well with an asymptotic Beta$(1,1)$ distribution (black dashed line), as is expected for Poisson-Dirichlet statistics with parameter $1$ (see chleboun2022poisson and feng2010poisson for details).

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Proposition 5
• proof
• Lemma 6
• proof
• Theorem 7