Constraint-Driven Condensation in Large Fluctuations of Linear Statistics

Condensation is the phenomenon whereby one of a sum of random variables contributes a finite fraction to the sum. It is manifested as an aggregation phenomenon in diverse physical systems such as coalescence in granular media, jamming in traffic, and gelation in networks. We show here that the same condensation scenario, which normally happens only if the underlying probability distribution has tails heavier than exponential, can occur for light-tailed distributions in the presence of additional constraints. We demonstrate this phenomenon on the sample variance, whose probability distribution conditioned on the particular value of the sample mean undergoes a phase transition. The transition is manifested by a change in behavior of the large deviation rate function.

Figure 1

Phase diagram in the $\mu -\sigma$ plane for $f(m)=r{e}^{-rm}$ and $p=1/2$. Shaded area $\sigma <{\mu }^2$ is forbidden owing to Jensen’s inequality.

Figure 2

Expression (11) from large deviation theory (LDT, dashed line) compared to $\mathrm{\Pi }(V;s)$ (ZRP, solid line). ${\mathrm{\Pi }}_L(V;s)$ was obtained numerically for $L=200$ and $r+s=-\ln (0.9)$ via recursion relation ${\mathrm{\Pi }}_L(V;s)=\sum _{v}w(v;s){\mathrm{\Pi }}_{L-1}(V-v;s)$ valid for integer random variables [11], where $w(v;s)$ is discrete version of Weibull distribution, $w(v;s)=\exp (-(r+s)\sqrt{v})-\exp (-(r+s)\sqrt{v+1})$, and $v$ is an integer. For large $v$, $w(v;s)$ reduces to the continuous Weibull distribution defined earlier.

Figure 3

Phase diagram in the $\mu -\sigma$ plane for $p=1/2$ and the Pareto distribution $f(m)=(\gamma -1)/m^{\gamma }$, $m\ge 1$, where $\gamma =7/2$. Shaded area $\sigma <{\mu }^2$ is forbidden owing to Jensen’s inequality.