Universal stable fixed points for the renormalisation transformation

Let $R$ be the renormalisation transformation on point fields of concentration points and shocks, and suppose the density decays as $n^{-\alpha}$ for a critical exponent $\alpha\in(0,1)$. Renormalisation fixed-point conjecture. For every $\alpha\in(0,1)$, $R$ has a fixed point with density scaling as $n^{-\alpha}$; these fixed points are stable except for one neutral direction corresponding to the parameter $\alpha$. Consequently, every random process with density decay $n^{-\alpha}$ converges under $R$ to the $\alpha$-fixed point, which supplies universal asymptotic statistics for all systems with exponent $\alpha$. The claim is a proposed renormalisation description of the KPZ and related scaling regimes, and no resolution is supplied.