  Nonequilibrium reaction networks (NRNs) underlie most biological functions.
Despite their diverse dynamic properties, NRNs share the signature
characteristics of persistent probability fluxes and continuous energy
dissipation even in the steady state. Dynamics of NRNs can be described at
different coarse-grained levels. Our previous work showed that the apparent
energy dissipation rate at a coarse-grained level follows an inverse power law
dependence on the scale of coarse-graining. The scaling exponent is determined
by the network structure and correlation of stationary probability fluxes.
However, it remains unclear whether and how the (renormalized) flux correlation
varies with coarse-graining. Following Kadanoff's real space renormalization
group (RG) approach for critical phenomena, we address this question by
developing a State-Space Renormalization Group (SSRG) theory for NRNs, which
leads to an iterative RG equation for the flux correlation function. In square
and hypercubic lattices, we solve the RG equation exactly and find two types of
fixed point solutions: a family of nontrivial fixed points where the
correlation exhibits power-law decay and a trivial fixed point where the
correlation vanishes beyond the nearest neighbors. The power-law fixed point is
stable if and only if the power exponent is less than the lattice dimension
$n$. Consequently, the correlation function converges to the power-law fixed
point only when the correlation in the fine-grained network decays slower than
$r^{-n}$ and to the trivial fixed point otherwise. If the flux correlation in
the fine-grained network contains multiple stable solutions with different
exponents, the RG iteration dynamics select the fixed point solution with the
smallest exponent. We also discuss a possible connection between the RG flows
of flux correlation with those of the Kosterlitz-Thouless transition.
