  The experimental measurement of correlation functions and critical exponents
in disordered systems is key to testing renormalization group (RG) predictions.
We mechanically unzip single DNA hairpins with optical tweezers, an
experimental realization of the diffusive motion of a particle in a
one-dimensional random force field, known as the Sinai model. We measure the
unzipping forces $F_w$ as a function of the trap position $w$ in equilibrium
and calculate the force-force correlator $\Delta_m(w)$, its amplitude, and
correlation length, finding agreement with theoretical predictions. We study
the universal scaling properties since the effective trap stiffness $m^2$
decreases upon unzipping. Fluctuations of the position of the base pair at the
unzipping junction $u$ scales as $u \sim m^{-\zeta}$, with a roughness exponent
$ \zeta=1.34\pm0.06$, in agreement with the analytical prediction $\zeta =
\frac{4}{3}$. Our study provides a single-molecule test of the functional RG
approach for disordered elastic systems in equilibrium.
