Test of replica theory: Thermodynamics of two-dimensional model systems with quenched disorder

We study the statistics of thermodynamic quantities in two related systems with quenched disorder: A (1+1)-dimensional planar lattice of elastic lines in a random potential and the two-dimensional random bond dimer model. The first system is examined by a replica-symmetric Bethe Ansatz (RBA) while the latter is studied numerically by a polynomial algorithm which circumvents slow glassy dynamics. We establish a mapping of the two models which allows for a detailed comparison of RBA predictions and simulations. Over a wide range of disorder strength, the effective lattice stiffness and cumulants of various thermodynamic quantities in both approaches are found to agree excellently. Our comparison provides a detailed quantitative confirmation of the replica approach and renders the planar line lattice a unique testing ground for concepts in random systems.