  In principle, the probability of configurations, determined by the system's
partition function or wave function, encapsulates essential information about
phases and phase transitions. Despite the exponentially large configuration
space, we show that the generic correlation of distances between
configurations, with a degree of freedom proportional to the lattice size, can
probe phase transitions using importance sampling procedures like Monte Carlo
simulations. The distribution of sampled distances varies significantly across
different phases, suggesting universal critical behavior for uncertainty and
participation entropy. For various classical spin models with different phases
and transitions, finite-size analysis based on these quantities accurately
identifies phase transitions and critical points. Notably, in all cases, the
critical exponent derived from the uncertainty of distances equals the
anomalous dimension governing real-space correlation decay. Thus, configuration
space correlations, defined by distance uncertainties, share the same decay
ratio as real-space correlations, determining the universality class of phase
transitions. This work applies to diverse lattice models with different local
degrees of freedom, e.g., two levels for Ising-like models, discrete
multi-levels for q-state clock models, and continuous local levels for the XY
model, offering a robust, alternative method for understanding complex phases
and transitions.
