Self-Similar Bridge Between Regular and Critical Regions

In statistical and nonlinear systems, two qualitatively distinct parameter regions are typically identified: the regular region, which is characterized by smooth behavior of key quantities; and the critical region, where these quantities exhibit singularities or strong fluctuations. Due to their starkly different properties, those regions are often perceived as being weakly related, if ever. However, here, we demonstrate that these regions are intimately connected, specifically showing how they have a relationship that can be explicitly revealed using self-similar approximation theory. The framework considered enables the prediction of observable quantities near the critical point based on information from the regular region, and vice versa. Remarkably, the method relies solely on asymptotic expansions with respect to a parameter, regardless of whether the expansion originates in the regular or critical region. The mathematical principles of self-similar theory remain consistent across both cases. We illustrate this consistency by extrapolating from the regular region to predict the existence, location, and critical indices of a critical point of an equation of state for a statistical system, even when no direct information about the critical region is available. Conversely, we explore extrapolation from the critical to the regular region in systems with discrete scale invariance, where log-periodic oscillations in observables introduce additional complexity. The findings provide insights and solutions applicable to diverse phenomena, including material fracture, stock market crashes, and earthquake forecasting.