Numerical Investigations of Non-Reciprocal and Frustrated Systems

This PhD thesis addresses two distinct problems in classical statistical physics: how vision-dependent interactions shape collective order, and how machine-learning techniques combined with renormalization-group ideas can improve the sampling of frustrated systems. The first and main part of the thesis focuses on non-reciprocal interactions in active matter, while the second part applies a wavelet-based approach to configuration sampling in statistical physics. In particular, the first part investigates two-dimensional XY models with vision-dependent interactions. By introducing and comparing non-reciprocal and reciprocal realizations of these models, we show that the emergence of long-range orientational order is not caused by non-reciprocity alone. Instead, it primarily originates from the coupling between the spin orientations and the underlying lattice—a consequence of the vision-cone interaction. This coupling generates effective angular anisotropies, which selects preferred orientations. One of the reciprocal models also displays an unexpected order-by-disorder transition, in which thermal fluctuations strengthen orientational order. We then consider non-reciprocal models with smooth anisotropic interaction kernels and show that the selected pinning directions depend on both the kernel and the kind of terms retained in the microscopic dynamics. In particular, as the interaction strength of a spin with the neighbors depends on its orientation relative to the bonds connecting to them, the Langevin dynamics naturally gives rise to two distinct contributions, which we called reactive and proactive. The reactive term describes the response of a spin to the instantaneous orientations of the currently perceived neighbors, whereas the proactive term accounts for the gradient of the interaction strength. We clarify their different roles in producing pinning along specific lattice directions and investigate how this pinning appears in both local and global equations of motion. Their possible implications for the stability of ordered phases and for off-lattice active systems are also discussed. The second part of the thesis applies the wavelet conditional renormalization group to the sampling of a frustrated soft-spin biaxial next-nearest-neighbor Ising model, for which conventional cluster algorithms cannot provide efficient sampling. The method decomposes microscopic configurations into coarse fields and wavelet coefficients and learns their conditional distributions independently at subsequent scales. At the Ising-like critical point, the conditional sampling dynamics shows no detectable critical slowing down. The reconstructed configurations reproduce local statistics and two-point correlation functions across most of the phase diagram, while discrepancies in the microscopic energy distribution reveal limitations associated with the finite-dimensional parametrization of the conditional energies. These limitations can, in principle, be reduced systematically by enriching the operator basis. This provides an interpretable route for improving the model, in contrast to more standard machine-learning approaches whose internal representations often remain difficult to understand.