Noise sensitivity and percolation beyond the classical framework

In this thesis, we study critical properties of models for which the classical framework of study is not applicable. For instance, the phenomenon of sharpness is well understood for Bernoulli percolation, but removing the independence hypothesis breaks down most of the standard techniques. A similar situation arises when the perturbations studied in noise sensitivity questions exhibit more local structures, differing from the typical i.i.d. resampling dynamics, causing the discrete Fourier analysis tools to fail. In pushing these hypotheses, we investigate new proof strategies and methodologies that can be applied in more general contexts, mirroring the complexities often observed in models arising from the physics literature. Chapter 2 focuses on a new approach to explore the sensitivity to small perturbations of real-valued observables. In particular, we study whether being above the median is a noise sensitive property for first-passage percolation. We are able to positively answer this question when we consider first-passage paths crossing an n x n-square horizontally and whose vertical fluctuations are smaller than n¹⧸²² In Chapter 3, we further develop the notions of dynamical noise sensitivity and stability for two different classes of perturbations in the voter model dynamics. In the first one, an exclusion process is performed on the edge selections, while in the second, independent Brownian motions are applied to such a sequence. In both cases, we prove stability of the consensus opinion, provided the noise is run for a short amount of time. Finally, in Chapter 4, we study the frog model as a long-range-dependent percolation model with parameters 𝜆 (the particle density) and t (the particle lifespan). We establish the existence of the phase transition with respect to each parameter for non-amenable graphs of bounded degrees and quasi-transitive graphs of superlinear polynomial growth. We also prove the sharpness of the phase transition for transitive graphs.