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ABSTRACT Given a reversible Markov chain on states, and another chain obtained by perturbing each row of by at most in total variation, we study the total variation distance between the two stationary distributions, . We show that for chains withcutoff, converges to 0, is asymptotically at most (with a sequence of perturbations for which convergence to this bound occurs), and converges to 1, respectively, if the product of and the mixing time of converges to 0, , and , respectively. This echoes recent results for specific random walks that exhibit cutoff, suggesting that cutoff is the key property underlying such results. Moreover, we show is maximized byrestart perturbations, for which “restarts” at a random state with probability at each step. Finally, we show thatpre‐cutoffis (almost) equivalent to a notion of “sensitivity to restart perturbations,” suggesting that chains with sharper convergence to stationarity are inherently less robust. 

Abstract We numerically study axion-U(1) inflation, focusing on the regime where the coupling between axions and gauge fields results in significant backreaction from the amplified gauge fields during inflation. These amplified gauge fields not only generate high-frequency gravitational waves (GWs), but also enhance spatial inhomogeneities in the axion field. GWs serve as key probe for constraining the coupling strength between the axion and gauge fields. We find that, when backreaction is important during inflation, the constraints on the coupling strength due to GW overproduction are relaxed compared to previous studies, in which backreaction matters only after inflation. Moreover, our results suggest that the probability density function (PDF) of axion fluctuations tends toward a Gaussian distribution even in cases where gauge field backreaction is important only after inflation. This aligns with previous studies where the same effect was observed for cases with strong backreaction during inflation. This finding can be crucial for future studies of primordial black hole (PBH) formation, which can further constrain the coupling strength. We also calculate the spectrum of the produced magnetic fields in this model and find that their strength is compatible with the observed lower limits.