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In this article, we establish the variational principle of the affinity exponent of Borel Anosov representations. We also establish such a principle of the Rauzy gasket. In Li-Pan-Xu, they obtain a dimension formula of the stationary measures on $\mathbb{P}(\mathbb{R}^3)$. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in $\mathrm{SL}_3(\mathbb{R})$ and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to $1.5$., Comment: We add an appendix where we prove the Hausdorff dimension of Rauzy gasket is at least $1.5$

We establish new results on the dimensional properties of measures and invariant sets associated to random walks and group actions by circle diffeomorphisms. This leads to several dynamical applications. Among the applications, we show, strengthening of a recent result of Deroin-Kleptsyn-Navas [24], that the minimal set of a finitely generated group of real-analytic circle diffeomorphisms, if exceptional, must have Hausdorff dimension less than one. Moreover, if the minimal set contains a fixed point of multiplicity k + 1 of an diffeomorphism of the group, then its Hausdorff dimension must be greater than k/(k + 1). These results generalize classical results about Fuchsian group actions on the circle to non-linear settings. This work is built on three novel components, each of which holds its own interest: a structure theorem for smooth random walks on the circle, several dimensional properties of smooth random walks on the circle and a dynamical generalization of the critical exponent of Fuchsian groups., Comment: In v3, we add an appendix consists of an example which illustrates the sharpness of our main theorem. Namely we construct an example show that if the $C^\omega$ assumption in (1). of Main theorem is replaced by $C^\infty$ assumption, then the Hausdorff dimension of the exceptional minimal set can indeed reach one