In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy–Smirnov formula. This theorem, together with the introduction of Schramm–Loewner Evolution and techniques developed over the years in percolation, allow precise descriptions of the critical and near-critical regimes of the model. This survey aims to describe the different steps leading to the proof that the infinite-cluster density $\theta(p)$ for site percolation on the triangular lattice behaves like $(p-p_{c})^{5/36+o(1)}$ as $p\searrow p_{c}=1/2$.

Consider a mathematical model with a finite number of random parameters. Variance based sensitivity analysis provides a framework to characterize the contribution of the individual parameters to the total variance of the model response. We consider the spectral methods for variance based sensitivity analysis which utilize representations of square integrable random variables in a generalized polynomial chaos basis. Taking a measure theoretic point of view, we provide a rigorous and at the same time intuitive perspective on the spectral methods for variance based sensitivity analysis. Moreover, we discuss approximation errors incurred by fixing inessential random parameters, when approximating functions with generalized polynomial chaos expansions.

Let $X_{1},X_{2},\ldots,$ be independent random variables with $EX_{i}=0$ and write $S_{n}=\sum_{i=1}^{n}X_{i}$ and $V_{n}^{2}=\sum_{i=1}^{n}X_{i}^{2}$. This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum $S_{n}/V_{n}$. Other self-normalized limit theorems are also briefly discussed.