We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent $\tau$. We investigate the case where $\tau\in(3,4)$, so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by $n^{-(\tau-2)/(\tau-1)}$, converge to hitting times of a “thinned” Lévy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812–854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1–59].

Our results should be contrasted to the case $\tau>4$, so that the third moment is finite. There, instead, the sizes of the components rescaled by $n^{-2/3}$ converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous [Ann. Probab. 25 (1997) 812–854] for the Erdős–Rényi random graph and extended to the present setting in Bhamidi, van der Hofstad and van Leeuwaarden [Electron. J. Probab. 15 (2010) 1682–1703] and Turova [(2009) Preprint].

For any two-dimensional nearest neighbor shift of finite type $X$ and any integer $n\geq1$, one can define the horizontal strip shift $H_{n}(X)$ to be the set of configurations on $\mathbb{Z}\times\{1,\ldots,n\}$ which do not contain any forbidden pairs of adjacent letters for $X$. It is always the case that the sequence $h^{\mathrm{top} }(H_{n}(X))/n$ of normalized topological entropies of the strip shifts converges to $h^{\mathrm{top} }(X)$, the topological entropy of $X$. In this paper, we combine ergodic theoretic techniques with methods from percolation theory and interacting particle systems to show that for the two-dimensional hard square shift $\mathcal{H}$, the sequence $h^{\mathrm{top} }(H_{n+1}(\mathcal{H}))-h^{\mathrm{top} }(H_{n}(\mathcal{H}))$ also converges to $h^{\mathrm{top} }(\mathcal{H})$, and that the rate of convergence is at least exponential. As a corollary, we show that $h^{\mathrm{top} }(\mathcal{H})$ is computable to any tolerance $\varepsilon$ in time polynomial in $1/\varepsilon$. We also show that this phenomenon is not true in general by defining a block gluing two-dimensional nearest neighbor shift of finite type $Y$ for which $h^{\mathrm{top} }(H_{n+1}(Y))-h^{\mathrm{top} }(H_{n}(Y))$ does not even approach a limit.

We consider continuous time random interlacements on ${\mathbb{Z}}^{d}$, $d\ge3$, and characterize the distribution of the corresponding stationary random field of occupation times. When $d=3$, we relate this random field to the two-dimensional Gaussian free field pinned at the origin by looking at scaled differences of occupation times of long rods by random interlacements at appropriately tuned levels. In the main asymptotic regime, a scaling factor appears in the limit, which is independent of the free field, and distributed as the time-marginal of a zero-dimensional Bessel process. For arbitrary $d\ge3$, we also relate the field of occupation times at a level tending to infinity, to the $d$-dimensional Gaussian free field.

We analyze from the viewpoint of an abstract Markov operator recent results by Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as a condition on a Wiener chaos to have a distribution close to Gaussian. In particular, we are led to introduce a notion of chaos associated to a Markov operator through its iterated gradients and present conditions on the (pure) point spectrum for a sequence of chaos eigenfunctions to converge to a Gaussian distribution. Convergence to gamma distributions may be examined similarly.

Relative complexity measures the complexity of a probability preserving transformation relative to a factor being a sequence of random variables whose exponential growth rate is the relative entropy of the extension. We prove distributional limit theorems for the relative complexity of certain zero entropy extensions: RWRSs whose associated random walks satisfy the $\alpha$-stable CLT ($1<\alpha\le2$). The results give invariants for relative isomorphism of these.

Suppose that $d\geq2$ and $\alpha\in(1,2)$. Let $D$ be a bounded $C^{1,1}$ open set in ${\mathbb{R}}^{d}$ and $b$ an ${\mathbb{R}}^{d}$-valued function on ${\mathbb{R}}^{d}$ whose components are in a certain Kato class of the rotationally symmetric $\alpha$-stable process. In this paper, we derive sharp two-sided heat kernel estimates for $\mathcal{L} ^{b}=\Delta^{\alpha/2}+b\cdot\nabla$ in $D$ with zero exterior condition. We also obtain the boundary Harnack principle for $\mathcal{L} ^{b}$ in $D$ with explicit decay rate.

We study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition. For any locally compact unimodular group $G$ and any positive function $\varrho\,\colon\,G\rightarrow [0,+\infty]$, we introduce a function $\Phi_{G,\varrho}$ which describes the fastest possible decay of $n\mapsto\phi^{(2n)}(e)$ when $\phi$ is a symmetric continuous probability density such that $\int\varrho\phi$ is finite. We estimate $\Phi_{G,\varrho}$ for a variety of groups $G$ and functions $\varrho$. When $\varrho$ is of the form $\varrho=\rho\circ\delta$ with $\rho\,\colon\,[0,+\infty)\rightarrow [0,+\infty)$, a fixed increasing function, and $\delta\,\colon\,G\rightarrow [0,+\infty)$, a natural word length measuring the distance to the identity element in $G$, $\Phi_{G,\varrho}$ can be thought of as a group invariant.

We consider a family of random trees satisfying a Markov branching property. Roughly, this property says that the subtrees above some given height are independent with a law that depends only on their total size, the latter being either the number of leaves or vertices. Such families are parameterized by sequences of distributions on partitions of the integers that determine how the size of a tree is distributed in its different subtrees. Under some natural assumption on these distributions, stipulating that “macroscopic” splitting events are rare, we show that Markov branching trees admit the so-called self-similar fragmentation trees as scaling limits in the Gromov–Hausdorff–Prokhorov topology.

The main application of these results is that the scaling limit of random uniform unordered trees is the Brownian continuum random tree. This extends a result by Marckert–Miermont and fully proves a conjecture by Aldous. We also recover, and occasionally extend, results on scaling limits of consistent Markov branching models and known convergence results of Galton–Watson trees toward the Brownian and stable continuum random trees.

In a recent and celebrated article, Smirnov [Ann. of Math. (2) 172 (2010) 1435–1467] defines an observable for the self-dual random-cluster model with cluster weight $q=2$ on the square lattice $\mathbb{Z} ^{2}$, and uses it to obtain conformal invariance in the scaling limit. We study this observable away from the self-dual point. From this, we obtain a new derivation of the fact that the self-dual and critical points coincide, which implies that the critical inverse temperature of the Ising model equals $\frac{1}{2}\log(1+\sqrt{2})$. Moreover, we relate the correlation length of the model to the large deviation behavior of a certain massive random walk (thus confirming an observation by Messikh [The surface tension near criticality of the 2d-Ising model (2006) Preprint]), which allows us to compute it explicitly.

In this paper we define distributions on moment spaces corresponding to measures on the real line with an unbounded support. We identify these distributions as limiting distributions of random moment vectors defined on compact moment spaces and as distributions corresponding to random spectral measures associated with the Jacobi, Laguerre and Hermite ensemble from random matrix theory. For random vectors on the unbounded moment spaces we prove a central limit theorem where the centering vectors correspond to the moments of the Marchenko–Pastur distribution and Wigner’s semi-circle law.