We study exit laws from large balls in $\mathbb{Z}^{d}$, $d\geq3$, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary segments. As an application, we prove transience of the random walks in random environment.

Our work includes the results on isotropic random walks in random environment of Bolthausen and Zeitouni [Probab. Theory Related Fields 138 (2007) 581–645]. Since several proofs in Bolthausen and Zeitouni (2007) were incomplete, a somewhat different approach was given in the first author’s thesis [Long-time behavior of random walks in random environment (2013) Zürich Univ.]. Here, we extend this approach to certain anisotropic walks and provide a further step towards a fully perturbative theory of random walks in random environment.

In dynamical critical site percolation on the triangular lattice or bond percolation on $\mathbb{Z}^{2}$, we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster. We show that at a typical time with respect to this measure, the percolation configuration has the law of Kesten’s incipient infinite cluster. In the most technical result of this paper, we show that, on the other hand, at the first exceptional time, the law of the configuration is different. We believe that the two laws are mutually singular, but do not show this. We also study the collapse of the infinite cluster near typical exceptional times and establish a relation between static and dynamic exponents, analogous to Kesten’s near-critical relation.

We study the nonlinear stochastic heat equation in the spatial domain $\mathbb{R}$, driven by space–time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on $\mathbb{R}$, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall’s lemma, by keeping tight control over moments in the Picard iteration scheme. Upper bounds on all $p$th moments $(p\ge2)$ are obtained as well as a lower bound on second moments. These bounds become equalities for the parabolic Anderson model when $p=2$. We determine the growth indices introduced by Conus and Khoshnevisan [Probab. Theory Related Fields 152 (2012) 681–701].

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite-dimensional unitary group and their $q$-deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE-eigenvalues distribution in the limit. We also investigate similar behavior for alternating sign matrices (equivalently, six-vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in $O(n=1)$ dense loop model.

Certain countably and finitely additive measures can be associated to a given nonnegative supermartingale. Under weak assumptions on the underlying probability space, existence and (non)uniqueness results for such measures are proven.

The coalescing Brownian flow on $\mathbb{R}$ is a process which was introduced by Arratia [Coalescing Brownian motions on the line (1979) Univ. Wisconsin, Madison] and Tóth and Werner [Probab. Theory Related Fields 111 (1998) 375–452], and which formally corresponds to starting coalescing Brownian motions from every space–time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. This result holds under a finite variance assumption and is thus optimal. In previous works by Fontes et al. [Ann. Probab. 32 (2004) 2857–2883], Newman et al. [Electron. J. Probab. 10 (2005) 21–60], the topology and state-space required a moment of order $3-\varepsilon$ for this convergence to hold. The proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work—in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the noncrossing property.

Self-destructive percolation with parameters $p,\delta$ is obtained by taking a site percolation configuration with parameter $p$, closing all sites belonging to infinite clusters, then opening every closed site with probability $\delta$, independently of the rest. Call $\theta(p,\delta)$ the probability that the origin is in an infinite cluster in the configuration thus obtained.

For two-dimensional lattices, we show the existence of $\delta>0$ such that, for any $p>p_{c}$, $\theta(p,\delta)=0$. This proves the conjecture of van den Berg and Brouwer [Random Structures Algorithms 24 (2004) 480–501], who introduced the model. Our results combined with those of van den Berg and Brouwer [Random Structures Algorithms 24 (2004) 480–501] imply the nonexistence of the infinite parameter forest-fire model. The methods herein apply to site and bond percolation on any two-dimensional planar lattice with sufficient symmetry.

The noise sensitivity of a Boolean function describes its likelihood to flip under small perturbations of its input. Introduced in the seminal work of Benjamini, Kalai and Schramm [Inst. Hautes Études Sci. Publ. Math. 90 (1999) 5–43], it was there shown to be governed by the first level of Fourier coefficients in the central case of monotone functions at a constant critical probability $p_{c}$.

Here we study noise sensitivity and a natural stronger version of it, addressing the effect of noise given a specific witness in the original input. Our main context is the Erdős–Rényi random graph, where already the property of containing a given graph is sufficiently rich to separate these notions. In particular, our analysis implies (strong) noise sensitivity in settings where the BKS criterion involving the first Fourier level does not apply, for example, when $p_{c}\to0$ polynomially fast in the number of variables.

Let $\theta_{1},\ldots,\theta_{n}$ be random variables from Dyson’s circular $\beta$-ensemble with probability density function $\operatorname{Const}\cdot\prod_{1\leq j<k\leq n}|e^{i\theta_{j}}-e^{i\theta_{k}}|^{\beta}$. For each $n\geq2$ and $\beta>0$, we obtain some inequalities on $\mathbb{E}[p_{\mu}(Z_{n})\overline{p_{\nu}(Z_{n})}]$, where $Z_{n}=(e^{i\theta_{1}},\ldots,e^{i\theta_{n}})$ and $p_{\mu}$ is the power-sum symmetric function for partition $\mu$. When $\beta=2$, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: $\lim_{n\to\infty}\mathbb{E}[p_{\mu}(Z_{n})\overline{p_{\nu}(Z_{n})}]=\delta_{\mu\nu}(\frac{2}{\beta})^{l(\mu)}z_{\mu}$ for any $\beta>0$ and partitions $\mu,\nu$; $\lim_{m\to\infty}\mathbb{E}[|p_{m}(Z_{n})|^{2}]=n$ for any $\beta>0$ and $n\geq2$, where $l(\mu)$ is the length of $\mu$ and $z_{\mu}$ is explicit on $\mu$. These results apply to the three important ensembles: COE ($\beta=1$), CUE ($\beta=2$) and CSE ($\beta=4$). We further examine the nonasymptotic behavior of $\mathbb{E}[|p_{m}(Z_{n})|^{2}]$ for $\beta=1,4$. The central limit theorems of $\sum_{j=1}^{n}g(e^{i\theta_{j}})$ are obtained when (i) $g(z)$ is a polynomial and $\beta>0$ is arbitrary, or (ii) $g(z)$ has a Fourier expansion and $\beta=1,4$. The main tool is the Jack function.

We introduce a technique using nonbacktracking random walk for estimating the spectral radius of simple random walk. This technique relates the density of nontrivial cycles in simple random walk to that in nonbacktracking random walk. We apply this to infinite Ramanujan graphs, which are regular graphs whose spectral radius equals that of the tree of the same degree. Kesten showed that the only infinite Ramanujan graphs that are Cayley graphs are trees. This result was extended to unimodular random rooted regular graphs by Abért, Glasner and Virág. We show that an analogous result holds for all regular graphs: the frequency of times spent by simple random walk in a nontrivial cycle is a.s. 0 on every infinite Ramanujan graph. We also give quantitative versions of that result, which we apply to answer another question of Abért, Glasner and Virág, showing that on an infinite Ramanujan graph, the probability that simple random walk encounters a short cycle tends to 0 a.s. as the time tends to infinity.

We prove pathwise nonuniqueness in the stochastic partial differential equations (SPDEs) for some one-dimensional super-Brownian motions with immigration. In contrast to a closely related case investigated by Mueller, Mytnik and Perkins [Ann. Probab. 42 (2014) 2032–2112], the solutions of the present SPDEs are assumed to be nonnegative and have very different properties including uniqueness in law. In proving possible separation of solutions, we derive delicate properties of certain correlated approximating solutions, which is based on a novel coupling method called continuous decomposition. In general, this method may be of independent interest in furnishing solutions of SPDEs with intrinsic adapted structure.

We prove two theorems concerning extreme values of general Gaussian fields. Our first theorem concerns the phenomenon of multiple peaks. Consider a centered Gaussian field whose sites have variance at most $1$, and let $\rho$ be the standard deviation of its supremum. A theorem of Chatterjee states that when such a Gaussian field is superconcentrated (i.e., $\rho\ll1$), it typically attains values near its maximum on multiple almost-orthogonal sites and is said to exhibit multiple peaks. We improve his theorem in two respects: (i) the number of peaks attained by our bound is of the order $\exp(c/\rho^{2})$ (as opposed to Chatterjee’s polynomial bound in $1/\rho$) and (ii) our bound does not assume that the correlations are nonnegative. We also prove a similar result based on superconcentration of the free energy. As primary applications, we infer that for the S–K spin glass model on the $n$-hypercube and directed polymers on $\mathbb{Z}_{n}^{2}$, there are polynomially (in $n$) many almost-orthogonal sites that achieve values near their respective maxima.

Our second theorem gives an upper bound on moderate deviation for the supremum of a general Gaussian field. While the Gaussian isoperimetric inequality implies a sub-Gaussian concentration bound for the supremum, we show that the exponent in that bound can be improved under the assumption that the expectation of the supremum is of the same order as that of the independent case.

The authors of [Ann. Henri Poincaré 16 (2015) 691–708] introduced a multi-species version of the Sherrington–Kirkpatrick model and suggested the analogue of the Parisi formula for the free energy. Using a variant of Guerra’s replica symmetry breaking interpolation, they showed that, under certain assumption on the interactions, the formula gives an upper bound on the limit of the free energy. In this paper we prove that the bound is sharp. This is achieved by developing a new multi-species form of the Ghirlanda–Guerra identities and showing that they force the overlaps within species to be completely determined by the overlaps of the whole system.