This work extends to dimension $d\geq3$ the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on $\mathbb{Z}^{d}$ evolving with the Metropolis dynamics under a fixed small positive magnetic field $h$ starting from the minus phase. When the inverse temperature $\beta$ goes to $\infty$, the relaxation time of the system, defined as the time when the plus phase has invaded the origin, behaves like $\exp(\beta\kappa_{d})$. The value $\kappa_{d}$ is equal to

\[\kappa_{d}=\frac{1}{d+1}(\Gamma_{1}+\cdots+\Gamma_{d}),\]

where $\Gamma_{i}$ is the energy of the $i$-dimensional critical droplet of the Ising model at zero temperature and magnetic field $h$.

Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed is zero. We investigate the total progeny of the killed branching random walk and give their precise tail distribution both in the critical and subcritical cases, which solves an open problem of Aldous [Power laws and killed branching random walks, http://www.stat.berkeley.edu/~aldous/Research/OP/brw.html].

A $U$-statistic of a Poisson point process is defined as the sum $\sum f(x_{1},\ldots,x_{k})$ over all (possibly infinitely many) $k$-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener–Itô chaos expansion of such a functional is computed and used to derive a formula for the variance. Central limit theorems for $U$-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable. As applications, the intersection process of Poisson hyperplanes and the length of a random geometric graph are investigated.

We study the biased random walk in positive random conductances on $\mathbb{Z}^{d}$. This walk is transient in the direction of the bias. Our main result is that the random walk is ballistic if, and only if, the conductances have finite mean. Moreover, in the sub-ballistic regime we find the polynomial order of the distance moved by the particle. This extends results obtained by Shen [Ann. Appl. Probab. 12 (2002) 477–510], who proved positivity of the speed in the uniformly elliptic setting.

We study the nonlinear operator of mapping the terminal value $\xi$ to the corresponding minimal supersolution of a backward stochastic differential equation with the generator being monotone in $y$, convex in $z$, jointly lower semicontinuous and bounded below by an affine function of the control variable $z$. We show existence, uniqueness, monotone convergence, Fatou’s lemma and lower semicontinuity of this operator. We provide a comparison principle for minimal supersolutions of BSDEs.

We prove that the distance between two reflected Brownian motions, driven by the same white noise, outside a sphere in a $3$-dimensional flat torus does not converge to $0$, a.s., if the radius of the sphere is sufficiently small, relative to the size of the torus.

We show that the top eigenvalue of an $n\times n$ random symmetric Toeplitz matrix, scaled by $\sqrt{2n\log n}$, converges to the square of the $2\to4$ operator norm of the sine kernel.

We establish the existence of free energy limits for several combinatorial models on Erdös–Rényi graph $\mathbb{G} (N,\lfloor cN\rfloor)$ and random $r$-regular graph $\mathbb{G} (N,r)$. For a variety of models, including independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. This resolves an open problem which was proposed by Aldous (Some open problems) as one of his six favorite open problems. It was also mentioned as an open problem in several other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999 (Canterbury) (1999) 239–298 Cambridge Univ. Press]; Bollobás and Riordan [Random Structures Algorithms 39 (2011) 1–38]; Janson and Thomason [Combin. Probab. Comput. 17 (2008) 259–264] and Aldous and Steele [In Probability on Discrete Structures (2004) 1–72 Springer].

Our approach is based on extending and simplifying the interpolation method of Guerra and Toninelli [Comm. Math. Phys. 230 (2002) 71–79] and Franz and Leone [J. Stat. Phys. 111 (2003) 535–564]. Among other applications, this method was used to prove the existence of free energy limits for Viana–Bray and K-SAT models on Erdös–Rényi graphs. The case of zero temperature was treated by taking limits of positive temperature models. We provide instead a simpler combinatorial approach and work with the zero temperature case (optimization) directly both in the case of Erdös–Rényi graph $\mathbb{G} (N,\lfloor cN\rfloor)$ and random regular graph $\mathbb{G} (N,r)$. In addition we establish the large deviations principle for the satisfiability property of the constraint satisfaction problems, coloring, K-SAT and NAE-K-SAT, for the $\mathbb{G} (N,\lfloor cN\rfloor)$ random graph model.

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{1}{2}\leq\alpha\leq\beta\leq1$, there is a group $\Gamma $ with measure $\mu $ equidistributed on a finite generating set such that

\[\liminf\frac{\log H_{\Gamma ,\mu }(n)}{\log n}=\alpha ,\qquad\limsup \frac{\log H_{\Gamma ,\mu }(n)}{\log n}=\beta .\]

The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk $Y_{n}$ on such groups are also evaluated, providing an example of group with return probability satisfying

\[\liminf\frac{{\log}|{\log P}(Y_{n}=_{\Gamma }1)|}{\log n}=\frac{1}{3},\qquad\limsup\frac{{\log}|{\log P}(Y_{n}=_{\Gamma }1)|}{\log n}=1\]

and drift satisfying

\[\liminf\frac{\log{\mathbb{E}}\|Y_{n}\|}{\log n}=\frac{1}{2},\qquad\limsup\frac{\log{\mathbb{E}}\|Y_{n}\|}{\log n}=1.\]

We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree $T$, and study the existence of the free energy density $\phi$, the limit of the log-partition function divided by the number of vertices $n$ as $n$ tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity $\phi$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$. By way of example we compute $\phi$ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on $\phi$.

In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on $T$. In the special case that $T$ has a Galton–Watson law, this formula coincides with the nonrigorous “Bethe prediction” obtained by statistical physicists using the “replica” or “cavity” methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.

We analyze the landscape of general smooth Gaussian functions on the sphere in dimension $N$, when $N$ is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large. We end the paper by discussing how these results can be interpreted in the language of spin glasses models.

We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the $1+1$-dimensional exactly solvable case with log-gamma distributed random weights. Along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions.

In this paper we extend the Bernstein, Prohorov and Bennett inequalities to the noncommutative setting. In addition we provide an improved version of the noncommutative Rosenthal inequality, essentially due to Nagaev, Pinelis and Pinelis, Utev for commutative random variables. We also present new best constants in Rosenthal’s inequality. Applying these results to random Fourier projections, we recover and elaborate on fundamental results from compressed sensing, due to Candes, Romberg and Tao.

Let $X_{n}$ be independent random elements in the Skorohod space $D([0,1];E)$ of càdlàg functions taking values in a separable Banach space $E$. Let $S_{n}=\sum_{j=1}^{n}X_{j}$. We show that if $S_{n}$ converges in finite dimensional distributions to a càdlàg process, then $S_{n}+y_{n}$ converges a.s. pathwise uniformly over $[0,1]$, for some $y_{n}\in D([0,1];E)$. This result extends the Itô–Nisio theorem to the space $D([0,1];E)$, which is surprisingly lacking in the literature even for $E=R$. The main difficulties of dealing with $D([0,1];E)$ in this context are its nonseparability under the uniform norm and the discontinuity of addition under Skorohod’s $J_{1}$-topology.

We use this result to prove the uniform convergence of various series representations of càdlàg infinitely divisible processes. As a consequence, we obtain explicit representations of the jump process, and of related path functionals, in a general non-Markovian setting. Finally, we illustrate our results on an example of stable processes. To this aim we obtain new criteria for such processes to have càdlàg modifications, which may also be of independent interest.

On the supercritical percolation cluster with parameter $p$, the distances between two distant points of the axis are asymptotically increased by a factor $1+\frac{1-p}{2}+o(1-p)$ with respect to the usual distance. The proof is based on an apparently new connection with the TASEP (totally asymmetric simple exclusion process).

The notion of $r$-crossing and $r$-nesting of a complete matching was introduced and a symmetry property was proved by Chen et al. [Trans. Amer. Math. Soc. 359 (2007) 1555–1575]. We consider random matchings of large size and study their maximal crossing and their maximal nesting. It is known that the marginal distribution of each of them converges to the GOE Tracy–Widom distribution. We show that the maximal crossing and the maximal nesting becomes independent asymptotically, and we evaluate the joint distribution for the Poissonized random matchings explicitly to the first correction term. This leads to an evaluation of the asymptotic of the covariance. Furthermore, we compute the explicit second correction term in the distribution function of two objects: (a) the length of the longest increasing subsequence of Poissonized random permutation and (b) the maximal crossing, and hence also the maximal nesting, of Poissonized random matching.

For a complete graph of size $n$, assign each edge an i.i.d. exponential variable with mean $n$. For $\lambda>0$, consider the length of the longest path whose average weight is at most $\lambda$. It was shown by Aldous [Combin. Probab. Comput. 7 (1998) 1–10] that the length is of order $\log n$ for $\lambda<1/\mathrm{e}$ and of order $n$ for $\lambda>1/\mathrm{e}$. Aldous [Open problems (2003) Preprint] posed the question on detailed behavior at and near criticality $1/\mathrm{e}$. In particular, Aldous asked whether there exist scaling exponents $\mu$, $\nu$ such that for $\lambda$ within $1/\mathrm{e}$ of order $n^{-\mu}$, the length for the longest path of average weight at most $\lambda$ has order $n^{\nu}$.

We answer this question by showing that the critical behavior is far richer: For $\lambda$ around $1/\mathrm{e}$ within a window of $\alpha(\log n)^{-2}$ with a small absolute constant $\alpha>0$, the longest path is of order $(\log n)^{3}$. Furthermore, for $\lambda\geq1/\mathrm{e}+\beta(\log n)^{-2}$ with $\beta$ a large absolute constant, the longest path is at least of length a polynomial in $n$. An interesting consequence of our result is the existence of a second transition point in $1/\mathrm{e}+[\alpha(\log n)^{-2},\beta(\log n)^{-2}]$. In addition, we demonstrate a smooth transition from subcritical to critical regime. Our results were not known before even in a heuristic sense.