Given a stationary and isotropic Poisson hyperplane process and a convex body K in ${\mathbb R}^d$, we consider the random polytope defined by the intersection of all closed half-spaces containing K that are bounded by hyperplanes of the process not intersecting K. We investigate how well the expected mean width of this random polytope approximates the mean width of K if the intensity of the hyperplane process tends to infinity.

We obtain simple quadratic recurrence formulas counting bipartite maps on surfaces with prescribed degrees (in particular, $2k$-angulations) and constellations. These formulas are the fastest known way of computing these numbers.

Our work is a natural extension of previous works on integrable hierarchies (2-Toda and KP), namely, the Pandharipande recursion for Hurwitz numbers (proved by Okounkov and simplified by Dubrovin–Yang–Zagier), as well as formulas for several models of maps (Goulden–Jackson, Carrell–Chapuy, Kazarian–Zograf). As for those formulas, a bijective interpretation is still to be found. We also include a formula for monotone simple Hurwitz numbers derived in the same fashion.

These formulas also play a key role in subsequent work of the author with T. Budzinski establishing the hyperbolic local limit of random bipartite maps of large genus.

We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spanning forests of transient proper plane graphs with bounded degrees and codegrees. We find that the same exponents hold universally over this entire class of graphs provided that measurements are made using the hyperbolic geometry of their circle packings rather than their usual combinatorial geometry.

We give a simple set of geometric conditions on curves $\unicode[STIX]{x1D702}$, $\widetilde{\unicode[STIX]{x1D702}}$ in $\mathbf{H}$ from $0$ to $\infty$ so that if $\unicode[STIX]{x1D711}:\mathbf{H}\rightarrow \mathbf{H}$ is a homeomorphism which is conformal off $\unicode[STIX]{x1D702}$ with $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702})=\widetilde{\unicode[STIX]{x1D702}}$ then $\unicode[STIX]{x1D711}$ is a conformal automorphism of $\mathbf{H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm–Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $\unicode[STIX]{x1D702}$ is a non-space-filling $\text{SLE}_{\unicode[STIX]{x1D705}}$ curve in $\mathbf{H}$ from $0$ to $\infty$, and $\unicode[STIX]{x1D711}$ is a homeomorphism which is conformal on $\mathbf{H}\setminus \unicode[STIX]{x1D702}$, and $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702})$, $\unicode[STIX]{x1D702}$ are equal in distribution, then $\unicode[STIX]{x1D711}$ is a conformal automorphism of $\mathbf{H}$. Applying this result for $\unicode[STIX]{x1D705}=4$ establishes that the welding operation for critical ($\unicode[STIX]{x1D6FE}=2$) LQG is well defined. Applying it for $\unicode[STIX]{x1D705}\in (4,8)$ gives a new proof that the welding of two independent $\unicode[STIX]{x1D705}/4$-stable looptrees of quantum disks to produce an $\text{SLE}_{\unicode[STIX]{x1D705}}$ on top of an independent $4/\sqrt{\unicode[STIX]{x1D705}}$-LQG surface is well defined.

We consider a random graph model that was recently proposed as a model for complex networks by Krioukov et al. (2010). In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has previously been shown that this model has various properties associated with complex networks, including a power-law degree distribution and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes N, which we think of as going to infinity, and $\alpha, \nu > 0$, which we think of as constant. Roughly speaking, $\alpha$ controls the power-law exponent of the degree sequence and $\nu$ the average degree. Earlier work of Kiwi and Mitsche (2015) has shown that, when $\alpha \lt 1$ (which corresponds to the exponent of the power law degree sequence being $\lt 3$), the diameter of the largest component is asymptotically almost surely (a.a.s.) at most polylogarithmic in N. Friedrich and Krohmer (2015) showed it was a.a.s. $\Omega(\log N)$ and improved the exponent of the polynomial in $\log N$ in the upper bound. Here we show the maximum diameter over all components is a.a.s. $O(\log N),$ thus giving a bound that is tight up to a multiplicative constant.

In Weil (2001) formulae were proved for stationary Boolean models Z in ℝd with convex or polyconvex grains, which express the densities (specific mean values) of mixed volumes of Z in terms of related mean values of the underlying Poisson particle process X. These formulae were then used to show that in dimensions 2 and 3 the densities of mixed volumes of Z determine the intensity γ of X. For d = 4, a corresponding result was also stated, but the proof given was incomplete, since in the formula for the density of the Euler characteristic V̅0(Z) of Z a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result has recently been proved by Hug, Rataj, and Weil (2013), (2018) and is based on flag measures of the convex bodies involved. Here, we show that such flag representations not only lead to a correct derivation of the four-dimensional result, but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter we make use of Alesker’s representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the nonstationary case.

For a fixed k ∈ {1, …, d}, consider arbitrary random vectors X0, …, Xk ∈ ℝd such that the (k + 1)-tuples (UX0, …, UXk) have the same distribution for any rotation U. Let A be any nonsingular d × d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed Xi satisfies \[|{\rm{conv}}(A{X_{\rm{0}}} \ldots ,A{X_k}){\rm{|}}\mathop {\rm{ = }}\limits^{\rm{D}} (|{P_\xi }\varepsilon |/{\kappa _k})|{\rm{conv}}\left( {{X_0}, \ldots ,{X_k}} \right)\], where ɛ:= {x ∈ ℝd : x┬ (A┬A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace ξ independent of X0, …, Xk, and κk is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: ck,d,p∫Ad,k |ɛ ∩ E|p+d+1μd,k(dE) = |ɛ|k+1∫Gd,k |PLɛ|pνd,k(dL), where $c_{k,d,p} = \big({\kappa_{d}^{k+1}}/{\kappa_k^{d+1}}\big) ({\kappa_{k(d+p)+k}}/{\kappa_{k(d+p)+d}})$. Here p > −1 and Ad,k and Gd,k are the affine and the linear Grassmannians equipped with their respective Haar measures. The p = 0 case reduces to an affine version of the integral formula of Furstenberg and Tzkoni (1971).

We study the geometry of the component of the origin in the uniform spanning forest of $\mathbb{Z}^{d}$ and give bounds on the size of balls in the intrinsic metric.

We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length $n$ decays exponentially with $n$ except at a particular value $p_{c}$ of the percolation parameter $p$ for which the decay is polynomial (of order $n^{-10/3}$). Moreover, the probability that the origin cluster has size $n$ decays exponentially if $p<p_{c}$ and polynomially if $p\geqslant p_{c}$.

The critical percolation value is $p_{c}=1/2$ for site percolation, and $p_{c}=(2\sqrt{3}-1)/11$ for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds.

Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at $p_{c}$, the percolation clusters conditioned to have size $n$ should converge toward the stable map of parameter $\frac{7}{6}$ introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c∕(λ−λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.

In this paper we prove asymptotic upper bounds on the variance of the number of vertices and the missed area of inscribed random disc-polygons in smooth convex discs whose boundary is C+2. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.

We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley‒Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges that do not contain triangles.

Let (Mt:t>0) be a Markov process of tessellations of ℝℓ, and let (𝒞t:t>0) be the process of their zero cells (zero polytopes), which has the same distribution as the corresponding process for Poisson hyperplane tessellations. In the present paper we describe the stationary zero cell process (at𝒞at:t∈ℝ),a>1, in terms of some regenerative structure and we show that it is a Bernoulli flow. An important application is to STIT tessellation processes.

In this paper we analyze the first-order behavior (that is, the right-sided derivative) of the volume of the dilation A⊕tQ as t converges to 0. Here A and Q are subsets of n-dimensional Euclidean space, A has finite perimeter, and Q is finite. If Q consists of two points only, n and n+u, say, this derivative coincides up to a sign with the directional derivative of the covariogram of A in direction u. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of A. We extend this result to finite sets Q and use it to determine the derivative of the contact distribution function with finite structuring element of a stationary random set at 0. The proofs are based on an approximation of the indicator function of A by smooth functions of bounded variation.

We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in ℝ2 with radii determined by an underlying stationary and ergodic random field φ:ℝ2→[0,∞), independent of the Poisson process. This setting, in which the random field is independent of the point process, is often referred to as geostatistical marking. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of ℝ2 does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.

We consider the random polytope Kn, defined as the convex hull of n points chosen independently and uniformly at random on the boundary of a smooth convex body in ℝd. We present both lower and upper variance bounds, a strong law of large numbers, and a central limit theorem for the intrinsic volumes of Kn. A normal approximation bound from Stein's method and estimates for surface bodies are among the tools involved.

Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ0,! such that there exists a point process η where Φ=Φ0,!∪η in distribution and Φ0,!∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n→∞. We also prove that for some DPPs, there exists an R∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √nR∗. This R∗ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high-dimensional Boolean models is given.

We consider a Markov chain of point processes such that each state is a superposition of an independent cluster process with the previous state as its centre process together with some independent noise process and a thinned version of the previous state. The model extends earlier work by Felsenstein (1975) and Shimatani (2010) describing a reproducing population. We discuss when closed-form expressions of the first- and second-order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.

A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.