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 decays exponentially with except at a particular value of the percolation parameter for which the decay is polynomial (of order ). Moreover, the probability that the origin cluster has size decays exponentially if and polynomially if .

The critical percolation value is for site percolation, and 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 , the percolation clusters conditioned to have size should converge toward the stable map of parameter introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

]]>The aim of this paper is to construct Calabi–Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold by a non-symplectic involution . We first compute the Hodge numbers of a Calabi–Yau constructed in this way in a general setting, and then we apply the results to several specific examples of non-symplectic involutions, producing Calabi–Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where is the Hilbert scheme of two points on a K3 surface , and the involution is induced by a non-symplectic involution on the K3 surface. In this case we compare the Calabi–Yau 4-fold , which is the crepant resolution of , with the Calabi–Yau 4-fold , constructed from through the Borcea–Voisin construction. We give several explicit geometrical examples of both these Calabi–Yau 4-folds, describing maps related to interesting linear systems as well as a rational map from to .

]]>We characterize the class of RFD -algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the -algebra is finite-dimensional, which is equivalent to the -algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of -algebras whose norms in finite-dimensional representations fit certain prescribed properties.

]]>Is there some absolute such that for any claw-free graph , the chromatic number of the square of satisfies , where is the clique number of ? Erdős and Nešetřil asked this question for the specific case where is the line graph of a simple graph, and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and, moreover, that it essentially reduces to the original question of Erdős and Nešetřil.

]]>We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence of topological groups n such that is a subgroup of and the latter induces the given topology on , for each . Let be the direct limit of the sequence in the category of topological groups. We show that induces the given topology on each whenever is an identity neighbourhood in for all identity neighbourhoods . If, moreover, each is complete, then is complete. We also show that the weak direct product is complete for each family of complete Lie groups . As a consequence, every strict direct limit of finite-dimensional Lie groups is complete, as well as the diffeomorphism group of a paracompact finite-dimensional smooth manifold and the test function group , for each and complete Lie group modelled on a complete locally convex space.

]]>We show that if the zeros of an automorphic -function are weighted by the central value of the -function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the -value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted -values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

]]>Let be a complex Enriques surface whose universal cover is birational to a general quartic Hessian surface. Using the result on the automorphism group of due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of . The list of elliptic fibrations on and the list of combinations of rational double points that can appear on a surface birational to are presented. As an application, a set of generators of the automorphism group of the generic Enriques surface is calculated explicitly.

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