299 research outputs found
Hamiltonicity in multitriangular graphs
The family of 5-valent polyhedral graphs whose faces are all triangles or 3s-gons, s â„ 9, is shown to contain non-hamiltonian graphs and to have a shortness exponent smaller than one
Planar Ising model at criticality: state-of-the-art and perspectives
In this essay, we briefly discuss recent developments, started a decade ago
in the seminal work of Smirnov and continued by a number of authors, centered
around the conformal invariance of the critical planar Ising model on
and, more generally, of the critical Z-invariant Ising model on
isoradial graphs (rhombic lattices). We also introduce a new class of
embeddings of general weighted planar graphs (s-embeddings), which might, in
particular, pave the way to true universality results for the planar Ising
model.Comment: 19 pages (+ references), prepared for the Proceedings of ICM2018.
Second version: two references added, a few misprints fixe
Magnetization in the zig-zag layered Ising model and orthogonal polynomials
We discuss the magnetization in the -th column of the zig-zag
layered 2D Ising model on a half-plane using Kadanoff-Ceva fermions and
orthogonal polynomials techniques. Our main result gives an explicit
representation of via Hankel determinants constructed from
the spectral measure of a certain Jacobi matrix which encodes the interaction
parameters between the columns. We also illustrate our approach by giving short
proofs of the classical Kaufman-Onsager-Yang and McCoy-Wu theorems in the
homogeneous setup and expressing as a Toeplitz+Hankel determinant for the
homogeneous sub-critical model in presence of a boundary magnetic field.Comment: minor updates + Section 5.3 added; 38 page
Conformal Invariance of Spin Correlations in the Planar Ising Model
We rigorously prove the existence and the conformal invariance of scaling
limits of the magnetization and multi-point spin correlations in the critical
Ising model on arbitrary simply connected planar domains. This solves a number
of conjectures coming from the physical and the mathematical literature. The
proof relies on convergence results for discrete holomorphic spinor observables
and probabilistic techniques.Comment: Changes in this version: the explicit formula for n-point spin
correlations is proved in full generality. The appendix is rewritten
completely and contains this new proof, the introduction is changed
accordingly, the presentation in Sections 2.5 and 2.7(2.8) is rearranged
slightly. 42 pages, 2 figure
Quantum Gravity and Matter: Counting Graphs on Causal Dynamical Triangulations
An outstanding challenge for models of non-perturbative quantum gravity is
the consistent formulation and quantitative evaluation of physical phenomena in
a regime where geometry and matter are strongly coupled. After developing
appropriate technical tools, one is interested in measuring and classifying how
the quantum fluctuations of geometry alter the behaviour of matter, compared
with that on a fixed background geometry.
In the simplified context of two dimensions, we show how a method invented to
analyze the critical behaviour of spin systems on flat lattices can be adapted
to the fluctuating ensemble of curved spacetimes underlying the Causal
Dynamical Triangulations (CDT) approach to quantum gravity. We develop a
systematic counting of embedded graphs to evaluate the thermodynamic functions
of the gravity-matter models in a high- and low-temperature expansion. For the
case of the Ising model, we compute the series expansions for the magnetic
susceptibility on CDT lattices and their duals up to orders 6 and 12, and
analyze them by ratio method, Dlog Pad\'e and differential approximants. Apart
from providing evidence for a simplification of the model's analytic structure
due to the dynamical nature of the geometry, the technique introduced can shed
further light on criteria \`a la Harris and Luck for the influence of random
geometry on the critical properties of matter systems.Comment: 40 pages, 15 figures, 13 table
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