Complex systems are very often organized under the form of networks where nodes and edges are embedded in space Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, neural networks, are all examples where space is relevant and where topology alone does not contain all the information Characterizing and understanding the structure and the evolution of spatial networks is thus crucial for many different fields ranging from urbanism to epidemiology An important consequence of space on networks is that there is a cost associated to the length of edges which in turn has dramatic effects on the topological structure of these networks We will expose thoroughly the current state of our understanding of how the spatial constraints affect the structure and properties of these networks We will review the most recent empirical observations and the most important models of spatial networks We will also discuss various processes which take place on these spatial networks, such as phase transitions, random walks, synchronization, navigation, resilience, and disease spread

Spatial Networks

This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.

Scaling and Renormalization in Statistical Physics

Vesicles consisting of a bilayer membrane of amphiphilic lipid molecules are remarkably flexible surfaces that show an amazing variety of shapes of different symmetry and topology. Owing to the fluidity of the membrane, shape transitions such as budding can be induced by temperature changes or the action of optical tweezers. Thermally excited shape fluctuations are both strong and slow enough to be visible by video microscopy. Depending on the physical conditions, vesicles adhere to and unbind from each other or a substrate. This article describes the systematic physical theory developed to understand the static and dynamic aspects of membrane and vesicle configurations. The preferred shapes arise from a competition between curvature energy, which derives from the bending elasticity of the membrane, geometrical constraints such as fixed surface area and fixed enclosed volume, and a signature of the bilayer aspect. These shapes of lowest energy are arranged into phase diagrams, which separate regi...

Configurations of fluid membranes and vesicles

The physics of Anderson transitions between localized and metallic phases in disordered systems is reviewed The term ``Anderson transition'' is understood in a broad sense, including both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states The emphasis is put on recent developments, which include: multifractality of critical wave functions, criticality in the power-law random banded matrix model, symmetry classification of disordered electronic systems, mechanisms of criticality in quasi-one-dimensional and two-dimensional systems and survey of corresponding critical theories, network models, and random Dirac Hamiltonians Analytical approaches are complemented by advanced numerical simulations

Anderson Transitions

Stochastic equations in infinite dimensions

Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫
 D
 ∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on D is the weak limit as e→0 of the measures $$\varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz,$$
 where dz is Lebesgue measure on D and h
 e
 (z) denotes the mean value of h on the circle of radius e centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819–826, 1988) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

Liouville quantum gravity and KPZ

By mapping the two-dimensional percolation problem on a Coulomb gas, we obtain the exact fractal dimension of the external perimeter (or "hull") of the infinite percolation cluster: ${D}_{H}=\frac{7}{4}$, in agreement with numerical estimates and a recent conjecture. We also determine an infinite set of exact exponents associated with various topologies of this hull. We argue finally that the different fractal dimensions observed recently by Grossman and Aharony, who modified the definition of the hull, are all equal to ${D}_{e}=\frac{4}{3}$.

Exact Determination of the Percolation Hull Exponent in Two Dimensions

Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the KPZ relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of the boundary of D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

Liouville Quantum Gravity and KPZ

Electromagnetic waves near perfect conductors. II. Casimir effect

We propose the exact values of the tricritical exponents of a collapsing polymer in two dimensions: \ensuremath{\nu}=(4/7, \ensuremath{\gamma}=(8/7, and \ensuremath{\varphi}=(3/7. They are obtained in a model of self-avoiding walk on a hexagonal lattice, with random forbidden hexagons, whose percolation threshold gives the exact tricritical point. The infinitely many exact tricritical exponents then derived from Coulomb gas methods are critical exponents of the O(n=1) Ising model below ${\mathrm{T}}_{\mathrm{c}}$. The numerical check is very good.

Bertrand Duplantier

Papers

Liouville quantum gravity and KPZ

Exact Determination of the Percolation Hull Exponent in Two Dimensions

Liouville Quantum Gravity and KPZ

Electromagnetic waves near perfect conductors. II. Casimir effect

Exact tricritical exponents for polymers at the FTHETA point in two dimensions.