Relative, local and global dimension in complex networks

Dimension is a fundamental property of objects and the space in which they are embedded. Yet ideal notions of dimension, as in Euclidean spaces, do not always translate to physical spaces, which can be constrained by boundaries and distorted by inhomogeneities, or to intrinsically discrete systems such as networks. To take into account locality, finiteness and discreteness, dynamical processes can be used to probe the space geometry and define its dimension. Here we show that each point in space can be assigned a relative dimension with respect to the source of a diffusive process, a concept that provides a scale-dependent definition for local and global dimension also applicable to networks. To showcase its application to physical systems, we demonstrate that the local dimension of structural protein graphs correlates with structural flexibility, and the relative dimension with respect to the active site uncovers regions involved in allosteric communication. In simple models of epidemics on networks, the relative dimension is predictive of the spreading capability of nodes, and identifies scales at which the graph structure is predictive of infectivity. We further apply our dimension measures to neuronal networks, economic trade, social networks, ocean flows, and to the comparison of random graphs. Defining the dimension in bounded, inhomogeneous or discrete physical systems may be challenging. The authors introduce here a dynamics-based notion of dimension by analysing diffusive processes in space, relevant for non-ideal physical systems and networks.

Nonperturbative Quantum Field Theory in Curved Spacetime

Kamran Salehi Vaziri

Quantum Field Theory(QFT) as one of the most promising frameworks to study high energy and condensed matter physics, has been mostly developed by perturbative methods. However, perturbative methods can only capture a small island of the space of QFTs.QFT in hyperbolic space can be used to link the conformal bootstrap and massive QFT. Conformal boundary correlators also can be studied by their general properties such as unitarity, crossing symmetry and analicity. On the other hand, by sending the curvature radius to infinity we reach to the flat-space limit in hyperbolic space. This allows us to use conformal bootstrap methods to study massive QFT in one higher dimension and calculate observables like scattering amplitudes or finding bounds on the couplings of theory. The main goal of my research during my Ph.D. would be to study QFTs in hyperbolic space to better understand strongly coupled QFTs.Hamiltonian truncation is a numerical method to study strongly coupled QFTs by imposing a UV cutoff. We use this method to study strongly coupled QFT in hyperbolic space background. For simplicity, we start with scalar field theory in 2-dimensional AdS. We expect to extract the spectrum of our theory as a function of AdS curvature and find the boundary correlation functions.

EPFL2022

Unique decomposition of homogeneous languages and application to isothetic regions

Nicolas René Jean Ninin

A language is said to be homogeneous when all its words have the same length. Homogeneous languages thus form a monoid under concatenation. It becomes freely commutative under the simultaneous actions of every permutation group G(n) on the collection of homogeneous languages of length n is an element of N. One recovers the isothetic regions from (Haucourt 2017, to appear (online since October 2017)) by considering the alphabet of connected subsets of the space vertical bar G vertical bar, viz the geometric realization of a finite graph G. Factoring the geometric model of a conservative program amounts to parallelize it, and there exists an efficient factoring algorithm for isothetic regions. Yet, from the theoretical point of view, one wishes to go beyond the class of conservative programs, which implies relaxing the finiteness hypothesis on the graph G. Provided that the collections of n-dimensional isothetic regions over G (denoted by R-n vertical bar G vertical bar) are co -unital distributive lattices, the prime decomposition of isothetic regions is given by an algorithm which is, unfortunately, very inefficient. Nevertheless, if the collections R-n vertical bar G vertical bar satisfy the stronger property of being Boolean algebras, then the efficient factoring algorithm is available again. We relate the algebraic properties of the collections R-n vertical bar G vertical bar to the geometric properties of the space I GI. On the way, the algebraic structure R-n vertical bar G vertical bar is proven to be the universal tensor product, in the category of semilattices with zero, of n copies of the algebraic structure R-1 vertical bar G vertical bar.

2019

Localization errors of the stochastic heat equation

David Jean-Michel Candil

In this thesis, we study the stochastic heat equation (SHE) on bounded domains and on the whole Euclidean space

\R^d.

We confirm the intuition that as the bounded domain increases to the whole space, both solutions become arbitrarily close to one another. Both vanishing Dirichlet and Neumann boundary conditions are considered.We first study the nonlinear SHE in any space dimension with multiplicative correlated noise and bounded initial data. We prove that the solutions to SHE on an increasing sequence of domains converge exponentially fast to the solution to SHE on

\R^d.

Uniform convergence on compact set is obtained for all

p

-moments. The conditions that need to be imposed on the noise are the same as those required to ensure existence of a random field solution. A Gronwall-type iteration argument is used together with uniform bounds on the solutions, which are surprisingly valid for the entire sequence of increasing domains.We then study SHE in space dimension

d\ge 2

with additive white noise and bounded initial data. Even though both solutions need to be considered as distributions, their difference is proved to be smooth. If fact, the order of smoothness depends only on the regularity of the boundary of the increasing sequence of domains. We prove that the Fourier transform, in the sense of distributions, of the solution to SHE on

\R^d

do not have any locally mean-square integrable representative. Therefore, convergence is studied in local versions of Sobolev spaces. Again, exponential rate is obtained.Finally, we study the Anderson model for SHE with correlated noise and initial data given by a measure. We obtain a special expression for the second moment of the difference of the solution on

\R^d

with that on a bounded domain. The contribution of the initial condition is made explicit. For example, exponentially fast convergence on compact sets is obtained for any initial condition with polynomial growth. More interestingly, from a given convergence rate, we can decide whether some initial data is admissible.

EPFL2022

Localization errors of the stochastic heat equation

David Jean-Michel Candil

In this thesis, we study the stochastic heat equation (SHE) on bounded domains and on the whole Euclidean space

\R^d.

\R^d.

Uniform convergence on compact set is obtained for all

p

d\ge 2

\R^d

\R^d

EPFL2022