# Mathematical Physics

**Coordinator :** Christoph Kopper

**Permanent staff :**

Dario Benedetti

Jean-René Chazottes

Razvan Gurau

Philippe Mounaix

**PhD students:**

**Post-Doc:**

**Emeritus at CNRS :**

Pierre Collet

Jacques Magnen

**Research activities**

The activities of the mathematical physics group cover a broad spectrum of research topics. There are two main direcions, field theory and statistical physics

**Field Theory **

• Random Tensors and Tensor Field Theory (R. Gurau).Over the past five years, random tensors have undergone a tremendous transformation. The contributions from the CPHT has been at the forefront of the research in this field. Our results are centered on non-perturbative aspects of models with quartic interactions and the study of the double scaling limit. A fundamental result, the classification of edge colored graphs by the degree, has been obtained in collaboration with LIX. An important recent result is the proof that a model for a symmetric traceless tensor in rank three has an 1/N expansion dominated by melonic graphs.

• Renormalization Group Flow Equations (Ch. Kopper, J. Magnen).The programme is intended to base renormalization theory entirely and with mathematical rigour on the flow equations of the renormalization group. Recent results concern the renormalizability proof of nonabelian (unbroken) gauge theories, as they appear in the QCD sector of the standard model, and the convergence of the perturbative operator product expansion for massless theories. Recent work by J. Magnen concerns an analysis of the Kadar-Parisi-Zhang equation with renormalization group methods.

• Mathematical Ecology (J.-R. Chazottes, P. Collet)establishes for example approximations of quasi-stationary distributions in birth-and-death processes. Our main recent achievement is the fine description of the quasi stationary behaviour of a class of birth-and-death processes describing a population made of d sub-populations of different types which interact with one another. The framework allows for many applications conceptually, theoretically and empirically.

• Dynamical Systems (J.-R. Chazottes, P. Collet).In this domain we analyse statistics of return times and integrability of discrete dynamical systems. We consider consequences of concentration inequalities for Gibbs measures and the evolution of concentration bounds under several types of stochastic dynamics. We also analyse non-uniformly hyperbolic dynamical systems.

• Stochastic Processes (J.-R. Chazottes, P. Collet, Ph. Mounaix).Recent results are on asymptotics of the heat kernel with specific boundary conditions in multi-cone domains, conditional random Gaussian fields, random walks, Lévy flights, diffusion processes and fluctuations of trajectories, time exponents for transition probabilities, survival probabilities and large time limits.