# His research

Hugo Duminil-Copin's research focuses on the mathematical branch of statistical physics. He studies phase transitions – i.e abrupt changes in the properties of matter, such as the transition from the gaseous to the liquid state of water - using probability theory. He uses probability theory to analyse mathematical models describing three distinct phenomena: material porosity (via percolation theory), ferromagnetism (via the Ising model) and polymers (via the study of self-avoiding walks).

### Material porosity

The idea is to understand what happens in a porous material like pumice (or coffee, hence the name of this research field). When water passes through such a material, what path does it take? Is it blocked, does it go straight through or does it follow tortuous paths? The regime of tortuous paths is known as the «critical phase» where a «phase transition» occurs between an «impermeable» state and a «unimpeded» one. To model this problem, mathematicians use - among other things - "random graphs" which simulate all possible paths and whose connectivity properties can be studied.

### Ferromagnetism

Magnets lose their magnetic properties as soon as they are heated above a certain temperature (known as the Curie temperature). As soon as they cool below this temperature, they become magnets again. As in the case of percolation, this is a phase transition between two states - one magnetised and the other not - separated by a critical temperature. What exactly happens at this temperature? To find out, mathematicians develop models (one of the most important being the Ising model) in which the material is considered to be an assembly of a multitude of small magnets whose alignment varies according to the temperature, i.e. the agitation. By making a number of assumptions, researchers can translate this model into mathematical language and study its properties.

### Polymers

The model of self-avoiding walks was introduced in 1948 by the chemist Paul Flory (Nobel Prize in Chemistry in 1974) in order to model the behaviour of polymers (such as DNA) immersed in a solvent. The central objects of this model are the trajectories - simply called walks - of walkers that are not allowed to pass through a place they already visited. This combinatorial problem can also be reformulated with the help of graphs and leads to complex geometric questions.