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Here are some of the topics I am interested in/have worked on presented in a few lines. Other topics will appear when I find more time to generate pictures. Displayed equations and text content are to be taken as only informal presentation, and one should always refer to the concerned articles for precise statements/hypotheses. Numbers in square brackets correspond to elements in my publication list.
Here also an AI-powered site producing short overview of papers, you can find some of mine there. I found the explanations and accuracy to be very good on the few summaries I read there.
Renewal theory for correlations of lattice models
A central ingredient in my research is the derivation and use of suitable renewal structures for the description of polymer models in polymer ensembles manifesting a ballistic behaviour of the polymer endpoint. Such structures are in particular used on polymer models corresponding to graphical expansion of correlation functions of classical lattice spin models to make rigorous a classical picture for correlation functions due to the physicists Ornstein and Zernike. The latter is based on assuming that correlation functions satisfy a suitable renewal equation, and predict the sharp (spatial) asymptotic of correlations decay. Another application is to 2D interfaces, which can be seen as interacting polymers.
The modern approach to derivations of this renewal picture has started in the late 1990's/early 2000's from the joint efforts of (in chronological order of apparition on the relevant series of papers) D. Ioffe, M. Campanino, and Y. Velenik. Several refinements/extensions have been made since then.
A true renewal picture for polymer models
Whilst previous works proved approximate renewal equations (process with "weak" infinite memory rather than Markov chain/random walk) satisfied by certain polymer models, in [1] we exploited ideas coming from perfect sampling to prove a true renewal equation (random walk picture) for suitable classes of polymer model under a mixing assumption, and applied it to FK percolation. These ideas were further simplified in [12] in the case of self-repulsive polymer models.
Asymptotic behaviour of correlations
The sharp asymptotic (as distance between points goes to infinity) of the (truncated) two-point function of many lattice spin model away from phase transition points is seen to have an asymptotic predicted by Ornstein and Zernike. In [3], I extended these asymptotic to Ising models in the presence of a positive magnetic field, and in [12] we extend these asymptotics to infinite range Ising models at T>T_c, h=0 with suitable decay of interaction strength.
In [2], we combined the arguments used for the two-point functions together with the random current representation of the Ising model to obtain up-to-constant asymptotic for the energy-energy correlation of the Ising model at T>T_c, and, more general to any even-even correlation.
In [8,11,13] we observed that for infinite range interactions decaying exactly exponentially, a phenomenon akin to condensation in the study of large deviations for sums of i.i.d. random variable with heavy tails was occurring. This phenomenon leads in particular to an anomalous (non-OZ) decay of the two-point function in the Ising model at very high temperature: the decay then becomes equivalent to the decay of the interaction. This issue is closely related to the failure of the so-called mas gap condition in the OZ theory.
2D Interfaces in lattice spin models
Many problems about the geometry of interfaces in dimension 2 can be studied by combining properties of the graphical representations of models such as Ising, Potts or Ashkin-Teller models (e.g.: FK percolation, Peierls contours expansion, random currents), and a renewal theory for polymer models (same as derived for graphical expansions of correlations).
Brownian behaviour of Interfaces in 2D Potts/Ising models.
In most situations, the interface between two stable substances/pure phases can be represented by a microscopic object of intrinsic with O(1). This object then experiences diffusive fluctuation in the direction orthogonal to its main extension direction, and scales, under diffusive limit, to the graph of a 1D Brownian motion.
With co-authors, we established this picture for the order-disorder interface of the Potts model on the square lattice in [18]. This represent the first non-perturbative treatment of an interface between phases not related by an explicit symmetry of the Hamiltonian.
In [5] we proved that the 2D Potts interface at T<T_c experiences entropic repulsion when forced, via suitable boundary conditions, to stay above a wall. Moreover, we obtain that the, diffusively re-scaled, interface converges to a Brownian excursion.
Our arguments take place in the random cluster model, and thus extend to non-integer q's. Along the way, we derived a method to prove entropic repulsion for models enjoying monotonicity (lattice FKG property).
Interfacial wetting in 2D Potts model
Interfacial wetting is a phenomenon occurring when forcing two stables substances, A, and B, to coexists in the presence of a third stable one, C. Then, if the microscopic contact energy between particles of phase A and B is larger than the sum of the contact energy between A and C and the contact energy between C and B, it becomes favourable for the system to use a mesoscopic layer of phase C to separate A from B.
In [22], we provided the first rigorous derivation of such a phenomenon in a lattice spin model: the 2D Potts model.
3D interfaces
The 3D interfaces of the Ising model remain mostly a mystery at the rigorous level. At the physical level, detailed predictions are available: flat interfaces should go from a fluctuating (logarithmic variance in the system size) to a localised (O(1) variance uniformly over system size) at a roughening temperature 0<T_R < T_c. Tilted interfaces should enjoy unbounded fluctuations for any temperature below the Curie one. At the rigorous level, only the behaviour of flat interfaces at very small T is understood, thanks to the celebrated work of Dobrushin (1972).
Effective models for 3D interfaces
A famous effective model for the Ising interfaces is the SOS model: a integer-height function model on the square lattice. This model was shown to undergo a roughening transition in a famous paper of Fröhlich and Spencer (1981): at high temperature the flat interface has logarithmic variance in the system size. In [19], we extended their method to arbitrary tilt of the interface. In [10] we provided a simpler non-quantitative argument proving delocalisation at high temperature for any tilt.
Universal properties of off-transitions systems
It is widely believed (and partially substantiated by rigorous results) that properties of classical spin models away from their transition point share some universal features: fast decay of correlations, fast spatial mixing, analyticity of local observable and thermodynamic quantities in the parameters, fast relaxation of Glauber-type dynamics... The current best attempt to formulate a general theory is due to Dobrushin and Shlosman in 1985, who proposed to define the off-transition regime via a list of (non-trivially) equivalent properties. It was since then understood that their criteria were too strong to apply to models such as the 3D Ising model with a small positive magnetic at low temperatures.
Factors of i.i.d. and analyticity
In [4,6], I proposed a way to relate an information theoretic flavoured notion, being a factor of i.i.d. with suitable properties to analyticity of thermodynamic functions and local observable in the parameters. This idea is then applied to the Ising model and FK percolation to prove, amongst other thing, that the pressure is analytic in the whole regime of parameters away from the transition line.
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