# Research interests

My research interests lie mainly in the applications of Probability Theory to the study of rigorous classical Statistical Mechanics, especially lattice random fields (spin systems, effective interface models, etc.), and random walks (self-interacting random walks, polymers, etc.). One recurrent theme in my research is the derivation of the large-scale asymptotics of various extended objects, such as interfaces, polymers, etc., and the associated phase transitions.
Below is a short summary of some of the results obtained (a more detailed account of papers [1] to [18] can be found in the research summary I wrote for my habilitation, see here; you may also have a look at these lecture notes).

## Ornstein-Zernike asymptotics

A celebrated heuristic theory proposed by Ornstein and Zernike in 1914 implies that the asymptotic form of the truncated two-point density correlation function of simple fluids away from the critical region is given by \[ G_\beta(r) = A_\beta r^{-(d-1)/2} e^{-\xi_\beta r}, \] where the value of the inverse correlation length $\xi_\beta$ depends only on the inverse temperature and the spatial dimension $d$. In several papers in collaboration with M. Campanino and D. Ioffe, we developed a non-perturbative approach to derive rigorously such estimates in various systems. Among the results obtained are:

- Derivation of the Ornstein-Zernike asymptotics for the 2-point function of finite-range Ising models above their critical temperature in any dimension [14,15]. The extension to arbitrary odd-odd correlation functions and additional results can be found in [16]. In the course of the proof, we also establish that the inverse correlation length is an analytic function of the direction.
- Extension of the above results to the class of random-cluster models, under an assumption of finite-volume exponential decay of connectivity, known to hold in many situations (and conjectured to always hold in the subcritical regime). In particular, we prove sharp Ornstein-Zernike asymptotics for the connectivity function and related quantities, the invariance principle for long clusters, the strict convexity and analyticity of the massgap. In two dimensions, using duality, our results imply invariance principle for interfaces and analyticity and strict convexity of the (Wulff) Equilibrium Crystal Shape, in subcritical models; in particular this holds for all 2D Potts models [24].
- Use of the Ornstein Zernike approach to study self-interacting random walks and polymers in the ballistic regime [25]. We consider random walks with drift, whose path measures are perturbed by general functionals of the local time at sites or bonds of either purely attractive or purely repulsive type. The results apply to the whole ballistic regime (except at the critical point), and implies, among other things, a local limit theorem for the endpoint, a functional CLT, analyticity of the Lyapunov exponents, etc. We also show that small perturbations of these pure models exhibit the same behavior; as an illustration of the latter, we consider a weakly reinforced random walk with drift. We also obtain local limit theorems for general local path observables.
- Analysis of self-attractive random walks and polymers in the ballistic regime in the critical case [34]. Proof that the collapsed/stretched phase transition is of first order in dimensions $d\geq 2$, and that the polymer is stretched at criticality, with well-defined macroscopic extension (LLN and CLT).
- Detailed analysis of the inverse correlation length of Potts models on $\mathbb{Z}^d$ in the presence of a line of modified coupling constant [43].

## Effective interface models

Up to now, the analysis of the interactions of rough interfaces with various external potentials (pinning, wetting, etc.) is too hard to be carried through rigorously for lattice spin systems. For this reason, it is useful to consider simplified models of interfaces, the so-called effective interface models. In these models, interfaces are represented as graphs of (random) functions from an underlying lattice to $\mathbb{R}$ or $\mathbb{Z}$. I have been interested, with various collaborators, in the behavior of such systems. Among the results obtained are:

- Proof of a strong form of localization of a 2D massless gradient model with a class of strictly convex (not necessarily Gaussian) interactions by an arbitrarily weak self-potential [ 6], including the proof of exponential decay of correlations for the pinned field [7].
- Precise analysis of the critical behavior of the transverse and longitudinal correlation lengths for a general class of Gaussian interfaces at the depinning transition [11].
- Proof of the existence of a wetting transition for various continuous gradient models [9]. Pathwise estimates showing that the interface is really pinned in the whole thermodynamically-defined partial wetting regime, and derivation of some informations on the rate of divergence of the average height at the wetting transition [26].
- Analysis of the phenomenon of entropic repulsion in the presence of an external potential; with special emphasis on the application to the description of critical prewetting. Effective interface models in dimensions $d+1$, $d\geq 2$ are studied in [17], while $1+1$ dimensional effective interface models are studied in [18]. Proof that the scaling limit in the $1+1$-dimensional case is given by Ferrari-Spohn diffusions [39]; the extension to a finite number of non-crossing paths has been addressed in [41]. A review with a more detailed discussion of the applications to wetting, prewetting and faceting in various models can be found in [42].
- Study of the effect on a 2-dimensional interface of a diluted pinning potential confined to a plane. The main point was to characterize explicitly the set of disorder configurations leading to pinning (the results are thus much stronger than almost sure statements) [20].
- Construction of a simple example of interface model in which the zero-temperature fluctuations are much bigger than the finite temperature ones [10].

## Polymers

The probabilistic analysis of simple effective models for polymers has been a very active field of study in recent years. The topics I have worked on, with various collaborators, include:

- Study of the effect on a polymer of a diluted pinning potential confined to a line, for both relevant cases of dimensions 1+1 and 1+2. The main point was to characterize explicitly the set of disorder configurations leading to pinning (the results are thus much stronger than almost sure statements) [20].
- Use of the Ornstein Zernike approach to study self-interacting random walks and polymers in the ballistic regime [25]. We consider random walks with drift, whose path measures are perturbed by general functionals of the local time at sites or bonds of either purely attractive or purely repulsive type. The results apply to the whole ballistic regime (except at the critical point), and implies, among other things, a local limit theorem for the endpoint, a functional CLT, analyticity of the Lyapunov exponents, etc. We also show that small perturbations of these pure models exhibit the same behavior; as an illustration of the latter, we consider a weakly reinforced random walk with drift. We also obtain local limit theorems for general local path observables. A review is given in [28], including a presentation of ongoing work on the order of the transition for self-attractive polymers, as well as diffusivity of high- dimensional polymers with weak quenched disorder (see below).
- Analysis of self-attractive random walks and polymers in the ballistic regime in the critical case [34]. Proof that the collapsed/stretched phase transition is of first order in dimensions $d\geq 2$, and that the polymer is stretched at criticality, with well-defined macroscopic extension (LLN and CLT).
- Proof of diffusivity (a.s. CLT) for non-directed models of a polymer in a quenched random environment at weak disorder, both for crossing and stretched ensembles [31], [33], [36].
- Analysis of a class of models describing semiflexible polymers. In particular, we have derived a functional CLT, and deduced from it bounds on the free energy of a semiflexible polymer confined to a tube [27].
- Analysis of the effect of a line of defects on the correlation length of a subcritical Bernoulli bond percolation model [35]. This can be considered as the first analysis of a non-effective version of the problem of a polymer pinned by a line, not relying on exact computations. This analysis was extended to FK-percolation (and thus Ising/Potts models in [43].

## Phase separation in lattice spin systems

These papers are about the properties of interfaces in lattice spin systems.

- Analysis of the wetting transition for the Ising model in the canonical ensemble (i.e. with fixed magnetization), microscopic derivation of the Winterbottom construction: the two-dimensional results can be found in [1, 3, 5], while the corresponding results for higher dimensions can be found in [12]; see also the general review about equilibrium crystal shapes [8].
- Analysis of a macroscopic manifestation of the wetting transition in the 2D Ising model in the grand-canonical ensemble (which turns out to be a nice example of a (multiply) reentrant phase transition) [4, 5].
- Analysis of the growth of the layer of minus spins (critical prewetting) in a 2D Ising model with a positive magnetic field but a wall favoring the minus phase, as the magnetic field goes to zero [17].
- Proof, based on the Ornstein-Zernike theory, of a finite-volume version of the celebrated Aizenman-Higuchi Theorem (stating that all Gibbs states of the 2d Ising model are convex combinations of $\mu^+_\beta$ and $\mu^-_\beta$) [32]. The extension to general $q$-state Potts models on $\mathbb{Z}^2$ at temperatures $T<T_c(q)$ can 7 be found in [37] (even the infinite-volume claim being new in this case).
- Proof that the interface of the Potts model on $\mathbb{Z}^2$ is localized (pinned) by a row of reduced coupling constants [43].

## Others

The following are various other topics I have worked on, with various collaborators.

- Derivation of a reverse Poincaré inequality for percolation models enjoying the finite energy and strong FKG properties. Among other things, it allows to obtain a quantitative version of the Burton-Keane uniqueness statement for this class of models, and an upper bound on the probability of the four-arm event for FK model with general $q\geq 1$ [40].
- Proof of the absence of continuous symmetry breaking (Mermin-Wagner type theorem) for 2-dimensional spin systems without assuming smoothness of the interaction [13]. Proof of an algebraic upper bound on the 2-point correlation functions in a general class of two-dimensional $O(N)$-symmetric spin systems with not necessarily smooth, possible long-range interactions [38].
- Extension of the random-cluster (or Fortuin-Kasteleyn) representation to a class of models including the Ashkin-Teller model, and study of some basic properties (duality in 2D, FKG, comparison inequalities) [2].
- Proof of the existence of an isotropic/nematic phase transition in a model of rods on a lattice with purely hard-core interactions [21].
- Analysis of the self-affine structure of the corrections to the Ergodic Theorem for the Pascal-adic transformation [19].
- Extension of Propp's generalized domino shuffling algorithm, specially tailored to deal with cases when some of the weights are zero. This allows one to tile efficiently a large class of planar graphs with dimers, by embedding them in a large enough Aztec diamond. Estimates on the size of this Aztec diamond are also given [22].
- Derivation of the scaling limit of the (kinetic version of the) prudent self-avoiding walk, including a computation of its speed (in $\mathbb{L}^1$-norm) [29]. The limiting process displays several atypical features; in particular, in spite of its being ballistic, the walker has no asymptotic direction (we compute the nontrivial law of the angle), and displays ageing properties.
- Analysis of a model of random permutations weighted according to the distribution of cycle lengths. Determination of the length of typical cycles for various classes of weights [30].