I am a mathematician working mostly at the intersection of Probability Theory and Statistical Physics. For my publication list, see below. Here is a short summary of my academic life up to now:

1993	Physics Diploma on some problem of Topological Field Theory at the University of Geneva, Switzerland.
1993-1997	PhD thesis at the Institute for Theoretical Physics, Swiss Federal Institute of Technology in Lausanne (ÉPF-L), Switzerland, under the supervision of Charles Pfister.
1997-1999	Scientific collaborator in the group of Jean-Dominique Deuschel, Mathematics Department of the Technische Universität Berlin, Germany, in the framework of the DFG Schwerpunkt project.
Oct. 1999-Feb. 2000	Four months stay at the Institute for Mathematical Research (FIM), Swiss Federal Institute of Technology in Zürich (ETH-Z) and at the Mathematics Department of Zürich University, Switzerland, at the invitation of Alain-Sol Sznitman and Erwin Bolthausen.
Feb. 2000-Sep. 2000	Scientific collaborator in the group of Dima Ioffe, Technion, Haifa, Israel. Financed by a 2-years "advanced researcher grant" of the Swiss National Science Foundation.
Oct. 2000-Sep. 2003	CNRS research fellow at the Laboratoire d'Analyse, Topologie et Probabilités, Université de Provence.
Oct. 2003-Sep. 2006	CNRS research fellow at the Laboratoire de Mathématiques Raphaël Salem, Université de Rouen.
Nov. 2003	Habilitation à Diriger des Recherches.
Oct. 2006-	Full professor, Section de Mathématiques, Université de Genève.

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. Here 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 below; 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 [35]. 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).

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].
• 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 [35]. 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 for non-directed models of a polymer in a quenched random environment at weak disorder, both for crossing and stretched ensembles [31], [33].
• 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].

Phase separation in lattice spin systems These papers are about the properties of interfaces in lattice spin systems. Consider, e.g., an Ising model in the phase coexistence regime. It is possible, through suitable boundary conditions and/or by fixing the total magnetization, to enforce coexistence of the two equilibrium phases in a given vessel. Thermodynamics predicts that the geometry of the domain thus obtained should be determined by minimizing the total surface tension (subject to the geometrical constraints imposed by the b.c. and/or fixed magnetization). In this series of papers with various collaborators, we were interested in deriving this variational principle from first principles. Our main emphasis is on the effect of boundary fields on the macroscopic geometry, and on the associated wetting transition. Among the results obtained are:

• 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].

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

• Proof of the absence of continuous symmetry breaking (Mermin-Wagner type theorem) for 2-dimensional spin systems without assuming smoothness of the interaction [13].
• 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].
• 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].
• Analysis of the effect of a line of defects on the correlation length of a subcritical Bernoulli bond percolation model [34]. 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.

• Vincent Beffara (ENS Lyon, France)
• Volker Betz (University of Warwick, UK)
• Thierry Bodineau (ENS Paris, France)
• Erwin Bolthausen (Universität Zürich, Switzerland)
• Massimo Campanino (Università di Bologna, Italy)
• Pietro Caputo (Università Roma 3, Italy)
• Loren Coquille (Université de Genève, Switzerland)
• Jean-Dominique Deuschel (TU Berlin, Germany)
• Sacha Friedli (UFMG, Brazil)
• Giambattista Giacomin (Université Paris 7, France)
• Ostap Hryniv (Durham University, UK)
• Dima Ioffe (Technion, Israel)
• Élise Janvresse (Université de Rouen, France)
• Charles Pfister (ÉPF-L, Switzerland).
• Thierry de la Rue (Université de Rouen, France)
• Senya Shlosman (CPT, Luminy, France)
• Daniel Ueltschi (University of Warwick, UK)
• Milo¨ Zahradník (Charles University, Prague, Czech Republic)

Papers

• [35] Self-Attractive Random Walks: The Case of Critical Drifts D. Ioffe and Y. Velenik Accepted for publication in Commun. Math. Phys. (2011). [Abstract, PDF, bibtex]
• [34] Subcritical Percolation with a Line of Defects S. Friedli, D. Ioffe and Y. Velenik Accepted for publication in the Annals of Probability (2011). [Abstract, PDF, bibtex, slides]
• [33] Stretched Polymers in Random Environment D. Ioffe and Y. Velenik Festschrift in honor of J. Gärtner and E. Bolthausen, to appear (2011). [Abstract, PDF, bibtex]
• [32] A Finite-Volume Version of Aizenman-Higuchi Theorem for the 2d Ising model L. Coquille and Y. Velenik Accepted for publication in Probab. Theory Relat. Fields (2010). [Abstract, PDF, Published version, bibtex]
• [31] Crossing Random Walks and Stretched Polymers at Weak Disorder D. Ioffe and Y. Velenik Accepted for publication in the Annals of Probability (2010). [Abstract, PDF, bibtex]
• [30] Random Permutations with Cycle Weights V. Betz, D. Ueltschi and Y. Velenik Ann. Appl. Probab. 21, 312-331 (2011). [Abstract, PDF, Published version, bibtex]
• [29] Scaling Limit of the Prudent Walk V. Beffara, S. Friedli and Y. Velenik Electron. Commun. Probab. 15, 44-58 (2010). [Abstract, PDF, Published version, bibtex]
• [28] The Statistical Mechanics of Stretched Polymers D. Ioffe and Y. Velenik Braz. J. Probab. Stat. 24 (2), 279-299 (2010). [Abstract, PDF, Published version, bibtex]
• [27] Some Rigorous Results about Semiflexible Polymers I. Free and Confined Polymers O. Hryniv and Y. Velenik Stoch. Proc. Appl. 119, 3081-3100 (2009). [Abstract, PDF, Published version, bibtex]
• [26] Wetting of Gradient Fields: Pathwise Estimates Y. Velenik Probab. Theory Relat. Fields 143, 379-399 (2009). [Abstract, PDF, Published version, bibtex]
• [25] Ballistic Phase of Self-Interacting Random Walks D. Ioffe and Y. Velenik In "Analysis and Stochastics of Growth Processes and Interface Models", P. Mörters et al. (eds), Oxford University Press 55-79 (2008). [Abstract, PDF, bibtex]
• [24] Fluctuation Theory of Connectivities for Subcritical Random Cluster Models M. Campanino, D. Ioffe and Y. Velenik Ann. Probab. 36, 1287-1321 (2008). [Abstract, PDF, Published version, bibtex, slides]
• [23] Localization and Delocalization of Random Interfaces Y. Velenik Probab. Surv. 3, 112-169 (2006). [Abstract, Published version, bibtex]
• [22] A Note on Domino Shuffling É. Janvresse, T. de la Rue and Y. Velenik Electron. J. Combin. 13 (1), R30, (2006). [Abstract, PDF, Published version, errata, bibtex]
• [21] Entropy-Driven Phase Transition in a Polydisperse Hard-Rods Lattice System D. Ioffe, Y. Velenik and M. Zahradník J. Stat. Phys. 122, no. 4, 761-786 (2006). [Abstract, PDF, published version, bibtex, slides]
• [20] Pinning by a Sparse Potential É. Janvresse, T. de la Rue and Y. Velenik Stoch. Proc. Appl. 115, No 8, 1323-1331 (2005). [Abstract, PDF, Published version, bibtex]
• [19] Self-Similar Corrections to the Ergodic Theorem for the Pascal-Adic Transformation É. Janvresse, T. de la Rue and Y. Velenik Stoch. Dyn. 5, No 1, 1-25 (2005). [Abstract, PDF, Published version, bibtex, slides, slides+comments]
• [18] Universality of Critical Behaviour in a Class of Recurrent Random Walks O. Hryniv and Y. Velenik Probab. Theory Relat. Fields 130, 222-258 (2004). [Abstract, PDF, Published version, bibtex]
• [17] Entropic Repulsion of an Interface in an External Field Y. Velenik Probab. Theory Relat. Fields 129, 83-112 (2004). [Abstract, PDF, Published version, bibtex]
• [16] Random Path Representation and Sharp Correlations Asymptotics at High-Temperatures M. Campanino, D. Ioffe and Y. Velenik Stochastic Analysis on Large Scale Interacting Systems, Advanced Studies in Pure Mathematics 39 (T. Funaki and H. Osada ed.), 29-52 (2004). [Abstract, PDF, bibtex]
• [15] Rigorous Non-Perturbative Ornstein-Zernike Theory for Ising Ferromagnets M. Campanino, D. Ioffe and Y. Velenik Europhys. Lett. 62, no. 2, 182-188 (2003). [Abstract, PDF, Published version, bibtex]
• [14] Ornstein-Zernike Theory for Finite-Range Ising Models Above $T_{\rm c}$ M. Campanino, D. Ioffe and Y. Velenik Probab. Theory Relat. Fields 125 305-349 (2003). [Abstract, PDF, Published version, bibtex]
• [13] 2D Models of Statistical Physics with Continuous Symmetry: The Case of Singular Interactions D. Ioffe, S. Shlosman and Y. Velenik Commun. Math. Phys. 226, 433-454 (2002). [Abstract, PDF, Published version, bibtex]
• [12] Winterbottom Construction for Finite-Range Ferromagnetic Models: an $L_1$ Approach T. Bodineau, D. Ioffe and Y. Velenik J. Stat. Phys. 105, 93-131 (2001). [Abstract, PDF, Published version, bibtex]
• [11] Critical Behavior of the Massless Free Field at the Depinning Transition E. Bolthausen and Y. Velenik Commun. Math. Phys. 223, 161-203 (2001). [Abstract, PDF, Published version, bibtex]
• [10] On Entropic Reduction of Fluctuations T. Bodineau, G. Giacomin and Y. Velenik J. Stat. Phys. 102, 1439-1445 (2001). [Abstract, PDF, Published version, bibtex]
• [9] A Note on Wetting Transition for Gradient Fields P. Caputo and Y. Velenik Stoch. Proc. Appl. 87, 107-113 (2000). [Abstract, PDF, Published version, bibtex]
• [8] Rigorous Probabilistic Analysis of Equilibrium Crystal Shapes T. Bodineau, D. Ioffe and Y. Velenik J. Math. Phys. 41, 1033-1098 (2000). [Abstract, PDF, Published version, bibtex]
• [7] A Note on the Decay of Correlations Under δ-Pinning D. Ioffe and Y. Velenik Probab. Theory Relat. Fields 116, 379-389 (2000). [Abstract, PDF, Published version, bibtex]
• [6] Non-Gaussian Surface Pinned by a Weak Potential J.-D. Deuschel and Y. Velenik Probab. Theory Relat. Fields 116, 359-377 (2000). [Abstract,PDF, Published version, bibtex]
• [5] Macroscopic Description of Phase Separation in the 2D Ising Model C.-E. Pfister and Y. Velenik in Mathematical Results in Statistical Mechanics, S. Miracle-Sole, J. Ruiz and V. Zagrebnov eds., World Scientific, 121-135 (1999). [Abstract, PDF, bibtex]
• [4] Interface, Surface Tension and Reentrant Pinning Transition in the 2D Ising model C.-E. Pfister and Y. Velenik Commun. Math. Phys. 204, 269-312 (1999). [Abstract, PDF, Published version, bibtex]
• [3] Large Deviations and Continuum Limit in the 2D Ising Model C.-E. Pfister and Y. Velenik Probab. Theory Relat. Fields 109, 435-506 (1997). [Abstract, PDF, Published version, bibtex]
• [2] Random-Cluster Representation of the Ashkin-Teller Model C.-E. Pfister and Y. Velenik J. Stat. Phys. 88, 1295-1331 (1997). [Abstract, PDF, Published version, bibtex]
• [1] Mathematical Theory of the Wetting Phenomenon in the 2D Ising Model C.-E. Pfister and Y. Velenik Helv. Phys. Acta 69, 949-973 (1996). [Abstract, PDF, Published version, bibtex]

Other

• Le modèle d'Ising, lecture notes (in french) for a course given at the University of Geneva (master level), 2008-2009. [file]
• Introduction aux champs aléatoires markoviens et gibbsiens, lecture notes (in french) for a course given at the University of Geneva (master level), 2006-2007. [file]
• Habilitation à Diriger des Recherches, Université de Provence (Nov. 2003). [PDF]
• PhD thesis, ÉPF-L 1712 (1997). [PDF. Errata: PDF]
• Petite collection d'informations utiles pour collectionneur compulsif (popularization; in french) S. Sardy and Y. Velenik Images des Mathématiques, CNRS (2010). [URL]
• Paninimania: sticker rarity and cost-effective strategy (popularization) S. Sardy and Y. Velenik Swiss Statistical Society, Bulletin nr. 6 (2010), 2-6. [PDF]
• La loi de Benford (popularization; in french) É. Janvresse, T. de la Rue and Y. Velenik Poster (2005). [PDF]
• Le phénomène du cercle arctique (popularization; in french) É. Janvresse, T. de la Rue and Y. Velenik Poster (2005). [PDF]
• Savez-vous jouer aux choux ? (popularization; in french) É. Janvresse, T. de la Rue and Y. Velenik Tangente 105 (2005).
• Les Brisures de Symétrie (popularization; in french) É. Janvresse and Y. Velenik Tangente 85 (2002).