A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs, with Xiaolin Zeng, accepted in J. Amer. Math. Soc. preprint.

Abstract : This paper concerns the Vertex reinforced jump process (VRJP), the Edge reinforced random walk (ERRW) and their link with a random Schrödinger operator. On infinite graphs, we define a 1-dependent random potential β extending that defined in [16] on finite graphs, and consider its associated random Schrödinger operator H_β. We construct a random function ψ as a limit of martingales, such that ψ=0 when the VRJP is recurrent, and ψ is a positive generalized eigenfunction of the random Schrödinger operator with eigenvalue 0, when the VRJP is transient. Then, we prove a representation of the VRJP on infinite graphs as a mixture of Markov jump processes involving the function ψ, the Green function of the random Schrödinger operator and an independent Gamma random variable. On ℤ^d, we deduce from this representation a zero-one law for recurrence or transience of the VRJP and the ERRW, and a functional central limit theorem for the VRJP and the ERRW at weak reinforcement in dimension d≥3, using estimates of [9,7]. Finally, we deduce recurrence of the ERRW in dimension d=2 for any initial constant weights (using the estimates of Merkl and Rolles, [12,14]), thus giving a full answer to the old question of Diaconis. We also raise some questions on the links between recurrence/transience of the VRJP and localization/delocalization of the random Schrödinger operator H_β.

Edge-reinforced random walk, Vertex-Reinforced Jump Process and the supersymmetric hyperbolic sigma model, with Pierre Tarrès, Journal of the European Math. Society, Volume 17, Issue 9, 2015, pp. 2353–2378.

Abstract : Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process that takes values in the vertex set of a graph G, which is more likely to cross edges it has visited before. We show that it can be interpreted as an annealed version of the Vertex-reinforced jump process (VRJP), conceived by Werner and first studied by Davis and Volkov (2002,2004), a continuous-time process favouring sites with more local time. We calculate, for any finite graph G, the limiting measure of the centred occupation time measure of VRJP, and interpret it as a supersymmetric hyperbolic sigma model in quantum field theory. This enables us to deduce that VRJP is recurrent in any dimension for large reinforcement, using a localisation result of Disertori and Spencer

Random Dirichlet environment viewed from the particle in dimension $d\ge 3$, Ann. Probab. 41 (2013), no. 2, 722–743,

Abstract : We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On Z^d, RWDE are parameterized by a 2d-tuple of positive reals called weights. In this paper, we characterize for d ≥ 3 the weights for which there exists an absolutely continuous invariant probability distribution for the process viewed from the particle. We can deduce from this result and from [27] a complete description of the ballistic regime for d ≥ 3.

Random Walks in Random Dirichlet Environment are transient in dimension $d\ge 3$, Probability Theory and Related Fields, October 2011, Volume 151, Issue 1, pp 297–317

We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On Z^d, RWDE are parameterized by a 2d-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension d ≥ 3. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for d ≥ 3.

Limit laws for transient random walks in random environment on Z, with Olivier Zindy, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2469–2508.

Abstract : We consider transient random walks in random environment on Z with zero asymptotic speed. A classical result of H. Kesten, M. V. Kozlov and F. L. Spitzer [Compositio Math. 30 (1975), 145--168] says that the hitting time of the level n converges in law, after a proper normalization, towards a positive stable law, but they did not obtain a description of its parameter. A different proof of this result is presented that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.

Spectral properties of self-similar lattices and iteration of rational maps. Mém. Soc. Math. Fr. (N.S.) No. 92 (2003), vi+104 pp.

Abstract : In this text we consider discrete Laplace operators defined on lattices based on finitely-ramified self-similar sets, and their continuous analogous defined on the self-similar sets themselves. We are interested in the spectral properties of these operators. The basic example is the lattice based on the Sierpinski gasket. We introduce a new renormalization map which appears to be a rational map defined on a smooth projective variety (more precisely, this variety is isomorphic to a product of three types of Grassmannians: complex Grassmannians, Lagrangian Grassmannians, orthogonal Grassmannians). We relate some characteristics of the dynamics of its iterates with some characteristics of the spectrum of our operator. More specifically, we give an explicit formula for the density of states in terms of the Green current of the map, and we relate the indeterminacy points of the map with the so-called Neuman-Dirichlet eigenvalues which lead to eigenfunctions with compact support on the unbounded lattice. Depending on the asymptotic degree of the map we can prove drastically different spectral properties of the operators. Our formalism is valid for the general class of finitely ramified self-similar sets (i.e. for the class of pcf self-similar sets of Kigami). Hence, this work aims at a generalization and a better understanding of the initial work of the physicists Rammal and Toulouse on the Sierpinski gasket.

Existence and uniqueness of diffusions on finitely ramified self-similar fractals. Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 5, 605–673.

Abstract : We give a criterion for the existence and uniqueness or the non-existence of the diffusions on a finitely ramified self-similar fractal. In classical examples this criterion is easy to apply and in particular, it gives the uniqueness of the diffusion on nested fractals (Lindstrøm proved the existence in [19] but the problem of uniqueness remained unsolved) and completly solves the problem of existence and uniqueness in the case of the Sierpinski gasket with inhomogeneous weights.

This problem also gives a solution to a non trivial problem of fixed point for a non-linear, non-expansive map of a cone with the Hilbert's projective metric (cf. [23]).