We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on nonamenable graphs. While we rely on the description of the LRRW as a mixture of Markov chains, the proof does not use the magic formula. We also derive analogous results for the vertex reinforced jump process. * Supported in part by NSERC † Supported in part at the Technion by a Landau fellowship ‡ Supported in part by the Israel Science Foundation Diaconis & Freedman [2, Theorem 7] showed that a recurrent partially exchangeable process is a mixture of Markov chains. Today the name random walk in random environment is more popular than mixture of Markov chains, but they mean the same thing: that there is some measure µ on the space of Markov chains (known as the "mixing measure") such that the process first picks a Markov chain using µ and then walks according to this Markov chain. In particular, this result applies to the LRRW whenever it is recurrent. There "recurrent" means that it returns to its starting vertex infinitely often. We find this result, despite its simple proof (it follows from de Finetti's theorem for exchangeable processes, itself not a very difficult theorem) to be quite deep. Even for LRRW on a graph with three vertices it gives non-trivial information. For general exchangeable processes recurrence is necessary; see [2, Example 19c] for an example of a partially exchangeable process which is not a mixture of Markov chains. For LRRW this cannot happen, it is a mixture of Markov chains even when it is not recurrent (see Theorem 4 below).On finite graphs, the mixing measure µ has an explicit expression, known fondly as the "magic formula". See [11] for a survey of the formula and the history of its discovery. During the last decade significant effort was invested to understand the magic formula, with the main target the recurrence of the process in two dimensions, a conjecture dating back to the 80s (see e.g. [16, §6]). Notably, Merkl and Rolles [14] showed, for any fixed a, that LRRW on certain "dilute" two dimensional graphs is recurrent, though the amount of dilution needed increases with a. Their approach did not work for Z 2 , but required stretching each edge of the lattice to a path of length 130 (or more). The proof uses the explicit form of the mixing measure, which turns out to be amenable to entropy arguments. These methods involve relative entropy arguments which also lead to the Mermin-Wagner theorem [10]. This connection suggests that the methods should not apply in higher dimension.An interesting variation on this theme is when each directed edge has a weight. When one crosses an edge one increases only the weight in the direction one has crossed. This process is also partially exchangeable, and is also described by a random walk in a random environment. On the plus ...

We consider a class of spin systems on Z d with vector valued spins (S x ) that interact via the pair-potentials J x,y S x · S y . The interactions are generally spreadout in the sense that the J x,y 's exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions d ≥ 3, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions d = 1, 2 for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique "state," then in any sequence of translation-invariant Gibbs states various observables converge to their meanfield values and the states themselves converge to a product measure.

We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $$\beta >0$$ β > 0 per edge. This is called the arboreal gas model, and the special case when $$\beta =1$$ β = 1 is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $$p=\beta /(1+\beta )$$ p = β / ( 1 + β ) conditioned to be acyclic, or as the limit $$q\rightarrow 0$$ q → 0 with $$p=\beta q$$ p = β q of the random cluster model. It is known that on the complete graph $$K_{N}$$ K N with $$\beta =\alpha /N$$ β = α / N there is a phase transition similar to that of the Erdős–Rényi random graph: a giant tree percolates for $$\alpha > 1$$ α > 1 and all trees have bounded size for $$\alpha <1$$ α < 1 . In contrast to this, by exploiting an exact relationship between the arboreal gas and a supersymmetric sigma model with hyperbolic target space, we show that the forest constraint is significant in two dimensions: trees do not percolate on $${\mathbb {Z}}^2$$ Z 2 for any finite $$\beta >0$$ β > 0 . This result is a consequence of a Mermin–Wagner theorem associated to the hyperbolic symmetry of the sigma model. Our proof makes use of two main ingredients: techniques previously developed for hyperbolic sigma models related to linearly reinforced random walks and a version of the principle of dimensional reduction.

This paper studies a generalization of the Curie-Weiss model (the Ising model on a complete graph) to quantum mechanics. Using a natural probabilistic representation of this model, we give a complete picture of the phase diagram of the model in the parameters of inverse temperature and transverse field strength. Further analysis computes the critical exponent for the vanishing of the order parameter in the approach to the critical curve and gives useful stability properties for a variational problem associated with the representation.

No abstract