Non-contractible loops in the dense O(n) loop model on the cylinder

A lattice model of critical dense polymers $O(0)$ is considered for the finite cylinder geometry. Due to the presence of non-contractible loops with a fixed fugacity $\xi$, the model is a generalization of the critical dense polymers solved by Pearce, Rasmussen and Villani. We found the free energy for any height $N$ and circumference $L$ of the cylinder. The density $\rho$ of non-contractible loops is found for $N \rightarrow \infty$ and large $L$. The results are compared with those obtained for the anisotropic quantum chain with twisted boundary conditions. Using the latter method we obtained $\rho$ for any $O(n)$ model and an arbitrary fugacity.Comment: arXiv admin note: text overlap with arXiv:0810.223

The traveling salesman problem, conformal invariance, and dense polymers

We propose that the statistics of the optimal tour in the planar random Euclidean traveling salesman problem is conformally invariant on large scales. This is exhibited in power-law behavior of the probabilities for the tour to zigzag repeatedly between two regions, and in subleading corrections to the length of the tour. The universality class should be the same as for dense polymers and minimal spanning trees. The conjectures for the length of the tour on a cylinder are tested numerically.Comment: 4 pages. v2: small revisions, improved argument about dimensions d>2. v3: Final version, with a correction to the form of the tour length in a domain, and a new referenc

Infinite canonical super-Brownian motion and scaling limits

We construct a measure valued Markov process which we call infinite canonical super-Brownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on $\Z^d$ when these objects are (a) critical; (b) mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We conjecture that it also arises as the scaling limit in various other models above the upper-critical dimension, such as the incipient infinite lattice tree above 8 dimensions, the incipient infinite cluster for unoriented percolation, uniform spanning trees above 4 dimensions, and invasion percolation above 6 dimensions. This paper also serves as a survey of recent results linking super-Brownian to scaling limits in statistical mechanics.Comment: 34 page

The ZZ-invariant massive Laplacian on isoradial graphs

We introduce a one-parameter family of massive Laplacian operators $(\Delta^{m(k)})_{k\in[0,1)}$ defined on isoradial graphs, involving elliptic functions. We prove an explicit formula for the inverse of $\Delta^{m(k)}$, the massive Green function, which has the remarkable property of only depending on the local geometry of the graph, and compute its asymptotics. We study the corresponding statistical mechanics model of random rooted spanning forests. We prove an explicit local formula for an infinite volume Boltzmann measure, and for the free energy of the model. We show that the model undergoes a second order phase transition at $k=0$, thus proving that spanning trees corresponding to the Laplacian introduced by Kenyon are critical. We prove that the massive Laplacian operators $(\Delta^{m(k)})_{k\in(0,1)}$ provide a one-parameter family of $Z$-invariant rooted spanning forest models. When the isoradial graph is moreover $\mathbb{Z}^2$-periodic, we consider the spectral curve of the characteristic polynomial of the massive Laplacian. We provide an explicit parametrization of the curve and prove that it is Harnack and has genus $1$. We further show that every Harnack curve of genus $1$ with $(z,w)\leftrightarrow(z^{-1},w^{-1})$ symmetry arises from such a massive Laplacian.Comment: 71 pages, 13 figures, to appear in Inventiones mathematica

From elongated spanning trees to vicious random walks

Given a spanning forest on a large square lattice, we consider by combinatorial methods a correlation function of $k$ paths ($k$ is odd) along branches of trees or, equivalently, $k$ loop--erased random walks. Starting and ending points of the paths are grouped in a fashion a $k$--leg watermelon. For large distance $r$ between groups of starting and ending points, the ratio of the number of watermelon configurations to the total number of spanning trees behaves as $r^{-\nu} \log r$ with $\nu = (k^2-1)/2$. Considering the spanning forest stretched along the meridian of this watermelon, we see that the two--dimensional $k$--leg loop--erased watermelon exponent $\nu$ is converting into the scaling exponent for the reunion probability (at a given point) of $k$ (1+1)--dimensional vicious walkers, $\tilde{\nu} = k^2/2$. Also, we express the conjectures about the possible relation to integrable systems.Comment: 27 pages, 6 figure

Traffic flow densities in large transport networks

We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flowing according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a link-density and a sub-ballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function.Comment: 20 pages, 7 figure