Related papers: Spanning Forests on Random Planar Lattices
We propose a matrix-model derivation of the scaling exponents of the critical and tricritical q-states Potts model coupled to gravity on a sphere. In close analogy with the $O(n)$ model, we reduce the determination of the one-loop-to-vacuum…
We investigate numerically and analytically Potts models on ``thin'' random graphs -- generic Feynman diagrams, using the idea that such models may be expressed as the N --> 1 limit of a matrix model. The thin random graphs in this limit…
We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-like structure of the lattice this model behaves…
The function $$W(aq,b)=\int\int_0^{2\pi}\ln[1-a\cos x-b\cos y-(1-a-b)\cos(x+y)]dxdy$$ which expresses the spanning-tree entropy for various two dimensional lattices, for example, is evaluated directly in terms of standard functions. It is…
Let $\mathcal{L}(T,\lambda)=\sum_{k=0}^n(-1)^{k}c_{k}(T)\lambda^{n-k}$ be the characteristic polynomial of its Laplacian matrix of a tree $T$. This paper studied some properties of the generating function of the coefficients sequence $(c_0,…
We present an exact solution of the $O(n)$ model on a random lattice. The coupling constant space of our model is parametrized in terms of a set of moment variables and the same type of universality with respect to the potential as observed…
While Flory theories provide an extremely useful framework for understanding the behavior of interacting, randomly branching polymers, the approach is inherently limited. Here we use a combination of scaling arguments and computer…
We compute in small temperature expansion the two-loop renormalization constants and the three-loop coefficient of the beta-function, that is the first non-universal term, for the sigma-model with O(N) invariance on the triangular lattice…
Random spanning trees are among the most prominent determinantal point processes. We give four examples of random spanning trees on ladder-like graphs whose rungs form stationary renewal processes or regenerative processes of order two,…
The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial…
We study the asymptotic expansion of the determinant of the graph Laplacian associated to discretizations of a half-translation surface endowed with a flat unitary vector bundle. By doing so, over the discretizations, we relate the…
For $n$-vertex, $d$-dimensional lattices $\Lambda$ with $d \ge 2$, the number of spanning trees $N_{ST}(\Lambda)$ grows asymptotically as $\exp(n z_\Lambda)$ in the thermodynamic limit. We present an exact closed-form result for the…
We define the notion of a spanning tree generating function (STGF) $\sum a_n z^n$, which gives the spanning tree constant when evaluated at $z=1,$ and gives the lattice Green function (LGF) when differentiated. By making use of known…
On a finite graph, there is a natural family of Boltzmann probability measures on cycle-rooted spanning forests, parametrized by weights on cycles. For a certain subclass of those weights, we construct Gibbs measures in infinite volume, as…
Uniform spanning trees are a statistical model obtained by taking the set of all spanning trees on a given graph (such as a portion of a cubic lattice in d dimensions), with equal probability for each distinct tree. Some properties of such…
Using generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically…
We obtain long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Enting's finite lattice method (FLM) with exact transfer matrix enumerations. The models encompass…
We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case…
Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any bounded-degree n-vertex graph, the union of two random spanning trees…
We define matrix models that converge to the generating functions of a wide variety of loop models with fugacity taken in sets with an accumulation point. The latter can also be seen as moments of a non-commutative law on a subfactor planar…