We show that the dimer model on a bipartite graph $\Gamma$ on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space ${Ł}_{\Gamma }$ of line bundles with connections on the graph $\Gamma$. The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs ${\Gamma }_1$ and ${\Gamma }_2$ areequivalentif the Newton polygons of the corresponding partition functions...

The sequence of random probability measures νn that gives a path of length n, $\frac{1}{n}$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment. Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.

There is enough evidence to re-examine disclinations and hedgehogs, the singularities often observed in nematic liquid crystals, in the light of a new theory that allows for local changes in the degree of orientation.

Let $\sum$ be a flat surface of genus $g$ with cone type singularities. Given a bipartite graph $\Gamma$ isoradially embedded in $\sum$, we define discrete analogs of the $2^{2g}$ Dirac operators on $\sum$. These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair $\Gamma \subset \sum$ for these discrete Dirac operators to be Kasteleyn matrices of the graph $\Gamma$. As a consequence, if these conditions are met, the partition function of the dimer...

Recently the renormalization group predictions on the effect of disorder on pinning models have been put on mathematical grounds. The picture is particularly complete if the disorder is relevant or irrelevant in the Harris criterion sense: the question addressed is whether quenched disorder leads to a critical behavior which is different from the one observed in the pure, i.e. annealed, system. The Harris criterion prediction is based on the sign of the specific heat exponent of the pure system,...

We study the continuous time version of the random walk pinning model, where conditioned on a continuous time random walk (Ys)s≥0 on ℤd with jump rate ρ > 0, which plays the role of disorder, the law up to time t of a second independent random walk (Xs)0≤s≤t with jump rate 1 is Gibbs transformed with weight eβLt(X,Y), where Lt(X, Y) is the collision local time between X and Y up to time t. As the inverse temperature β varies, the model undergoes a localization–delocalization transition at...

Two new time-dependent versions of div-curl results in a bounded domain $\Omega \subset {ℝ}^3$ are presented. We study a limit of the product $v_k{w}_k$, where the sequences $v_k$ and $w_k$ belong to ${Ł}_2\left(\Omega \right)$. In Theorem 2.1 we assume that $\nabla \times v_k$ is bounded in the $L_p$-norm and $\nabla ·w_k$ is controlled in the $L_r$-norm. In Theorem 2.2 we suppose that $\nabla \times w_k$ is bounded in the $L_p$-norm and $\nabla ·w_k$ is controlled in the $L_r$-norm. The time derivative of $w_k$ is bounded in both cases in the norm of $-1\left(\Omega \right)$. The convergence (in the sense of distributions) of $v_k{w}_k$ to the product $vw$ of weak limits...

In this note, we prove a version of the conjectured duality for Schramm-Loewner Evolutions, by establishing exact identities in distribution between some boundary arcs of chordal ${\mathrm{SLE}}_{\kappa }$, $\kappa \>4$, and appropriate versions of ${\mathrm{SLE}}_{\widehat{\kappa }}$, $\widehat{\kappa }=16/\kappa$.

Computer simulation of a few thousands of particles moving (approximately) according to the energy and momentum conservation laws on a tessellation of squares in discrete time steps and interacting according to the predator-prey scheme is analyzed. The population dynamics are described by the basic Lotka-Volterra interactions (multiplication of preys, predation and multiplication of predators, death of predators), but the spatial effects result in differences between the system evolution and the...

Using the Maxwell-Higgs model, we prove that linearly unstable symmetric vortices in the Ginzburg-Landau theory are dynamically unstable in the H1 norm (which is the natural norm for the problem).In this work we study the dynamic instability of the radial solutions of the Ginzburg-Landau equations in R2 (...)