Related papers: The Pfaffian Sign Theorem for the Dimer Model on a…
A. Albouy and R. Moeckel in 2000 found some interesting inequalities related to the inverse problem for collinear (Moulton) central configurations: the Pfaffian of a certain matrix is positive since all coefficients of some polynomials are…
The Frobenius-Schur indicator is a parameter $\kappa_a=\pm 1$ assigned to each self-dual particle $a$ in a TQFT. If $\kappa_a$ is negative then straightening out a timelike zig-zag in the worldline of a particle of type $a$ can incur a…
We explicitly show that the Rokhsar-Kivelson dimer model on the triangular lattice is a liquid with topological order. Using the Pfaffian technique, we prove that the difference in local properties between the two topologically degenerate…
Fermionic Gaussian operators are foundational tools in quantum many-body theory, numerical simulation of fermionic dynamics, and fermionic linear optics. While their structure is fully determined by two-point correlations, evaluating their…
We consider fermionic fully-packed loop and quantum dimer models which serve as effective low-energy models for strongly correlated fermions on a checkerboard lattice at half and quarter filling, respectively. We identify a large number of…
A half-tree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a half-tree theorem for the Pfaffian principal minors of a skew-symmetric matrix whose column sum is zero; introducing an…
Euclidean dense matter generically suffers from the fermion sign problem. However, we argue that the sign problem is absent if one considers only low-energy degrees of freedom. Specifically, the low energy effective theory of dense QCD has…
We present a KAM theorem for presymplectic dynamical systems. The theorem has a " a posteriori " format. We show that given a Diophantine frequency $\omega$ and a family of presymplectic mappings, if we find an embedded torus which is…
We introduce a new symmetry class of domino tilings of the Aztec diamond, called the off-diagonal symmetry class, which is motivated by the off-diagonally symmetric alternating sign matrices introduced by Kuperberg in 2002. We use the…
We prove an analog of the Caffarelli-Kohn-Nirenberg theorem for weak solutions of a system of PDE that model a viscoelastic fluid in the presence of an energy damping mechanism. The system was recently introduced as a possible method of…
We consider the equation $\Delta_x u+u_{yy}+f(u)=0,\ x=(x_1,\dots,x_N)\in\mathbb{R}^N,\ y\in \mathbb{R},$ where $N\geq 2$ and $f$ is a sufficiently smooth function satisfying $f(0)=0$, $f'(0)<0$, and some natural additional conditions. We…
This work shows that, in one dimension, due to its topology, a closed-loop product of short-time propagators is always positive, despite the fact that each anti-symmetric free fermion propagator can be of either sign.
We consider the class of autonomous systems $\dot x=f(x)$, where $x \in {\bf R}^{2n}$, $f \in C^1({\bf R}^{2n})$ whose phase portrait is a Cartesian product of $n$ two-dimensional {\em centres}. We also consider perturbations of this…
Consider a semi-infinite skew-symmetric moment matrix, $m_{\iy}$ evolving according to the vector fields $\pl m / \pl t_k=\Lb^k m+m \Lb^{\top k} ,$ where $\Lb$ is the shift matrix. Then the skew-Borel decomposition $ m_{\iy}:= Q^{-1} J…
Recent work by Forsg{\aa}rd indicates that not every convex lattice polygon arises as the characteristic polygon of an affine dimer or, equivalently, an admissible oriented line arrangement on the torus in general position. We begin the…
Let S be a flat surface of genus g with cone type singularities. Given a bipartite graph G isoradially embedded in S, we define discrete analogs of the 2^{2g} Dirac operators on S. These discrete objects are then shown to converge to the…
We study asymptotics of the dimer model on large toric graphs. Let $\mathbb L$ be a weighted $\mathbb{Z}^2$-periodic planar graph, and let $\mathbb{Z}^2 E$ be a large-index sublattice of $\mathbb{Z}^2$. For $\mathbb L$ bipartite we show…
We study the Pfaff lattice, introduced by us in the context of a Lie algebra splitting of gl(infinity) into sp(infinity) and lower-triangular matrices. We establish a set of bilinear identities, which we show to be equivalent to the Pfaff…
In this paper we investigate the effect of nonlinear damping on the Lugiato-Lefever equation $$ \i \partial_t a = -(\i-\zeta) a - da_{xx} -(1+\i\kappa)|a|^2a +\i f $$ on the torus or the real line. For the case of the torus it is shown that…
We show that the number Z of q-edge-colourings of a simple regular graph of degree q is deducible from functions describing dimers on the same graph, viz. the dimer generating function or equivalently the set of connected dimer correlation…