Related papers: T-systems, networks and dimers
Let $T$ be a set, of {\em types}, and let $\iota,o:T\to\oZ_+$. A {\em $T$-diagram} is a locally ordered directed graph $G$ equipped with a function $\tau:V(G)\to T$ such that each vertex $v$ of $G$ has indegree $\iota(\tau(v))$ and…
We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for…
A finite dimensional operator that commutes with some symmetry group admits quotient operators, which are determined by the choice of associated representation. Taking the quotient isolates the part of the spectrum supporting the chosen…
We present an application of the Grassmann algebra to the problem of the monomer-dimer statistics on a two-dimensional square lattice. The exact partition function, or total number of possible configurations, of a system of dimers with a…
The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…
We study the implicit bias of gradient flow (i.e., gradient descent with infinitesimal step size) on linear neural network training. We propose a tensor formulation of neural networks that includes fully-connected, diagonal, and…
The dimer model is a classical statistical mechanics model which is exactly solvable in two dimensions, but about which little is known in higher dimensions. In analogy with large $N$ limits in lattice gauge theory, we study a large $N$…
We use the combination of ideas and results from the theory of graph limits and nonlinear evolution equations to provide a rigorous mathematical justification for taking continuum limit for certain nonlocally coupled networks and to extend…
Chern-Simons topological field theories TFTs are the only TFTs that have already found application in the description of some exotic strongly-correlated electron systems and the corresponding concept of topological quantum computing. Here,…
Based on the thermodynamic variation, we rigorously derive the sharp-interface model for solid-state dewetting on a flat substrate in the form of cylindrical symmetry. The governing equations for the model belong to fourth-order geometric…
A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot-Guldberg Lie algebra,…
Let F be a smooth real manifold with a linear connection in the tangent bundle. How can we extend the coefficients of the connection to bi-differential operators that incorporate the original structure at zero order? Take a constant mapping…
The periodic discrete Toda equation defined over finite fields has been studied. We obtained the finite graph structures constructed by the network of states where edges denote possible time evolutions. We simplify the graphs by introducing…
We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and Yang-Baxter maps, which are set-theoretical solutions to the quantum Yang-Baxter equation. In particular, we clarify the structure…
We prove an explicit expression for the solutions of the discrete Schwarzian octahedron recurrence, also known as the discrete Schwarzian KP equation (dSKP), as the ratio of two partition functions. Each one counts weighted oriented dimer…
Modeling real processes often results in several suitable models. In order to be able to distinguish, or discriminate, which model best represents a phenomenon, one is interested, e.g., in so-called T-optimal designs. These consist of the…
In this paper we investigate equilibria of continuous differential equation models of network dynamics. The motivation comes from gene regulatory networks where each directed edge represents either down- or up-regulation, and is modeled by…
This paper explores the connection between perfect t-embeddings and the octahedron equation in the setting of the two-periodic Aztec diamond. In particular, we show that the positions of both the t-embedding and the corresponding origami…
A method of classification of integrable equations on quad-graphs is discussed based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators…
The divisor theory of graphs views a finite connected graph $G$ as a discrete version of a Riemann surface. Divisors on $G$ are formal integral combinations of the vertices of $G$, and linear equivalence of divisors is determined by the…