arXiv subjects
Publications and source records attributed to Richard Kenyon.
We prove a new integrability principle for gradient variational problems in $\mathbb{R}^2$, showing that solutions are explicitly parameterized by $κ$-harmonic functions, that is, functions which are harmonic for the laplacian with varying conductivity $κ$, where $κ$ is the square root of the Hessian determinant of the surface tension.
We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile ``orientations'' determine the tiling. We associate to a tiling an underlying bipartite planar graph $G$ and its corresponding Kasteleyn matrix $K$. If $G$ has quadrilateral faces, we show that $K$ is the differential of the map from edge intercepts to tile areas, and extract some geometric and probabilistic consequences.
Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural probability measure on such configurations (it is the uniform measure on standard tilings of the corresponding "blow-up" of $G$). In the limit of large multiplicities we compute asymptotic growth rate of the number of multinomial tilings. We show that the individual tile densities tend to a Gaussian field with respect to an associated discrete Laplacian. We also find an exact discrete Coulomb gas limit when we vary the multiplicities. For tilings of ${\mathbb Z}^d$ with translates of a single tile and a small density of defects, we study a crystallization phenomena when the defect density tends to zero, and give examples of naturally occurring quasicrystals in this framework.
Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case $m=n$; we study here combinatorial and probabilistic aspects of this generalized case. We construct a family of bijections between parking functions $\text{PF}(m, n)$ with $m$ cars and $n$ spots and spanning forests $\mathscr{F}(n+1, n+1-m)$ with $n+1$ vertices and $n+1-m$ distinct trees having specified roots. This leads to a bijective correspondence between $\text{PF}(m, n)$ and monomial terms in the associated Tutte polynomial of a disjoint union of $n-m+1$ complete graphs. We present an identity between the "inversion enumerator" of spanning forests with fixed roots and the "displacement enumerator" of parking functions. The displacement is then related to the number of graphs on $n+1$ labeled vertices with a fixed number of edges, where the graph has $n+1-m$ disjoint rooted components with specified roots. We investigate various probabilistic properties of a uniform parking function, giving a formula for the law of a single coordinate. As a side result we obtain a recurrence relation for the displacement enumerator. Adapting known results on random linear probes, we further deduce the covariance between two coordinates when $m=n$.
A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a PV number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.
We establish a correspondence between the dimer model on a bipartite graph and a circle pattern with the combinatorics of that graph, which holds for graphs that are either planar or embedded on the torus. The set of positive face weights on the graph gives a set of global coordinates on the space of circle patterns with embedded dual. Under this correspondence, which extends the previously known isoradial case, the urban renewal (local move for dimer models) is equivalent to the Miquel move (local move for circle patterns). As a consequence the Miquel dynamics on circle patterns is governed by the octahedron recurrence. As special cases of these circle pattern embeddings, we recover harmonic embeddings for resistor networks and s-embeddings for the Ising model.
For every irreducible automorphism $ϕ\in\text{SL}_3({\mathbb Z})$ of the $3$-torus, for which the product of the expanding eigenvalues is positive, we construct a pseudo-Anosov mapping $f$ of an associated surface, semi-conjugate and almost-isomorphic to $ϕ$, whose stretch factor is the product of the expanding eigenvalues of $ϕ$. This shows that any norm-$1$ cubic Pisot number occurs as the stretch factor of a pseudo-Anosov mapping, proving a conjecture of Fried in degree $3$. A similar construction works for the $4$-torus on condition that $ϕ$ has exactly two eigenvalues outside the unit circle (and whose product is positive). Furthermore for any irreducible hyperbolic automorphism $ϕ\in\text{SL}_n({\mathbb Z})$ of the $n$-torus, $n\ge 4$, we construct a pseudo-Anosov mapping semiconjugate and almost-isomorphic to any sufficiently large power of $ϕ$.
The third flight of the High-Resolution Coronal Imager (Hi-C 2.1) occurred on May 29, 2018, the Sounding Rocket was launched from White Sands Missile Range in New Mexico. The instrument has been modified from its original configuration (Hi-C 1) to observe the solar corona in a passband that peaks near 172 Angstrom and uses a new, custom-built low-noise camera. The instrument targeted Active Region 12712, and captured 78 images at a cadence of 4.4 sec (18:56:22 - 19:01:57 UT; 5 min and 35 sec observing time). The image spatial resolution varies due to quasi-periodic motion blur from the rocket; sharp images contain resolved features of at least 0.47 arcsec. There are coordinated observations from multiple ground- and space-based telescopes providing an unprecedented opportunity to observe the mass and energy coupling between the chromosphere and the corona. Details of the instrument and the data set are presented in this paper.
We study "holomorphic quadratic differentials" on graphs. We relate them to the reactive power in an LC circuit, and also to the chromatic polynomial of a graph. Specifically, we show that the chromatic polynomial $χ$ of a graph $G$, at negative integer values, can be evaluated as the degree of a certain rational mapping, arising from the defining equations for a holomorphic quadratic differential. This allows us to give an explicit integral expression for $χ(-k)$.
We study the map from conductances to edge energies for harmonic functions on finite graphs with Dirichlet boundary conditions. We prove that for any compatible acyclic orientation and choice of energies there is a unique choice of conductances such that the associated harmonic function realizes those orientations and energies. We call the associated function enharmonic. For rational energies and boundary data the Galois group of ${\mathbb Q}^{tr}$ (the totally real algebraic numbers) over ${\mathbb Q}$ permutes the enharmonic functions, acting on the set of compatible acyclic orientations. A consequence is the non-tileability of certain polygons by rational-area rectangles. For planar graphs there is an enharmonic conjugate function, together these form the real and imaginary parts of a "fixed energy" analytic function. In the planar scaling limit for ${\mathbb Z}^2$ (and the fixed south/west orientation), these functions satisfy a nonlinear analog of the Cauchy-Riemann equations, namely \begin{eqnarray*}u_xv_y &=& 1\\u_yv_x&=&-1.\end{eqnarray*} We give an analog of the Riemann mapping theorem for these functions, as well as a variational approach to finding solutions in both the discrete and continuous settings.
Square ice is a statistical mechanics model for two-dimensional ice, widely believed to have a conformally invariant scaling limit. We associate a Peano (space filling) curve to a square ice configuration, and more generally to a so-called 6-vertex model configuration, and argue that its scaling limit is a space-filling version of the random fractal curve SLE$_κ$, Schramm--Loewner evolution with parameter $κ$, where $4<κ\leq 12+8\sqrt{2}$. For square ice, $κ=12$. At the "free-fermion point" of the 6-vertex model, $κ=8+4\sqrt{3}$. These unusual values lie outside the classical interval $2\le κ\le 8$.
We define natural probability measures on cycle-rooted spanning forests (CRSFs) on graphs embedded on a surface with a Riemannian metric. These measures arise from the Laplacian determinant and depend on the choice of a unitary connection on the tangent bundle to the surface. We show that, for a sequence of graphs $(G_n)$ conformally approximating the surface, the measures on CRSFs of $G_n$ converge and give a limiting probability measure on finite multicurves (finite collections of pairwise disjoint simple closed curves) on the surface, independent of the approximating sequence. Wilson's algorithm for generating spanning trees on a graph generalizes to a cycle-popping algorithm for generating CRSFs for a general family of weights on the cycles. We use this to sample the above measures. The sampling algorithm, which relates these measures to the loop-erased random walk, is also used to prove tightness of the sequence of measures, a key step in the proof of their convergence. We set the framework for the study of these probability measures and their scaling limits and state some of their properties.
Given a substitution tiling $T$ of the plane with subdivision operator $τ$, we study the conformal tilings $\mathcal{T}_n$ associated with $τ^n T$. We prove that aggregate tiles within $\mathcal{T}_n$ converge in shape as $n\rightarrow \infty$ to their associated Euclidean tiles in $T$.
We generalize the uniform spanning tree to construct a family of determinantal measures on essential spanning forests on periodic planar graphs in which every component tree is bi-infinite. Like the uniform spanning tree, these measures arise naturally from the laplacian on the graph. More generally these results hold for the "massive" laplacian determinant which counts rooted spanning forests with weight $M$ per finite component. These measures typically have a form of conformal invariance, unlike the usual rooted spanning tree measure. We show that the spectral curve for these models is always a simple Harnack curve, this fact controls the decay of edge-edge correlations in these models. We compute a limit shape theory in these settings, where the limit shapes are defined by measured foliations of fixed isotopy type.
Based on numerical simulation and local stability analysis we describe the structure of the phase space of the edge/triangle model of random graphs. We support simulation evidence with mathematical proof of continuity and discontinuity for many of the phase transitions. All but one of themany phase transitions in this model break some form of symmetry, and we use this model to explore how changes in symmetry are related to discontinuities at these transitions.
We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by the "peano curve" surrounding the tree of left-most paths to the sink, converges in law with respect to the peanosphere topology to a $\sqrt{4/3}$-Liouville quantum gravity surface decorated by an independent Schramm-Loewner evolution with parameter $κ=12$ (i.e., SLE$_{12}$). This result is universal in the sense that it holds for bipolar-oriented triangulations, quadrangulations, $k$-angulations, and maps in which face sizes are mixed.
We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of $k$-star subgraphs, $k\ge 2$ fixed. We prove that under such constraints graphs are "multipodal": asymptotically in the number of vertices there is a partition of the vertices into $M < \infty$ subsets $V_1, V_2, \ldots, V_M$, and a set of well-defined probabilities $g_{ij}$ of an edge between any $v_i \in V_i$ and $v_j \in V_j$. For $2\le k\le 30$ we determine the phase space: the combinations of edge and $k$-star densities achievable asymptotically. For these models there are special points on the boundary of the phase space with nonunique asymptotic (graphon) structure; for the 2-star model we prove that the nonuniqueness extends to entropy maximizers in the interior of the phase space.
The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges. What does a typical random graph look like, if drawn from an exponential model subject to such constraints? Will there be a similar phase transition phenomenon (as one varies the parameters) as that which occurs in the unconstrained exponential model? We present some general results for this constrained model and then apply them to get concrete answers in the edge-triangle model with fixed density of edges.