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Charles Radin

Publications and source records attributed to Charles Radin.

At least 37 records · Page 2Linked to original sources
arXiv2017

The phases of large networks with edge and triangle constraints

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.

math.CO↗
arXiv2016

Wulff shape for equilibrium phases

We use surface tension to distinguish between phases with isotropic internal structure from phases which are microscopically anisotropic. There are many interesting open problems, especially in two dimensions, and in phase coexistence.

math-ph↗
arXiv2016

Multipodal Structure and Phase Transitions in Large Constrained Graphs

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.

math.CO↗
arXiv2016

A symmetry breaking transition in the edge/triangle network model

Our general subject is the emergence of phases, and phase transitions, in large networks subjected to a few variable constraints. Our main result is the analysis, in the model using edge and triangle subdensities for constraints, of a sharp transition between two phases with different symmetries, analogous to the transition between a fluid and a crystalline solid.

math.CO↗
arXiv2016

Phases in Large Combinatorial Systems

This is a status report on a companion subject to extremal combinatorics, obtained by replacing extremality properties with emergent structure, `phases'. We discuss phases, and phase transitions, in large graphs and large permutations, motivating and using the asymptotic formalisms of graphons for graphs and permutons for permutations. Phase structure is shown to emerge using entropy and large deviation techniques.

math.CO↗
arXiv2015

Bipodal structure in oversaturated random graphs

We study the asymptotics of large simple graphs constrained by the limiting density of edges and the limiting subgraph density of an arbitrary fixed graph $H$. We prove that, for all but finitely many values of the edge density, if the density of $H$ is constrained to be slightly higher than that for the corresponding Erdős-Rényi graph, the typical large graph is bipodal with parameters varying analytically with the densities. Asymptotically, the parameters depend only on the degree sequence of $H$.

math.CO↗
arXiv2015

Permutations with fixed pattern densities

We study scaling limits of random permutations ("permutons") constrained by having fixed densities of a finite number of patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In particular, we compute (exactly or numerically) the limit shapes with fixed \hbox{12} density, with fixed \hbox{12} and \hbox{123} densities, with fixed \hbox{12} density and the sum of \hbox{123} and \hbox{213} densities, and with fixed \hbox{123} and \hbox{321} densities. In the last case we explore a particular phase transition. To obtain our results, we also provide a description of permutons using a dynamic construction.

math.CO↗
arXiv2014

The Asymptotics of Large Constrained Graphs

We show, through local estimates and simulation, that if one constrains simple graphs by their densities $\varepsilon$ of edges and $τ$ of triangles, then asymptotically (in the number of vertices) for over $95\%$ of the possible range of those densities there is a well-defined typical graph, and it has a very simple structure: the vertices are decomposed into two subsets $V_1$ and $V_2$ of fixed relative size $c$ and $1-c$, and there are well-defined probabilities of edges, $g_{jk}$, between $v_j\in V_j$, and $v_k\in V_k$. Furthermore the four parameters $c, g_{11}, g_{22}$ and $g_{12}$ are smooth functions of $(\varepsilon,τ)$ except at two smooth `phase transition' curves.

math.CO↗
arXiv2013

Phase transitions in a complex network

We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle density and a range of edge density, though we can only prove they are local, not global, maxima of the entropy density. With this assumption we then prove that the resulting entropy density must lose its analyticity in various regimes. In particular this implies the existence of a phase transition between distinct heterogeneous multipartite phases at low triangle density, and a phase transition between these phases and the disordered phase at high triangle density.

math-ph↗
arXiv2012

A Revolutionary Material

This is an expository introduction, for a general mathematics audience, to the modeling of the fluid/solid phase transition and in particular to complications created by the discovery of quasicrystals. One goal is to elucidate certain features of the modeling which are ripe for mathematical investigation.

math-ph↗
arXiv2011

Emergent structures in large networks

We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to multipartite structure, separated by a phase transition from a region of disordered graphs.

math.PR↗
arXiv2011

Rigidity in Solids

We address the question of whether solids may be distinguished from fluids by their response to shear stress

cond-mat.stat-mech↗
arXiv2011

Sound speed in water-saturated glass beads as a function of frequency and porosity

Sound propagation in water-saturated granular sediments is known to depend on the sediment porosity, but few data in the literature address both the frequency and porosity dependency. To begin to address this deficiency, a fluidized bed technique was used to control the porosity of an artificial sediment composed of glass spheres of 265 μm diameter. Time-of-flight measurements and the Fourier phase technique were utilized to determine the sound speed for frequencies from 300 to 800 kHz and porosities from 0.37 to 0.43. A Biot-based model qualitatively describes the porosity dependence.

cond-mat.soft↗
arXiv2011

Shearing of frictional sphere packings

We measure shear response in packings of glass beads by pulling a thin, rough, metal plate vertically through a bed of volume fraction phi, which is set, before the plate is pulled, in the range 0.575 to 0.628. The yield stress is velocity independent over 4 decades and increases exponentially with phi, with a transition at phi approximately 0.595. An analysis of the measured force fluctuations indicates that the shear modulus is significantly smaller than the bulk modulus.

cond-mat.soft↗
arXiv2011

First Order Phase Transition in a Model of Quasicrystals

We introduce a family of two-dimensional lattice models of quasicrystals, using a range of square hard cores together with a soft interaction based on an aperiodic tiling set. Along a low temperature isotherm we find, by Monte Carlo simulation, a first order phase transition between disordered and quasicrystalline phases.

cond-mat.stat-mech↗
arXiv2010

First Order Phase Transition of a Long Polymer Chain

We consider a model consisting of a self-avoiding polygon occupying a variable density of the sites of a square lattice. A fixed energy is associated with each $90^\circ$-bend of the polygon. We use a grand canonical ensemble, introducing parameters $μ$ and $β$ to control average density and average (total) energy of the polygon, and show by Monte Carlo simulation that the model has a first order, nematic phase transition across a curve in the $β$-$μ$ plane.

cond-mat.soft↗