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Lorenzo Sadun

Publications and source records attributed to Lorenzo Sadun.

At least 55 records · Page 3Linked to original sources
arXiv2017

Surface effects in dense random graphs with sharp edge constraint

We show that the random number $T_n$ of triangles in a random graph on $n$ vertices, with a strict constraint on the total number of edges, admits an expansion $T_n = an^3 + bn^2 + F_n$, where $a$ and $b$ are numbers, with the mean $\langle F_n \rangle = O(n)$ and the standard deviation $σ(T_n) =σ(F_n)= O(n^{3/2})$. The presence of a `surface term' $bn^2$ has a significance analogous to the macroscopic surface effects of materials, and is missing in the model where the edge constraint is removed. We also find the surface effect in other graph models using similar edge constraints.

math.CO↗
arXiv2017

A Characterization of Rotation Number on One-Dimensional Tiling Spaces

Identity-homotopic self-homeomorphisms of a space of non-periodic 1-dimensional tiling are generalizations of orientation-preserving self-homeomorphisms of circles. We define the analogue of rotation numbers for such maps. In constrast to the classical situation, additional assumptions are required to make rotation numbers globally well-defined and independent of initial conditions. We prove that these conditions are sufficient, and provide counterexamples when these conditions are not met.

math.DS↗
arXiv2017

Small cocycles, fine torus fibrations, and a ${\mathbb Z}^2$ subshift with neither

Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam and Skau conjectured that all minimal, free ${\mathbb Z}^d$ actions on Cantor sets admit "small cocycles." These represent classes in $H^1$ that are mapped to small vectors in ${\mathbb R}^d$ by the Ruelle-Sullivan (RS) map. We show that there exist ${\mathbb Z}^d$ actions where no such small cocycles exist, and where the image of $H^1$ under RS is ${\mathbb Z}^d$. Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of "virtual eigenvalues," i.e. elements of ${\mathbb R}^d$ that become topological eigenvalues of the tiling flow after an arbitrarily small change in the shapes and sizes of the tiles.

math.DS↗
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

What's wrong with the Platonic ideal of space and time?

This is a chapter for an upcoming book, "Space, Time, and the Limits of Human Understanding", edited by Shyam Wuppuluri and Giancarlo Ghirardi. I first present the classical picture of flat space and ever-flowing time, as generally accepted until about 1900, discuss the challenges posed by modern physics and the adjustments required by special relativity, general relativity and quantum mechanics, and finally consider (and reject) a recent hypothesis that space-time is essentially discrete. Ultimately, none of the simple answers to "what is space-time?" stand up to reality. We have more work ahead of us!

physics.pop-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

Lecture Notes on Differential Forms

This is a series of lecture notes, with embedded problems, aimed at students studying differential topology. Many revered texts, such as Spivak's "Calculus on Manifolds" and Guillemin and Pollack's "Differential Topology" introduce forms by first working through properties of alternating tensors. Unfortunately, many students get bogged down with the whole notion of tensors and never get to the punch lines: Stokes' Theorem, de Rham cohomology, Poincare duality, and the realization of various topological invariants (e.g. the degree of a map) via forms, none of which actually require tensors to make sense! In these notes, we'll follow a different approach, following the philosophy of Amy's Ice Cream: Life is uncertain. Eat dessert first. We're first going to define forms on $\mathbb{R}^n$ via unmotivated formulas, develop some proficiency with calculation, show that forms behave nicely under changes of coordinates, and prove Stokes' Theorem. This approach has the disadvantage that it doesn't develop deep intuition, but the strong advantage that the key properties of forms emerge quickly and cleanly. Only then, in Chapter 3, will we go back and show that tensors with certain (anti)symmetry properties have the exact same properties as the formal objects that we studied in Chapters 1 and 2. This allows us to re-interpret all of our old results from a more modern perspective, and move onwards to using forms to do topology.

math.AT↗
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

Finitely Balanced Sequences and Plasticity of 1-Dimensional Tilings

We relate a balancing property of letters for bi-infinite sequences to the invariance of the resulting 1-dimensional tiling dynamics under changes in the lengths of the tiles. If the language of the sequence space is finitely balanced, then all length changes in the corresponding tiling space result in topological conjugacies, up to an overall rescaling.

math.DS↗
arXiv2014

Gaps problems and frequencies of patches in cut and project sets

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.

math.DS↗
arXiv2014

Cohomology of Hierarchical Tilings

This is a chapter in an upcoming book on aperiodic order. We go over different versions of tiling cohomology (\v Cech, pattern-equivariant, PV, quotient) with emphasis on the inverse limit constructions used to compute these cohomologies. We then consider the uses of tiling cohomology to distinguish spaces, to understand deformations, and to help understand maps between tiling spaces. The emphasis of this chapter is on substitution tilings and their generalizations, but the underlying ideas apply equally well to cut-and-project tilings and to tilings defined by local matching rules.

math.DS↗
arXiv2014

Pattern Equivariant Cohomology and Theorems of Kesten and Oren

In 1966 Harry Kesten settled the Erd\H os-Szüsz conjecture on the local discrepancy of irrational rotations. His proof made heavy use of continued fractions and Diophantine analysis. In this paper we give a purely topological proof Kesten's theorem (and Oren's generalization of it) using the pattern equivariant cohomology of aperiodic tiling spaces.

math.DS↗
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

Meyer sets, topological eigenvalues, and Cantor fiber bundles

We introduce two new characterizations of Meyer sets. A repetitive Delone set in $\R^d$ with finite local complexity is topologically conjugate to a Meyer set if and only if it has $d$ linearly independent topological eigenvalues, which is if and only if it is topologically conjugate to a bundle over a $d$-torus with totally disconnected compact fiber and expansive canonical action. "Conjugate to" is a non-trivial condition, as we show that there exist sets that are topologically conjugate to Meyer sets but are not themselves Meyer. We also exhibit a diffractive set that is not Meyer, answering in the negative a question posed by Lagarias, and exhibit a Meyer set for which the measurable and topological eigenvalues are different.

math.DS↗
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↗
arXiv2007

Topology of (some) tiling spaces without finite local complexity

A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have "fault lines", along which tiles can slide past one another. We investigate the topology of a certain class of tiling spaces of this type. We show that they can be written as inverse limits of CW complexes, and their Cech cohomology is related to properties of the fault lines.

math.DS↗