arXiv subjects

Lorenzo Sadun

Publications and source records attributed to Lorenzo Sadun.

At least 19 recordsLinked to original sources

arXiv2026

Covers of Tiling Spaces

We study the ways that one tiling space can be a finite regular cover of another. We classify all of the finite regular covers of a tiling space via its structure as an inverse limit space. If the tiling space $\Omega$ can be written as an inverse limit $\varprojlim \Gamma_n$, then the \'etale fundamental group of $\Omega$, which is defined via a limit of covers, is isomorphic to the inverse limit $\hat \pi_1(\Omega) := \varprojlim \hat \pi_1(\Gamma_n)$ of the profinite completions of the fundamental groups $\pi_1(\Gamma_n)$. This isomorphism allows us to construct all covers of tiling spaces and to use those covers to distinguish spaces that have identical cohomology groups.

math.AT↗

arXiv2025

Emergence in graphs with near-extreme constraints

We consider entropy-optimal graphons associated with extreme and near-extreme constraints on the densities of edges and triangles. We prove that the optimizers for near-extreme constraints are unique and multipodal and are perturbations of the previously known unique optimzers for extreme constraints. This proves the existence of infinitely many phases. We determine the podal structures in these phases and prove the existence of phase transitions between them.

math.PR↗

arXiv2025

Tripodal structure in undersaturated random graphs

We numerically investigate typical graphs in a region of the Strauss model of random graphs with constraints on the densities of edges and triangles. This region, where typical graphs had been expected to be bipodal but turned out to be tripodal, involves edge densities $e$ below $e_0 = (3-\sqrt{3})/6 \approx 0.2113$ and triangle densities $t$ slightly below $e^3$. We determine the extent of this region in $(e,t)$ space and show that there is a discontinuous phase transition at the boundary between this region and a bipodal phase. We further show that there is at least one phase transition within this region, where the parameters describing typical graphs change discontinuously.

math.CO↗

arXiv2025

Dynamics and topology of the Hat family of tilings

The recently discovered Hat tiling admits a 4-dimensional family of shape deformations, including the 1-parameter family already known to yield alternate monotiles. The continuous hulls resulting from these tilings are all topologically conjugate dynamical systems, and hence have the same dynamics and topology. We construct and analyze a self-similar element of this family called the CAP tiling, and we use it to derive properties of the entire family. The CAP tiling has pure-point dynamical spectrum, which we compute explicitly, and comes from a natural cut-and-project scheme with 2-dimensional Euclidean internal space. All other members of the Hat family, in particular the original version constructed from 30-60-90 right triangles, are obtained via small modifications of the projection from this cut-and-project scheme.

math.DS↗

arXiv2024

On the long-range order of the Spectre tilings

The Spectre is an aperiodic monotile for the Euclidean plane that is truly chiral in the sense that it tiles the plane without any need for a reflected tile. The topological and dynamical properties of the Spectre tilings are very similar to those of the Hat tilings. Specifically, the Spectre sits within a complex $2$-dimensional family of tilings, most of which involve two shapes rather than one. All tilings in the family give topologically conjugate dynamics, up to an overall rescaling and rotation. They all have pure point dynamical spectrum with continuous eigenfunctions and may be obtained from a $4:2$ dimensional cut-and-project scheme with regular windows of Rauzy fractal type. The diffraction measure of any Spectre tiling is pure point as well. For fixed scale and orientation, varying the shapes is MLD equivalent to merely varying the projection direction. These properties all follow from the first \v{C}ech cohomology being as small as it possibly could be, leaving no room for shape changes that alter the dynamics.

math.DS↗

arXiv2024

How big is a tiling's return module?

The rank of a tiling's return module depends on the geometry of its tiles and is not a topological invariant. However, the rank of the first \v Cech cohomology $\check H^1(Ω)$ gives upper and lower bounds for the size of the return module. For all sufficiently large patches, the rank of the return module is at most the same as the rank of the cohomology. For a generic choice of tile shapes and an arbitrary reference patch, the rank of the return module is at least the rank of $\check H^1(Ω)$. Therefore, for generic tile shapes and sufficiently large patches, the rank of the return module is equal to the rank of $\check H^1(Ω)$.

math.DS↗

arXiv2023

Existence of a symmetric bipodal phase in the edge-triangle model

In the edge-triangle model with edge density close to 1/2 and triangle density below 1/8 we prove that the unique entropy-maximizing graphon is symmetric bipodal. We also prove that,for any edge density $e$ less than $e_0 = (3-\sqrt{3})/6 \approx 0.2113$ and triangle density slightly less than $e^3$, the entropy-maximizing graphon is not symmetric bipodal.

math.PR↗

arXiv2023

Optimal graphons in the edge-2star model

In the edge-2star model with hard constraints we prove the existence of an open set of constraint parameters, bisected by a line segment on which there are nonunique entropy-optimal graphons related by a symmetry. At each point in the open set but off the line segment there is a unique entropy-optimizer, bipodal and varying analytically with the constraints. We also show that throughout another open set, containing a different portion of the same line of symmetry, there is instead a unique optimal graphon, varying analytically with the parameters. We explore the extent of these open sets, determining the point at which a symmetric graphon ceases to be a local maximizer of the entropy. Finally, we prove some foundational theorems in a general setting, relating optimal graphons to the Boltzmann entropy and the generic structure of large constrained random graphs.

math.PR↗

arXiv2022

Global solutions with infinitely many blowups in a mean-field neural network

We recently introduced idealized mean-field models for networks of integrate-and-fire neurons with impulse-like interactions -- the so-called delayed Poissonian mean-field models. Such models are prone to blowups: for a strong enough interaction coupling, the mean-field rate of interaction diverges in finite time with a finite fraction of neurons spiking simultaneously. Due to the reset mechanism of integrate-and-fire neurons, these blowups can happen repeatedly, at least in principle. A benefit of considering Poissonian mean-field models is that one can resolve blowups analytically by mapping the original singular dynamics onto uniformly regular dynamics via a time change. Resolving a blowup then amounts to solving the fixed-point problem that implicitly defines the time change, which can be done consistently for a single blowup and for nonzero delays. Here we extend this time-change analysis in two ways: First, we exhibit the existence and uniqueness of explosive solutions with a countable infinity of blowups in the large interaction regime. Second, we show that these delayed solutions specify "physical" explosive solutions in the limit of vanishing delays, which in turn can be explicitly constructed. The first result relies on the fact that blowups are self-sustaining but nonoverlapping in the time-changed picture. The second result follows from the continuity of blowups in the time-changed picture and incidentally implies the existence of periodic solutions. These results are useful to study the emergence of synchrony in neural network models.

math.PR↗

arXiv2022

Typical large graphs with given edge and triangle densities

The analysis of large simple graphs with extreme values of the densities of edges and triangles has been extended to the statistical structure of typical graphs of fixed intermediate densities, by the use of large deviations of Erdoes-Renyi graphs. We prove that the typical graph exhibits sharp singularities as the constraining densities vary between different curves of extreme values, and we determine the precise nature of the singularities. The extension to graphs with fixed densities of edges and k-cycles for odd k>3 is straightforward and we note the simple changes in the proof.

math.PR↗

arXiv2022

Moderate Deviations in Cycle Count

We prove moderate deviations bounds for the lower tail of the number of odd cycles in a $\calG(n, m)$ random graph. We show that the probability of decreasing triangle density by $t^3$, is $\exp(-Θ(n^2 t^2))$ whenever $n^{-3/4} \ll t^3 \ll 1$, while for $k \ge 5$ we give the same estimate for the probability of decreasing the $k$-cycle density by $t^k$, but for the larger range $n^{-1} \ll t^k \ll 1$. When $m \ge \frac 12 \binom n2$, we also find the leading coefficient in the exponent. This complements results of Goldschmidt et al., who showed that for $n^{-3/2} \ll t^k \ll n^{-1}$, the probability is $\exp(-Θ(n^3 t^{2k}))$. That is, deviations of order smaller than $n^{-1}$ behave like small deviations, and deviations of order larger than $n^{-3/4}$ (for triangles) or $n^{-1}$ (for $k$-cycles with $k \ge 5$) behave like large deviations. For triangles, we conjecture that a sharp change between the two regimes occurs for deviations of size $n^{-3/4}$, which we associate with a single large negative eigenvalue of the adjacency matrix becoming responsible for almost all of the cycle deficit. Our results can be interpreted as finite size effects in phase transitions in constrained random graphs.

math.PR↗

arXiv2021

Topological Mixing of Random Substitutions

We investigate topological mixing of compatible random substitutions. For primitive random substitutions on two letters whose second eigenvalue is greater than one in modulus, we identify a simple, computable criterion which is equivalent to topological mixing of the associated subshift. This generalises previous results on deterministic substitutions. In the case of recognisable, irreducible Pisot random substitutions, we show that the associated subshift is not topologically mixing. Without recognisability, we rely on more specialised methods for excluding mixing and we apply these methods to show that the random Fibonacci substitution subshift is not topologically mixing.

math.DS↗

arXiv2021

Number of bounded distance equivalence classes in hulls of repetitive Delone sets

Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many. A very similar result is proven in arXiv:2011.00106 [math.MG].

math.DS↗

arXiv2020

Rotation numbers and rotation classes on one-dimensional tiling spaces

We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a 1-dimensional tiling space $Ω$ with finite local complexity and study self-maps $F$ that are homotopic to the identity and whose displacements are strongly pattern equivariant (sPE). In place of the familiar rotation number we define a cohomology class $[μ]$. We prove existence and uniqueness results for this class, develop a notion of irrationality, and prove an analogue of Poncaré's Theorem: If $[μ]$ is irrational, then $F$ is semi-conjugate to uniform translation on a space $Ω_μ$ of tilings that is homeomorphic to $Ω$. In such cases, $F$ is semi-conjugate to uniform translation on $Ω$ itself if and only if $[μ]$ lies in a certain subspace of the first cohomology group of $Ω$.

math.DS↗

arXiv2019

Nucleation during phase transitions in random networks

We analyze the 3-parameter family of random networks which are uniform on networks with fixed number of edges, triangles, and nodes (between 33 and 66). We find precursors of phase transitions which are known to be present in the asymptotic node regime as the edge and triangle numbers are varied, and focus on one of the discontinuous ones. By use of a natural edge flip dynamics we determine nucleation barriers as a random network crosses the transition, in analogy to the process a material undergoes when frozen or melted, and characterize some of the stochastic properties of the network nucleation.

math.CO↗

arXiv2018

Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

Suppose that we have a repetitive and aperiodic tiling $\bf T$ of $\mathbb{R}^n$, and two mass distributions $f_1$ and $f_2$ on $\mathbb{R}^n$, each pattern equivariant with respect to $\bf T$. Under what circumstances is it possible to do a bounded transport from $f_1$ to $f_2$? When is it possible to do this transport in a strongly or weakly pattern-equivariant way? We reduce these questions to properties of the \v Cech cohomology of the hull of $\bf T$, properties that in most common examples are already well-understood.

math.DS↗

arXiv2018

Isomorphism of Hierarchical Structures

We consider hierarchical structures such as Fibonacci sequences and Penrose tilings, and examine the consequences of different choices for the definition of isomorphism. In particular we discuss the role such a choice plays with regard to matching rules for such structures.

math-ph↗

arXiv2018

Growth Rates in the Quaquaversal Tiling

Conway and Radin's "quaquaversal" tiling of R^3 is known to exhibit statistical rotational symmetry in the infinite volume limit. A finite patch, however, cannot be perfectly isotropic, and we compute the rates at which the anisotropy scales with size. In a sample of volume N, tiles appear in O(N^{1/6}) distinct orientations. However, the orientations are not uniformly populated. A small (O(N^{1/84})) set of these orientations account for the majority of the tiles. Furthermore, these orientations are not uniformly distributed on SO(3). Sample averages of functions on SO(3) seem to approach their ergodic limits as N^{-1/336}. Since even macroscopic patches of a quaquaversal tiling maintain noticable anisotropy, a hypothetical physical quasicrystal whose structure was similar to the quaquaversal tiling could be identified by anisotropic features of its electron diffraction pattern.

math-ph↗