Limit Shape Research Articles

Piecewise assembled acoustic arrays based on reconfigurable tessellated structures.

Physically reconfigurable, tessellated acoustic arrays inspired by origami structures have recently been leveraged to adaptively guide acoustic energy. Yet, the prior work only examined tessellated arrays composed from uniform folding patterns, so that the limited folding-induced shape change prohibits broad acoustic field tailoring. To explore a wider range of opportunities by origami-inspired acoustic arrays, here, piecewise geometries are assembled from multiple folding patterns so that acoustic transducer elements are reconfigured in more intricate ways upon array folding. An analytical model of assembled geometries and resulting acoustic wave radiation from the oscillating facets is formulated. Using the theoretical tool, parametric investigations are undertaken to study the adaptation of acoustic energy transmission caused by folding and modularity of the array assembly. A proof-of-concept specimen is fabricated and experiments are conducted to validate the theoretical model and to investigate the efficacy of the piecewise acoustic array concept. The total findings reveal that the assembly of tessellated acoustic arrays may emulate the wave radiation emitted by ideal acoustic sources of intricate shapes. Moreover, by exploitation of origami folding actions, the shape adaptations of the proposed arrays permit straightforward wave guiding opportunities for diverse application needs.

The size of the boundary in first-passage percolation

First-passage percolation is a random growth model defined using i.i.d. edge-weights $(t_{e})$ on the nearest-neighbor edges of $\mathbb{Z}^{d}$. An initial infection occupies the origin and spreads along the edges, taking time $t_{e}$ to cross the edge $e$. In this paper, we study the size of the boundary of the infected (“wet”) region at time $t$, $B(t)$. It is known that $B(t)$ grows linearly, so its boundary $\partial B(t)$ has size between $ct^{d-1}$ and $Ct^{d}$. Under a weak moment condition on the weights, we show that for most times, $\partial B(t)$ has size of order $t^{d-1}$ (smooth). On the other hand, for heavy-tailed distributions, $B(t)$ contains many small holes, and consequently we show that $\partial B(t)$ has size of order $t^{d-1+\alpha }$ for some $\alpha >0$ depending on the distribution. In all cases, we show that the exterior boundary of $B(t)$ [edges touching the unbounded component of the complement of $B(t)$] is smooth for most times. Under the unproven assumption of uniformly positive curvature on the limit shape for $B(t)$, we show the inequality $\#\partial B(t)\leq (\log t)^{C}t^{d-1}$ for all large $t$.

Corona Limits of Tilings: Periodic Case

We study the limit shape of successive coronas of a tiling, which models the growth of crystals. We define basic terminologies and discuss the existence and uniqueness of corona limits, and then prove that corona limits are completely characterized by directional speeds. As an application, we give another proof that the corona limit of a periodic tiling is a centrally symmetric convex polyhedron [see Zhuravlev (St Petersbg Math J 13(2):201–220, 2002) and Maleev and Shutov (Layer-by-layer growth model for partitions, packings, and graphs, Tranzit-X, Vladimir, 2011)].

Limit Shapes of the Stochastic Six Vertex Model

It is shown that limit shapes for the stochastic 6-vertex model on a cylinder with the uniform boundary state on one end are solutions to the Burger type equation. Solutions to these equations are studied for step initial conditions. When the circumference goes to infinity the solution corresponding to critical initial densities coincides with the one found by Borodin, Corwin, and Gorin.

Low Solids Emulsion Gels Based on Nanocellulose for 3D-Printing.

Multiphase (emulsion) gels with internal phase fractions between 0.1 and 0.5 were formulated at low loadings of cellulose nanofibrils (CNF), alginate, and polylactide (PLA). Their properties (rheology and morphology) fitted those of inks used for direct ink writing (DIW). The effect of formulation and composition variables were elucidated after printing cubic scaffolds and other solid designs. The distinctive microstructures that were developed allowed high printing fidelity and displayed limited shrinkage after room temperature and freeze-drying (0 and 5% shrinkage in the out-of-plane and in-plane directions upon freeze-drying, respectively). The CNF added in the continuous phase was shown to be critical to achieve rheology control as an effective interfacial stabilizer and to ensure the printability of the ink toward high structural reliability. We found that the extent of shape retention of the dried scaffolds resulted from the tightly locked internal structure. The PLA that was initially added in the nonpolar or organic phase (0 to 12%) was randomly embedded in the entire scaffold, providing a strong resistance to shrinkage during the slow water evaporation at ambient temperature. No surface collapse or lateral deformation of the dried scaffolds occurred, indicating that the incorporation of PLA limited drying-induced shape failure. It also reduced compression strain by providing better CNF skeletal support, improving the mechanical strength. Upon rewetting, the combination of the hydrophilicity imparted by CNF and alginate together with the highly porous structure of the 3D material and the internal microchannels contributed to high water absorption via capillary and other phenomena (swelling % between ∼400 and 900%). However, no shape changes occurred compared to the initial 3D-printed shape. The swelling of the scaffolds correlated inversely with the PLA content in the precursor emulsion gel, providing a means to regulate the interaction with water given its low surface energy. Overall, the results demonstrate that by compatibilization of the CNF-based hydrophilic and the PLA-based hydrophobic components, it is possible to achieve shape control and retention upon 3D printing, opening the possibility of adopting low-solids inks for DIW into dry objects. The dryable CNF-based 3D structural materials absorb water while being able to support load (high elastic modulus) and maintain the shape upon hydration.

Partial sum of matrix entries of representations of the symmetric group and its asymptotics

Many aspects of the asymptotics of Plancherel distributed partitions have been studied in the past fifty years, in particular the limit shape, the distribution of the longest rows, connections with random matrix theory and characters of the representation matrices of the symmetric group. Regarding the latter, we extend a celebrated result of Kerov on the asymptotic of Plancherel distributed characters by studying partial trace and partial sum of a representation matrix. We decompose each of these objects into a main term and a reminder, and for each such a decomposition we prove a central limit theorem for the main term. We apply these results to prove a law of large numbers for the partial sum. Our main tool is the expansion of symmetric functions evaluated on Jucys–Murphy elements.

Surfaces expanding by non-concave curvature functions

In this paper, we first investigate the flow of convex surfaces in the space form $\mathbb{R}^3(\kappa)~(\kappa=0,1,-1)$ expanding by $F^{-\alpha}$, where $F$ is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power $\alpha\in(0,1]$ for $\kappa=0,-1$ and $\alpha=1$ for $\kappa=1$. By deriving that the pinching ratio of the flow surface $M_t$ is no greater than that of the initial surface $M_0$, we prove the long time existence and the convergence of the flow. No concavity assumption of $F$ is required. We also show that for the flow in $\mathbb{H}^3$ with $\alpha\in (0,1)$, the limit shape may not be necessarily round after rescaling.

Limit Shapes via Bijections

We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes continuously in the plane. We start with bijections outlined in [43], and extend them to include limit shapes with different scaling functions.

Two-dimensional anisotropic KPZ growth and limit shapes

A series of recent works focused on two-dimensional (2D) interface growth models in the so-called anisotropic KPZ (AKPZ) universality class, that have a large-scale behavior similar to that of the Edwards–Wilkinson equation. In agreement with the scenario conjectured by Wolf (1991 Phys. Rev. Lett. 67 1783–6), in all known AKPZ examples the function giving the growth velocity as a function of the slope ρ has a Hessian with negative determinant (‘AKPZ signature’). While up to now negativity was verified model by model via explicit computations, in this work we show that it actually has a simple geometric origin in the fact that the hydrodynamic PDEs associated to these non-equilibrium growth models preserves the Euler–Lagrange equations determining the macroscopic shapes of certain equilibrium 2D interface models. In the case of growth processes defined via dynamics of dimer models on planar lattices, we further prove that the preservation of the Euler–Lagrange equations is equivalent to harmonicity of with respect to a natural complex structure.

Limit Shapes for Gibbs Ensembles of Partitions

We explicitly compute limit shapes for several grand canonical Gibbs ensembles of partitions of integers. These ensembles appear in models of aggregation and are also related to invariant measures of zero range and coagulation-fragmentation processes. We show, that all possible limit shapes for these ensembles fall into several distinct classes determined by the asymptotics of the internal energies of aggregates.

Gaussian fluctuations of Jack-deformed random Young diagrams

We introduce a large class of random Young diagrams which can be regarded as a natural one-parameter deformation of some classical Young diagram ensembles; a deformation which is related to Jack polynomials and Jack characters. We show that each such a random Young diagram converges asymptotically to some limit shape and that the fluctuations around the limit are asymptotically Gaussian.

Geodesics Toward Corners in First Passage Percolation

For stationary first passage percolation in two dimensions, the existence and uniqueness of semi-infinite geodesics directed in particular directions or sectors has been considered by Damron and Hanson (Commun. Math. Phys., 2014), Ahlberg and Hoffman (preprint, 2016), and others. However the main results do not cover geodesics in the direction of corners of the limit shape $\mathcal{B}$, where two facets meet. We construct an example with the following properties: (i) the limiting shape is an octagon, (ii) semi-infinite geodesics exist only in the four axis directions, and (iii) in each axis direction there are multiple such geodesics. Consequently, the set of points of $\partial \mathcal{B}$ which are in the direction of some geodesic does not have all of $\mathcal{B}$ as its convex hull.

Baire categorical aspects of first passage percolation

In the previous decades, the theory of first passage percolation became a highly important area of probability theory. In this work, we will observe what can be said about the corresponding structure if we forget about the probability measure defined on the product space of edges and simply consider topology in the terms of residuality. We focus on interesting questions arising in the probabilistic setup that make sense in this setting, too. We will see that certain classical almost sure events, as the existence of geodesics have residual counterparts, while the notion of the limit shape or time constants gets as chaotic as possible.

KPZ and Airy limits of Hall–Littlewood random plane partitions

Dans cet article, nous considerons une distribution de probabilite $\mathbb{P}^{q,t}_{\mathrm{HL}}$ sur les partitions planes, qui apparait comme une generalisation a un parametre de la mesure standard $q^{\mathrm{volume}}$. Cette generalisation est etroitement reliee aux classiques polynomes multivaries de Hall–Littlewood, et a ete introduite pour la premiere fois par Vuletic dans (Trans. Am. Math. Soc. 361 (2009) 2789–2804). Nous montrons que lorsque la partition plane devient grande ($q$ tend vers $1$, alors que le parametre de Hall–Littlewood $t$ est fixe), la partie inferieure proprement renormalisee de la partition plane converge vers une forme limite deterministe, et que les fluctuations a un point autour de la forme limite sont asymptotiquement donnees par la distribution du GUE Tracy–Widom. Par contre, si $t$ converge vers sa propre valeur critique $1$, les fluctuations convergent cette fois vers l’equation unidimensionnelle de Kardar–Parisi–Zhang (KPZ) avec les conditions initiales a courte bande (narrow wedge data). La partie algebrique de notre argument est etroitement reliee au formalisme des processus de Macdonald (Probab. Theory Relat. Fields 158 (1) (2014) 225–400). La partie analytique consiste en une analyse asymptotique detaillee des determinants de Fredholm associes.

Rolling of 3D Printed Dual‐Layer Beam into a Cylinder by Ethanol Absorption

Abstract4D printing is an extension to 3D printing whereby a printed shape programmatically undergoes shape transformation through external stimulations. There is an increasing interest in this field because of its potential applications. However, many demonstrated shape transformations work on simple and limited geometrical shapes. In this work, the formation of a cylinder from a printed flat dual‐layer beam when exposed to ethanol is demonstrated. The newly formed cylinder retains its shape even when the ethanol is removed through the use of a locking mechanism. The proposed method can be used for building medical stents or various sensors and actuators that require cylindrical shapes.

Symbiotic N2-Fixer Community Composition, but Not Diversity, Shifts in Nodules of a Single Host Legume Across a 2-Million-Year Dune Chronosequence.

Long-term soil age gradients are useful model systems to study how changes in nutrient limitation shape communities of plant root mutualists because they represent strong natural gradients of nutrient availability, particularly of nitrogen (N) and phosphorus (P). Here, we investigated changes in the dinitrogen (N2)-fixing bacterial community composition and diversity in nodules of a single host legume (Acacia rostellifera) across the Jurien Bay chronosequence, a retrogressive 2 million-year-old sequence of coastal dunes representing an exceptionally strong natural soil fertility gradient. We collected nodules from plants grown in soils from five chronosequence stages ranging from very young (10s of years; associated with strong N limitation for plant growth) to very old (> 2,000,000years; associated with strong P limitation), and sequenced the nifH gene in root nodules to determine the composition and diversity of N2-fixing bacterial symbionts. A total of 335 unique nifH gene operational taxonomic units (OTUs) were identified. Community composition of N2-fixing bacteria within nodules, but not diversity, changed with increasing soil age. These changes were attributed to pedogenesis-driven shifts in edaphic conditions, specifically pH, exchangeable manganese, resin-extractable phosphate, nitrate and nitrification rate. A large number of common N2-fixing bacteria genera (e.g. Bradyrhizobium, Ensifer, Mesorhizobium and Rhizobium) belonging to the Rhizobiaceae family (α-proteobacteria) comprised 70% of all raw sequences and were present in all nodules. However, the oldest soils, which show some of the lowest soil P availability ever recorded, harboured the largest proportion of unclassified OTUs, suggesting a unique set of N2-fixing bacteria adapted to extreme P limitation. Our results show that N2-fixing bacterial composition varies strongly during long-term ecosystem development, even within the same host, and therefore rhizobia show strong edaphic preferences.

Convex cones, integral zonotopes, limit shape

Given a convex cone C in Rd, an integral zonotope T is the sum of segments [0,vi] (i=1,…,m) where each vi∈C is a vector with integer coordinates. The endpoint of T is k=∑1mvi. Let T(C,k) be the family of all integral zonotopes in C whose endpoint is k∈C. We prove that, for large k, the zonotopes in T(C,k) have a limit shape, meaning that, after suitable scaling, the overwhelming majority of the zonotopes in T(C,k) are very close to a fixed convex set. We also establish several combinatorial properties of a typical zonotope in T(C,k).

Generalized Procrustes analysis of an ontogenetic series of modified crania: Evaluating the technique of modification in the Migration Period of Europe (4th-7th century AD).

The arrival of the Huns into Europe in the fourth century AD increased the occurrence of intentional cranial modification among European nomads. It has been postulated that the Huns used a two-bandage cranial binding technique to differentiate themselves from surrounding nomadic groups, including those from Georgia. This study examines this hypothesis by comparing Migration Period (4th to 7th century AD) juvenile crania, which retain strong impressions of bindings, with adult modified crania from Hungary and Georgia. Twelve surface landmarks and 251 semi-landmarks were used to study ontogenetic trajectories in 9 juvenile and 16 adult modified skulls from 8 Hungarian sites and 21 adult skulls from two Georgian sites. Generalized Procrustes analysis, linear regression of Procrutes distance on dental age and log centroid size, and warping the principal components (PCs) in shape space helped to identify cranial shape changes. The PCs provide significant separation of the juvenile and adult groups from Georgia and Hungary. Variation in modified cranial shape was limited in Hungary compared to Georgia. There was stronger correlation between juvenile and adult modified cranial shape in Hungary than in Georgia. Warping along the first axis reveals the trajectory from marked flattening of the frontal and occipital regions in juveniles to diminished flattening in the same regions in adult crania, corresponding with one binding. Another depression extending from the post-bregmatic region to the temporal region, similarly strong in juveniles but diminishing in adults, marks the second binding. Hungarian crania were modified with two bindings with limited shape variation, whereas the Georgian crania had greater variation in shape being also modified with antero-posterior bindings. The findings from this study alongside contemporary historical sources help to understand the role of intentional cranial modification as a mark of social identity among nomads in the Migration Period of Europe.

Eigenvalue versus perimeter in a shape theorem for self-interacting random walks

We study paths of time-length $t$ of a continuous-time random walk on $\mathbb{Z}^{2}$ subject to self-interaction that depends on the geometry of the walk range and a collection of random, uniformly positive and finite edge weights. The interaction enters through a Gibbs weight at inverse temperature $\beta$; the “energy” is the total sum of the edge weights for edges on the outer boundary of the range. For edge weights sampled from a translation-invariant, ergodic law, we prove that the range boundary condensates around an asymptotic shape in the limit $t\to\infty$ followed by $\beta\to\infty$. The limit shape is a minimizer (unique, modulo translates) of the sum of the principal harmonic frequency of the domain and the perimeter with respect to the first-passage percolation norm derived from (the law of) the edge weights. A dense subset of all norms in $\mathbb{R}^{2}$, and thus a large variety of shapes, arise from the class of weight distributions to which our proofs apply.

Dimension improvement in Dhar's refutation of the Eden conjecture

We consider the Eden model on the d-dimensional hypercubical unoriented lattice, for large d. Initially, every lattice point is healthy, except the origin which is infected. Then, each infected lattice point contaminates any of its neighbours with rate 1. The Eden model is equivalent to first passage percolation, with exponential passage times on edges. The Eden conjecture states that the limit shape of the Eden model is a Euclidean ball. By pushing the computations of Dhar [5] a little further with modern computers and efficient implementation we obtain improved bounds for the speed of infection. This shows that the Eden conjecture does not hold in dimension superior to 22 (the lowest known dimension was 35).