Limit Shape Research Articles

Growing integer partitions with uniform marginals and the equivalence of partition ensembles

We present an explicit construction of a Markovian random growth process on integer partitions such that given it visits some level n, it passes through any partition λ of n with equal probabilities. The construction has continuous time, but we also investigate its discrete time jump chain. The jump probabilities are given by explicit but complicated expressions, so we find their asymptotic behavior as the partition becomes large. This allows us to explain how the limit shape is formed.Using the known connection of the considered probabilistic objects to Poisson point processes, we give an alternative description of the partition growth process in these terms. Then we apply the constructed growth process to find sufficient conditions for a phenomenon known as equivalence of two ensembles of random partitions for a finite number of partition characteristics. This result allows to show that counts of odd and even parts in a random partition of n are asymptotically independent as n→∞ and to find their limiting distributions, which are, somewhat surprisingly, different.

Towards Autonomous Retail Stocking and Picking: Methods Enabling Robust Vacuum-Based Robotic Manipulation in Densely Packed Environments.

With the advent of robotics and artificial intelligence, the potential for automating tasks within human-centric environments has increased significantly. This is particularly relevant in the retail sector where the demand for efficient operations and the shortage of labor drive the need for rapid advancements in robot-based technologies. Densely packed retail shelves pose unique challenges for robotic manipulation and detection due to limited space and diverse object shapes. Vacuum-based grasping technologies offer a promising solution but face challenges with object shape adaptability. The study proposes a framework for robotic grasping in retail environments, an adaptive vacuum-based grasping solution, and a new evaluation metric-termed grasp shear force resilience-for measuring the effectiveness and stability of the grasp during manipulation. The metric provides insights into how retail objects behave under different manipulation scenarios, allowing for better assessment and optimization of robotic grasping performance. The study's findings demonstrate the adaptive suction cups' ability to successfully handle a wide range of object shapes and sizes, which, in some cases, overcome commercially available solutions, particularly in adaptability. Additionally, the grasp shear force resilience metric highlights the effects of the manipulation process, such as in shear force and shake, on the manipulated object. This offers insights into its interaction with different vacuum cup grasping solutions in retail picking and restocking scenarios.

High moments of the SHE in the clustering regimes

We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under the delta(-like) initial condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar–Parisi–Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper [75] to prove an n-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape.

Minimization of detent force in PMLSM by end magnetic regulating module with free topologies

In this paper, a novel method for suppressing the detent force of permanent magnet linear synchronous motors is proposed. First, the end force is adjusted to offset the cogging force while the cogging force remains unchanged. On the other hand, the proposed method can arbitrarily change the shape of the end magnetic regulating module (EMRM), thus allowing the end force to obtain a wider adjustment range, which is different from the conventional limited shape optimization. More interestingly, compared to the traditional approach for suppressing the end force, which is to suppress the end force to the lowest level, the proposed method does not necessarily suppress the end force to the lowest level, but rather adjusts it to a reasonable level, so that it can offset the cogging force. This leads to the fact that the optimized end magnetic regulating module is effective in adjusting the end force and may even increase the end force, which is different from the conventional idea of suppressing the detent force. Next, the optimal EMRM's topologies are solved using the optimization algorithm, which replaces the traditional low-dimensional single-direction optimization and performs multi-direction global search. Finally, the prototype with optimal EMRM's topology and the testing platform are established and the experimental results validate the effectiveness of the proposed method.

Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests

We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the 4×4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\ imes 4$$\\end{document} problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size n≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 4$$\\end{document}, which appear new for n≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 5$$\\end{document}. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.

The Secret Ballet Inside Multivesicular Bodies.

Lipid bilayers possess the capacity for self-assembly due to the amphipathic nature of lipid molecules, which have both hydrophobic and hydrophilic regions. When confined, lipid bilayers exhibit astonishing versatility in their forms, adopting diverse shapes that are challenging to observe through experimental means. Exploiting this adaptability, lipid structures motivate the development of bio-inspired mechanomaterials and integrated nanobio-interfaces that could seamlessly merge with biological entities, ultimately bridging the gap between synthetic and biological systems. In this work, we demonstrate how, in numerical simulations of multivesicular bodies, a fascinating evolution unfolds from an initial semblance of order toward states of higher entropy over time. We observe dynamic rearrangements in confined vesicles that reveal unexpected limit shapes of distinct geometric patterns. We identify five structures as the basic building blocks that systematically repeat under various conditions of size and composition. Moreover, we observe more complex and less frequent shapes that emerge in confined spaces. Our results provide insights into the dynamics of multivesicular systems, offering a richer understanding of how confined lipid bodies spontaneously self-organize.

Elliptic Analogue of the Vershik–Kerov Limit Shape

We review the limit shape problem for the Plancherel measure and its generalizations found in supersymmetric gauge theory instanton count. We focus on the measure, interpolating between the Plancherel measure and the uniform measure, a \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U(1)$$\\end{document} case of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}=2^{*}$$\\end{document} gauge theory. We give the formula for its limit shape in terms of elliptic functions, generalizing the trigonometric “arcsin” law of Vershik–Kerov and Logan–Schepp.

The Role of Convection in the Limit Shape of the Critical Front Profile for Born-Infeld Diffusion Models

AbstractIn this paper, we deal with models with Born-Infeld (or relativistic) type diffusion and monostable reaction, investigating the effect of the introduction of a convection term on the limit shape of the critical front profile for vanishing diffusion. We first provide an estimate of the critical speed and then, through a careful analysis of an equivalent first-order problem, we show that different convection terms may lead either to a complete sharpening of the limit profile or to its complete regularization, presenting some related numerical simulations.

TailorMe: Self‐Supervised Learning of an Anatomically Constrained Volumetric Human Shape Model

AbstractHuman shape spaces have been extensively studied, as they are a core element of human shape and pose inference tasks. Classic methods for creating a human shape model register a surface template mesh to a database of 3D scans and use dimensionality reduction techniques, such as Principal Component Analysis, to learn a compact representation. While these shape models enable global shape modifications by correlating anthropometric measurements with the learned subspace, they only provide limited localized shape control. We instead register a volumetric anatomical template, consisting of skeleton bones and soft tissue, to the surface scans of the CAESAR database. We further enlarge our training data to the full Cartesian product of all skeletons and all soft tissues using physically plausible volumetric deformation transfer. This data is then used to learn an anatomically constrained volumetric human shape model in a self‐supervised fashion. The resulting TailorMe model enables shape sampling, localized shape manipulation, and fast inference from given surface scans.

Tailoring Photoweldable Shape Memory Polyurethane with Intrinsic Photothermal / Fluorescence Via Engineering Metal‐Phenolic Systems

AbstractMultifunctional shape memory polymers are significant for next‐generation intelligent systems as smart materials. However, their practical applications are limited by toughness, manufacturability, and functional integration strategies. Besides, while 4D printing is used to construct deformable devices, the limited accessible shapes and difficulty of integrating multiple materials hindered the development of multifunctional devices. Herein, metal‐phenolic coordination is used as crosslinks to tailor tough shape memory polyurethanes with intrinsic photothermal effect and/or tunable fluorescence. The reversible dissociation of metal‐phenolic complexes upon loading provides an efficient energy dissipation for the polymer network. The photothermal/fluorescent properties of target polyurethane can be effortlessly programmed by changing the coordinated metals and functional ligands. The obtained fluorescent polyurethane displays tunable fluorescence through the reversible dissociation‐association of dynamic bonds upon thermal stimulus. Moreover, relying on light‐triggered hybrid exchange reactions among various metal‐phenolic coordination bonds, heterogeneous interfaces can be merged to manufacture exquisite 3D shape‐shifting devices and integrate more advanced functions. As demonstrations, intrinsic photothermal‐responsive shape memory stent, and 3D fluorescent encryption device are produced based on photo‐welding instead of 4D printing. This study not only provides a convenient supramolecular strategy for constructing multifunctional smart materials but also demonstrates a feasible method for making personalized integrated architectures.

Building an evidence-based multi-level system of quality disability support for adults with acquired neurological disability

Purpose People with acquired neurological disability have the right to quality disability support, yet there is still limited user insights shaping disability services. This research aimed to develop a comprehensive understanding of quality support for adults with acquired neurological disability based on lived experiences. Methods Using a constructivist grounded theory approach, this project engaged 12 adults with acquired neurological disability, 12 support workers, and 10 close others in individual interviews. Each perspective was analysed independently and subsequently merged to construct a holistic model of quality support grounded in lived experience. Results The model of quality support encompasses interconnected themes spanning dyadic relationships, support worker competency, team dynamics, provider practices, and sector-wide considerations. Key elements include recognising the individual’s expertise in their support needs, nurturing effective support relationships, fostering a capable support team, and upholding sector accountability. Authentic choice and personal agency emerged as paramount factors in achieving quality support. Conclusions The findings align with disability rights and individualised funding principles, emphasising the centrality of people with lived experience in shaping support services. The research underscores the need to prioritise authentic choice and individual empowerment, offering valuable insights for both practice and policy development in the disability support sector.

Noncommutative Geometry of Random Surfaces

We associate a noncommutative curve to a periodic, bipartite, planar dimer model with polygonal boundary. It determines the inverse Kasteleyn matrix and hence all correlations. It may be seen as a quantization of the limit shape construction of Kenyon and the author. We also discuss various directions in which this correspondence may be generalized.

Sexual selection and mate limitation shape evolution of species' range limits.

Understanding what processes shape the formation of species' geographic range limits is one central objective linking ecology and evolutionary biology. One potentially key process is sexual selection; yet, theory examining how sexual selection could shape eco-evolutionary dynamics in marginal populations is still lacking. In species with separate sexes, range limits could be shaped by limitations in encountering mates at low densities. Sexual selection could therefore modulate mate limitation and resulting extinction-colonization dynamics at range margins, through evolution of mate encounter ability and/or mate competition traits, and their demographic consequences. We use a spatially explicit eco-genetic model to reveal how different forms of sexual selection can variably affect emerging range limits. Larger ranges emerged when sexual selection acted exclusively on traits increasing mate encounter probability, thus reducing female's mate limitation toward the range margins. In contrast, sexual selection via mate competition narrowed range limits due to increased trait-dependent mortality in males and elevated mate limitation for females. When mate encounter coevolved with mate competition, their combined effects on range limits depended on the mating system (polygyny vs. monogamy). Our results demonstrate that evolution of species' ranges may be importantly shaped by feedbacks between sexual selection and spatial population demography and dynamics.

Coalescence of Geodesics and the BKS Midpoint Problem in Planar First-Passage Percolation

We consider first-passage percolation on \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{Z}^{2}$\\end{document} with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics.The result leads to a quantitative resolution of the Benjamini–Kalai–Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge.We further prove that the limit shape assumption is satisfied for a specific family of distributions.Lastly, related to the 1965 Hammersley–Welsh highways and byways problem, we prove that the expected fraction of the square {−n,…,n}2 which is covered by infinite geodesics starting at the origin is at most an inverse power of n. This result is obtained without explicit limit shape assumptions.

Interface fluctuations associated with split Fermi seas

We consider the asymptotic behaviour of a family of unidimensional lattice fermion models, which are in exact correspondence with certain probability laws on partitions and on unitary matrices. These models exhibit limit shapes, and in the case where the bulk of these shapes are described by analytic functions, the fluctuations around their interfaces have been shown to follow a universal Tracy–Widom distribution or its higher-order analogue. Non-differentiable bulk limit shape functions arise when a gap appears in some quantum numbers of the model, in other words when the Fermi sea is split. We show that split Fermi seas give rise to new interface fluctuations, governed by integer powers of universal distributions. This breakdown in universality is analogous to the behaviour of a random Hermitian matrix when the support of its limiting eigenvalue distribution has multiple cuts, with oscillations appearing in the limit of the two-point correlation function. We show that when the Fermi sea is split in the lattice fermion model, there are multiple cuts in the eigenvalue support of the corresponding unitary matrix model.

Fully angularly resolved 3D microrheology with optical tweezers

Microrheology with optical tweezers (MOT) is an all-optical technique that allows the user to investigate a materials’ viscoelastic properties at microscopic scales, and is particularly useful for those materials that feature complex microstructures, such as biological samples. MOT is increasingly being employed alongside 3D imaging systems and particle tracking methods to generate maps showing not only how properties may vary between different points in a sample but also how at a single point the viscoelastic properties may vary with direction. However, due to the diffraction limited shape of focussed beams, optical traps are inherently anisotropic in 3D. This can result in a significant overestimation of the fluids’ viscosity in certain directions. As such, the rheological properties can only be accurately probed along directions parallel or perpendicular to the axis of trap beam propagation. In this work, a new analytical method is demonstrated to overcome this potential artefact. This is achieved by performing principal component analysis on 3D MOT data to characterise the trap, and then identify the frequency range over which trap anisotropy influences the data. This approach is initially applied to simulated data for a Newtonian fluid where the trap anisotropy induced maximum error in viscosity is reduced from ~ 150% to less than 6%. The effectiveness of the method is corroborated by experimental MOT measurements performed with water and gelatine solutions, thus confirming that the microrheology of a fluid can be extracted reliably across a wide frequency range and in any arbitrary direction. This work opens the door to fully spatially and angularly resolved 3D mapping of the rheological properties of soft materials over a broad frequency range.

Boundary asymptotics of non-intersecting Brownian motions: Pearcey, Airy and a transition

We study n non-intersecting Brownian motions, corresponding to the eigenvalues of an n × n Hermitian Brownian motion. At the boundary of their limit shape we find that only three universal processes can arise: the Pearcey process close to merging points, the Airy line ensemble at edges and a novel determinantal process describing the transition from the Pearcey process to the Airy line ensemble. The three cases are distinguished by a remarkably simple integral condition. Our results hold under very mild assumptions, in particular we do not require any kind of convergence of the initial configuration as n→∞ . Applications to largest eigenvalues of macro- and mesoscopic bulks and to random initial configurations are given.

Conceptual design of poloidal horseshoe limiter layout for JA DEMO

In the development of the JA DEMO fusion reactor, the proposed poloidal horseshoe limiter aims to mitigate the surface heat load localized on the breeding blanket. The limiter protrudes from the first wall toward the plasma to shade the blanket from the charged particles, which account for the majority of the surface heat load. To concretize the limiter design, this study presents a limiter layout that implements a concept that allows for sufficient cooling and remote maintenance. The limiter shape was determined by evaluating the surface heat load using 3-D magnetic field line tracing. A fully poloidal limiter with a width of 300 mm, which corresponds to 2.4 % of the FW area, was selected. The limiter heating, consisting of surface heat load and nuclear heating, was evaluated to discuss the required cooling, and the concept of a limiter cassette and modules for cooling was proposed. For the limiter replacement method, the outboard limiter is designed to be extracted from the upper port with the mounting segment. The inboard limiter is then grasped and transported to reach the upper port after the outboard limiter extraction. Both the inboard and outboard limiter are expected to be replaced without affecting other in-vessel components in this study.

Limit shapes from harmonicity: dominos and the five vertex model

We discuss how to construct limit shapes for the domino tiling model (square lattice dimer model) and five-vertex model, in appropriate polygonal domains. Our methods are based on the harmonic extension method of Kenyon and Prause (2022 Duke Math J. 171 3003–22).

Drivers and mechanisms that contribute to microbial β‐diversity patterns and range sizes in mountains across a climatic variability gradient

Microbial communities are highly diverse, yet the mechanisms underlying microbial community assembly are not well understood. Janzen's mountain passes hypothesis proposed that climatic barriers and dispersal limitation shape communities to a greater extent in mountains with lower climatic variability and overlap, permitting higher levels of species coexistence. Here, we investigate changes in microbial community dissimilarities, distributional range sizes and ecological processes along elevational gradients in three montane ecosystems representing a climatic variability gradient. We found that climate, climatic variability and spatial distance play dominating roles in affecting microbial β‐diversity patterns and range sizes along elevational gradients. Janzen's mountain passes hypothesis can be applied to microbial community assembly: mountains with lower climatic variability and higher climatic difference between elevations exhibited higher β‐diversity, higher endemism, lower range sizes and steeper distance–decay trends. However, microbial communities experience clear climate‐driven limited range sizes and dispersal processes, and show typical endemic patterns, in all mountain ecosystems. Our results emphasize the importance of dispersal and climatic niche processes in shaping montane biodiversity. As a result, changes in climate may significantly impact soil biodiversity in montane ecosystems by altering the effects of dispersal limitation and climatic variability on bacterial and fungal community composition along elevational gradients.Keywords: community assembly, dispersal limitation, β‐diversity, montane biodiversity, microbial community