Vertex models represent confluent tissue by polygonal or polyhedral tilings of space, with the individual cells interacting via force laws that depend on both the geometry of the cells and the ...topology of the tessellation. This dependence on the connectivity of the cellular network introduces several complications to performing molecular-dynamics-like simulations of vertex models, and in particular makes parallelizing the simulations difficult. cellGPU addresses this difficulty and lays the foundation for massively parallelized, GPU-based simulations of these models. This article discusses its implementation for a pair of two-dimensional models, and compares the typical performance that can be expected between running cellGPU entirely on the CPU versus its performance when running on a range of commercial and server-grade graphics cards. By implementing the calculation of topological changes and forces on cells in a highly parallelizable fashion, cellGPU enables researchers to simulate time- and length-scales previously inaccessible via existing single-threaded CPU implementations.
Program Title: cellGPU
Program Files doi:http://dx.doi.org/10.17632/6j2cj29t3r.1
Licensing provisions: MIT
Programming language: CUDA/C++
Nature of problem: Simulations of off-lattice “vertex models” of cells, in which the interaction forces depend on both the geometry and the topology of the cellular aggregate.
Solution method: Highly parallelized GPU-accelerated dynamical simulations in which the force calculations and the topological features can be handled on either the CPU or GPU.
Additional comments: The code is hosted at https://gitlab.com/dmsussman/cellGPU, with documentation additionally maintained at http://dmsussman.gitlab.io/cellGPUdocumentation
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In his work on the twenty vertex model, Di Francesco (2021) found a determinant formula for the number of configurations in a specific such model, and he conjectured a closed form product formula for ...the evaluation of this determinant. We prove this conjecture here. Moreover, we actually generalize this determinant evaluation to a one-parameter family of determinant evaluations, and we present many more determinant evaluations of similar type — some proved, some left open as conjectures.
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In this paper, we consider two models in the Kardar–Parisi–Zhang (KPZ) universality class, the asymmetric simple exclusion process (ASEP) and the stochastic six-vertex model. We introduce a new class ...of initial data (which we call shape generalized step Bernoulli initial data) for both of these models that generalizes the step Bernoulli initial data studied in a number of recent works on the ASEP. Under this class of initial data, we analyze the current fluctuations of both the ASEP and stochastic six-vertex model and establish the existence of a phase transition along a characteristic line, across which the fluctuation exponent changes from 1/2 to 1/3. On the characteristic line, the current fluctuations converge to the general (rank k) Baik–Ben–Arous–Péché distribution for the law of the largest eigenvalue of a critically spiked covariance matrix. For k = 1, this was established for the ASEP by Tracy and Widom; for k > 1 (and also k = 1, for the stochastic six-vertex model), the appearance of these distributions in both models is new.
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The genus-zero five-vertex model Kenyon, Richard; Prause, István
Probability and mathematical physics (Berkeley, Calif. : Online),
12/2022, Volume:
3, Issue:
4
Journal Article
We present an approach to understand geometric-incompatibility–induced rigidity in underconstrained materials, including subisostatic 2D spring networks and 2D and 3D vertex models for dense ...biological tissues. We show that in all these models a geometric criterion, represented by a minimal length ℓ̄min, determines the onset of prestresses and rigidity. This allows us to predict not only the correct scalings for the elastic material properties, but also the precise magnitudes for bulk modulus and shear modulus discontinuities at the rigidity transition as well as the magnitude of the Poynting effect. We also predict from first principles that the ratio of the excess shear modulus to the shear stress should be inversely proportional to the critical strain with a prefactor of 3. We propose that this factor of 3 is a general hallmark of geometrically induced rigidity in underconstrained materials and could be used to distinguish this effect from nonlinear mechanics of single components in experiments. Finally, our results may lay important foundations for ways to estimate ℓ̄min from measurements of local geometric structure and thus help develop methods to characterize large-scale mechanical properties from imaging data.
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Abstract
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).
This work is dedicated to the consideration of the construction of a representation of braid group generators from vertex models with N-states, which provides a great way to study the knot invariant. ...An algebraic formula is proposed for the knot invariant when different spins (N−1)/2 are located on all components of the knot. The work summarizes procedure outputting braid generator representations from three-partite vertex model. This representation made it possible to study the invariant of a knot with multi-colored links, where the components of the knot have different spins. The formula for the invariant of knot with a multi-colored link is studied from the point of view of the braid generators obtained from the R-matrices of three-partite vertex models. The resulting knot invariant 52 corresponds to the Jones polynomial and HOMFLY-PT.
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10.
Free fermionic Schur functions Naprienko, Slava
Advances in mathematics (New York. 1965),
01/2024, Volume:
436
Journal Article
Peer reviewed
Open access
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