On the distribution of angles between the N shortest vectors in a random lattice

We determine the joint distribution of the lengths of, and angles between, the N shortest lattice vectors in a random n-dimensional lattice as n tends to infinity. Moreover we interpret the result in terms of eigenvalues and eigenfunctions of the Laplacian on flat tori. Finally we discuss the limit distribution of any finite number of successive minima of a random n-dimensional lattice as n tends to infinity.Comment: 17 page

Applied Symmetry for Crystal Structure Prediction

This thesis presents an original open-source Python package called PyXtal (pronounced pi-crystal ) that generates random symmetric crystal structures for use in crystal structure prediction (CSP). The primary advantage of PyXtal over existing structure generation tools is its unique symmetrization method. For molecular structures, PyXtal uses an original algorithm to determine the compatibility of molecular point group symmetry with Wyckoff site symmetry. This allows the molecules in generated structures to occupy special Wyckoff positions without breaking the structure\u27s symmetry. This is a new feature which increases the space of search-able structures and in turn improves CSP performance. It is shown that using already-symmetric initial structures results in a higher probability of finding the lowest-energy structure. Ultimately, this lowers the computational time needed for CSP. Structures can be generated for any symmetry group of 0, 1, 2, or 3 dimensions of periodicity. Either atoms or rigid molecules may be used as building blocks. The generated structures can be optimized with VASP, LAMMPS, or other computational tools. Additional options are provided for the lattice and inter-atomic distance constraints. Results for carbon and silicon crystals, water ice crystals, and molybdenum clusters are presented as usage examples

Faster tuple lattice sieving using spherical locality-sensitive filters

To overcome the large memory requirement of classical lattice sieving algorithms for solving hard lattice problems, Bai-Laarhoven-Stehl\'{e} [ANTS 2016] studied tuple lattice sieving, where tuples instead of pairs of lattice vectors are combined to form shorter vectors. Herold-Kirshanova [PKC 2017] recently improved upon their results for arbitrary tuple sizes, for example showing that a triple sieve can solve the shortest vector problem (SVP) in dimension $d$ in time $2^{0.3717d + o(d)}$, using a technique similar to locality-sensitive hashing for finding nearest neighbors. In this work, we generalize the spherical locality-sensitive filters of Becker-Ducas-Gama-Laarhoven [SODA 2016] to obtain space-time tradeoffs for near neighbor searching on dense data sets, and we apply these techniques to tuple lattice sieving to obtain even better time complexities. For instance, our triple sieve heuristically solves SVP in time $2^{0.3588d + o(d)}$. For practical sieves based on Micciancio-Voulgaris' GaussSieve [SODA 2010], this shows that a triple sieve uses less space and less time than the current best near-linear space double sieve.Comment: 12 pages + references, 2 figures. Subsumed/merged into Cryptology ePrint Archive 2017/228, available at https://ia.cr/2017/122

Tradeoffs for nearest neighbors on the sphere

We consider tradeoffs between the query and update complexities for the (approximate) nearest neighbor problem on the sphere, extending the recent spherical filters to sparse regimes and generalizing the scheme and analysis to account for different tradeoffs. In a nutshell, for the sparse regime the tradeoff between the query complexity $n^{\rho_q}$ and update complexity $n^{\rho_u}$ for data sets of size $n$ is given by the following equation in terms of the approximation factor $c$ and the exponents $\rho_q$ and $\rho_u$: $$c^2\sqrt{\rho_q}+(c^2-1)\sqrt{\rho_u}=\sqrt{2c^2-1}.$$ For small $c=1+\epsilon$, minimizing the time for updates leads to a linear space complexity at the cost of a query time complexity $n^{1-4\epsilon^2}$. Balancing the query and update costs leads to optimal complexities $n^{1/(2c^2-1)}$, matching bounds from [Andoni-Razenshteyn, 2015] and [Dubiner, IEEE-TIT'10] and matching the asymptotic complexities of [Andoni-Razenshteyn, STOC'15] and [Andoni-Indyk-Laarhoven-Razenshteyn-Schmidt, NIPS'15]. A subpolynomial query time complexity $n^{o(1)}$ can be achieved at the cost of a space complexity of the order $n^{1/(4\epsilon^2)}$, matching the bound $n^{\Omega(1/\epsilon^2)}$ of [Andoni-Indyk-Patrascu, FOCS'06] and [Panigrahy-Talwar-Wieder, FOCS'10] and improving upon results of [Indyk-Motwani, STOC'98] and [Kushilevitz-Ostrovsky-Rabani, STOC'98]. For large $c$, minimizing the update complexity results in a query complexity of $n^{2/c^2+O(1/c^4)}$, improving upon the related exponent for large $c$ of [Kapralov, PODS'15] by a factor $2$, and matching the bound $n^{\Omega(1/c^2)}$ of [Panigrahy-Talwar-Wieder, FOCS'08]. Balancing the costs leads to optimal complexities $n^{1/(2c^2-1)}$, while a minimum query time complexity can be achieved with update complexity $n^{2/c^2+O(1/c^4)}$, improving upon the previous best exponents of Kapralov by a factor $2$.Comment: 16 pages, 1 table, 2 figures. Mostly subsumed by arXiv:1608.03580 [cs.DS] (along with arXiv:1605.02701 [cs.DS]

Introducing Quantum Ricci Curvature

Motivated by the search for geometric observables in nonperturbative quantum gravity, we define a notion of coarse-grained Ricci curvature. It is based on a particular way of extracting the local Ricci curvature of a smooth Riemannian manifold by comparing the distance between pairs of spheres with that of their centres. The quantum Ricci curvature is designed for use on non-smooth and discrete metric spaces, and to satisfy the key criteria of scalability and computability. We test the prescription on a variety of regular and random piecewise flat spaces, mostly in two dimensions. This enables us to quantify its behaviour for short lattices distances and compare its large-scale behaviour with that of constantly curved model spaces. On the triangulated spaces considered, the quantum Ricci curvature has good averaging properties and reproduces classical characteristics on scales large compared to the discretization scale.Comment: 43 pages, 27 figure

Normal stress anisotropy and marginal stability in athermal elastic networks

Hydrogels of semiflexible biopolymers such as collagen have been shown to contract axially under shear strain, in contrast to the axial dilation observed for most elastic materials. Recent work has shown that this behavior can be understood in terms of the porous, two-component nature and consequent time-dependent compressibility of hydrogels. The apparent normal stress measured by a torsional rheometer reflects only the tensile contribution of the axial component $\sigma_{zz}$ on long (compressible) timescales, crossing over to the first normal stress difference, $N_1 = \sigma_{xx}-\sigma_{zz}$ at short (incompressible) times. While the behavior of $N_1$ is well understood for isotropic viscoelastic materials undergoing affine shear deformation, biopolymer networks are often anisotropic and deform nonaffinely. Here, we numerically study the normal stresses that arise under shear in subisostatic, athermal semiflexible polymer networks. We show that such systems exhibit strong deviations from affine behavior and that these anomalies are controlled by a rigidity transition as a function of strain

Numerical Jordan-Wigner approach for two dimensional spin systems

We present a numerical self consistent variational approach based on the Jordan-Wigner transformation for two dimensional spin systems. We apply it to the study of the well known quantum (S=1/2) antiferromagnetic XXZ system as a function of the easy-axis anisotropy \Delta on a periodic square lattice. For the SU(2) case the method converges to a N\'eel ordered ground state irrespectively of the input density profile used and in accordance with other studies. This shows the potential utility of the proposed method to investigate more complicated situations like frustrated or disordered systems.Comment: Revtex, 8 pages, 4 figure