Rigidity of spherical codes

A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity of spherical codes, particularly kissing configurations. One surprise is that the kissing configuration of the Coxeter-Todd lattice is not jammed, despite being locally jammed (each individual cap is held in place if its neighbors are fixed); in this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic lattice in three dimensions. By contrast, we find that many other packings have jammed kissing configurations, including the Barnes-Wall lattice and all of the best kissing configurations known in four through twelve dimensions. Jamming seems to become much less common for large kissing configurations in higher dimensions, and in particular it fails for the best kissing configurations known in 25 through 31 dimensions. Motivated by this phenomenon, we find new kissing configurations in these dimensions, which improve on the records set in 1982 by the laminated lattices.Comment: 39 pages, 8 figure

Susceptibility and Percolation in 2D Random Field Ising Magnets

The ground state structure of the two-dimensional random field Ising magnet is studied using exact numerical calculations. First we show that the ferromagnetism, which exists for small system sizes, vanishes with a large excitation at a random field strength dependent length scale. This {\it break-up length scale} $L_b$ scales exponentially with the squared random field, $\exp(A/\Delta^2)$. By adding an external field $H$ we then study the susceptibility in the ground state. If $L>L_b$, domains melt continuously and the magnetization has a smooth behavior, independent of system size, and the susceptibility decays as $L^{-2}$. We define a random field strength dependent critical external field value $\pm H_c(\Delta)$, for the up and down spins to form a percolation type of spanning cluster. The percolation transition is in the standard short-range correlated percolation universality class. The mass of the spanning cluster increases with decreasing $\Delta$ and the critical external field approaches zero for vanishing random field strength, implying the critical field scaling (for Gaussian disorder) $H_c \sim (\Delta -\Delta_c)^\delta$, where $\Delta_c = 1.65 \pm 0.05$ and $\delta=2.05\pm 0.10$. Below $\Delta_c$ the systems should percolate even when H=0. This implies that even for H=0 above $L_b$ the domains can be fractal at low random fields, such that the largest domain spans the system at low random field strength values and its mass has the fractal dimension of standard percolation $D_f = 91/48$. The structure of the spanning clusters is studied by defining {\it red clusters}, in analogy to the ``red sites'' of ordinary site-percolation. The size of red clusters defines an extra length scale, independent of $L$.Comment: 17 pages, 28 figures, accepted for publication in Phys. Rev.

Quantum-fluctuation-induced repelling interaction of quantum string between walls

Quantum string, which was brought into discussion recently as a model for the stripe phase in doped cuprates, is simulated by means of the density-matrix-renormalization-group method. String collides with adjacent neighbors, as it wonders, owing to quantum zero-point fluctuations. The energy cost due to the collisions is our main concern. Embedding a quantum string between rigid walls with separation d, we found that for sufficiently large d, collision-induced energy cost obeys the formula \sim exp (- A d^alpha) with alpha=0.808(1), and string's mean fluctuation width grows logarithmically \sim log d. Those results are not understood in terms of conventional picture that the string is `disordered,' and only the short-wave-length fluctuations contribute to collisions. Rather, our results support a recent proposal that owing to collisions, short-wave-length fluctuations are suppressed, but instead, long-wave-length fluctuations become significant. This mechanism would be responsible for stabilizing the stripe phase

Short-term Memory of Deep RNN

The extension of deep learning towards temporal data processing is gaining an increasing research interest. In this paper we investigate the properties of state dynamics developed in successive levels of deep recurrent neural networks (RNNs) in terms of short-term memory abilities. Our results reveal interesting insights that shed light on the nature of layering as a factor of RNN design. Noticeably, higher layers in a hierarchically organized RNN architecture results to be inherently biased towards longer memory spans even prior to training of the recurrent connections. Moreover, in the context of Reservoir Computing framework, our analysis also points out the benefit of a layered recurrent organization as an efficient approach to improve the memory skills of reservoir models.Comment: This is a pre-print (pre-review) version of the paper accepted for presentation at the 26th European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning (ESANN), Bruges (Belgium), 25-27 April 201