Constructing and exploring wells of energy landscapes

Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features ``wells of duration $T$ and altitude $y$''. We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) which have been developed during the last two decades; we therefore propose a minimal appendix exposing them at the end of this paper to bridge the possible gap.Comment: 28 pages, submitted to J. Math. Phys -

A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley-Lieb algebra EPTL_N(\beta, \alpha) introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley-Lieb algebra. Beside the defining parameters \beta=u^2+u^{-2} with u=e^{i\lambda/2} (weight of contractible loops) and \alpha (weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix T_N(\lambda, \nu) depends on the anisotropy \nu and the spectral parameter \lambda that fixes the model. (The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6\lambda^2/(\pi(\lambda-\pi)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner i_N^d between these link representations and the eigenspaces of S^z of the XXZ models. This map is shown to be an isomorphism for generic values of u and v and the critical curves in the plane of these parameters for which i_N^d fails to be an isomorphism are given.Comment: Replacement of "The Gram matrix as a connection between periodic loop models and XXZ Hamiltonians", 31 page

Boundary states for a free boson defined on finite geometries

Langlands recently constructed a map that factorizes the partition function of a free boson on a cylinder with boundary condition given by two arbitrary functions in the form of a scalar product of boundary states. We rewrite these boundary states in a compact form, getting rid of technical assumptions necessary in his construction. This simpler form allows us to show explicitly that the map between boundary conditions and states commutes with conformal transformations preserving the boundary and the reality condition on the scalar field.Comment: 16 pages, LaTeX (uses AMS components). Revised version; an analogy with string theory computations is discussed and references adde

Order recall in verbal short-term memory: The role of semantic networks

In their recent article, Acheson, MacDonald, and Postle (Journal of Experimental Psychology: Learning, Memory, and Cognition 37:44-59, 2011) made an important but controversial suggestion: They hypothesized that (a) semantic information has an effect on order information in short-term memory (STM) and (b) order recall in STM is based on the level of activation of items within the relevant lexico-semantic long-term memory (LTM) network. However, verbal STM research has typically led to the conclusion that factors such as semantic category have a large effect on the number of correctly recalled items, but little or no impact on order recall (Poirier & Saint-Aubin, Quarterly Journal of Experimental Psychology 48A:384-404, 1995; Saint-Aubin, Ouellette, & Poirier, Psychonomic Bulletin & Review 12:171-177, 2005; Tse, Memory 17:874-891, 2009). Moreover, most formal models of short-term order memory currently suggest a separate mechanism for order coding-that is, one that is separate from item representation and not associated with LTM lexico-semantic networks. Both of the experiments reported here tested the predictions that we derived from Acheson et al. The findings show that, as predicted, manipulations aiming to affect the activation of item representations significantly impacted order memory

Proposal for a CFT interpretation of Watts' differential equation for percolation

G. M. T. Watts derived that in two dimensional critical percolation the crossing probability Pi_hv satisfies a fifth order differential equation which includes another one of third order whose independent solutions describe the physically relevant quantities 1, Pi_h, Pi_hv. We will show that this differential equation can be derived from a level three null vector condition of a rational c=-24 CFT and motivate how this solution may be fitted into known properties of percolation.Comment: LaTeX, 20p, added references, corrected typos and additional content

Geometric Exponents, SLE and Logarithmic Minimal Models

In statistical mechanics, observables are usually related to local degrees of freedom such as the Q < 4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic Loewner evolution (SLE_kappa), one can consider observables related to nonlocal degrees of freedom such as paths or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to values of conformal dimensions not found among the finite sets of values allowed by the rational minimal models. Working in the context of a loop gas with loop fugacity beta = -2 cos(4 pi/kappa), we use Monte Carlo simulations to measure the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the SLE parameter kappa = 4p'/p is rational with p < p', we argue that the geometric exponents are related to conformal dimensions found in the infinitely extended Kac tables of the logarithmic minimal models LM(p,p'). These theories describe lattice systems with nonlocal degrees of freedom. We present results for critical dense polymers LM(1,2), critical percolation LM(2,3), the logarithmic Ising model LM(3,4), the logarithmic tricritical Ising model LM(4,5) as well as LM(3,5). Our results are compared with rigourous results from SLE_kappa, with predictions from theoretical physics and with other numerical experiments. Throughout, we emphasize the relationships between SLE_kappa, geometric exponents and the conformal dimensions of the underlying CFTs.Comment: Added reference

Statistical properties of the low-temperature conductance peak-heights for Corbino discs in the quantum Hall regime

A recent theory has provided a possible explanation for the ``non-universal scaling'' of the low-temperature conductance (and conductivity) peak-heights of two-dimensional electron systems in the integer and fractional quantum Hall regimes. This explanation is based on the hypothesis that samples which show this behavior contain density inhomogeneities. Theory then relates the non-universal conductance peak-heights to the ``number of alternating percolation clusters'' of a continuum percolation model defined on the spatially-varying local carrier density. We discuss the statistical properties of the number of alternating percolation clusters for Corbino disc samples characterized by random density fluctuations which have a correlation length small compared to the sample size. This allows a determination of the statistical properties of the low-temperature conductance peak-heights of such samples. We focus on a range of filling fraction at the center of the plateau transition for which the percolation model may be considered to be critical. We appeal to conformal invariance of critical percolation and argue that the properties of interest are directly related to the corresponding quantities calculated numerically for bond-percolation on a cylinder. Our results allow a lower bound to be placed on the non-universal conductance peak-heights, and we compare these results with recent experimental measurements.Comment: 7 pages, 4 postscript figures included. Revtex with epsf.tex and multicol.sty. The revised version contains some additional discussion of the theory and slightly improved numerical result

Conformal Curves in Potts Model: Numerical Calculation

We calculated numerically the fractal dimension of the boundaries of the Fortuin-Kasteleyn clusters of the $q$-state Potts model for integer and non-integer values of $q$ on the square lattice. In addition we calculated with high accuracy the fractal dimension of the boundary points of the same clusters on the square domain. Our calculation confirms that this curves can be described by SLE$_{\kappa}$.Comment: 11 Pages, 4 figure

SLE local martingales in logarithmic representations

A space of local martingales of SLE type growth processes forms a representation of Virasoro algebra, but apart from a few simplest cases not much is known about this representation. The purpose of this article is to exhibit examples of representations where L_0 is not diagonalizable - a phenomenon characteristic of logarithmic conformal field theory. Furthermore, we observe that the local martingales bear a close relation with the fusion product of the boundary changing fields. Our examples reproduce first of all many familiar logarithmic representations at certain rational values of the central charge. In particular we discuss the case of SLE(kappa=6) describing the exploration path in critical percolation, and its relation with the question of operator content of the appropriate conformal field theory of zero central charge. In this case one encounters logarithms in a probabilistically transparent way, through conditioning on a crossing event. But we also observe that some quite natural SLE variants exhibit logarithmic behavior at all values of kappa, thus at all central charges and not only at specific rational values.Comment: 40 pages, 7 figures. v3: completely rewritten, new title, new result