Field theory conjecture for loop-erased random walks

We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in epsilon=4-d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d=4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d=4, which can be used as a further test of the conjecture.Comment: 5 page

Note on SLE and logarithmic CFT

It is discussed how stochastic evolutions may be linked to logarithmic conformal field theory. This introduces an extension of the stochastic Loewner evolutions. Based on the existence of a logarithmic null vector in an indecomposable highest-weight module of the Virasoro algebra, the representation theory of the logarithmic conformal field theory is related to entities conserved in mean under the stochastic process.Comment: 10 pages, LaTeX, v2: version to be publishe

Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions

We show that in the loop-erased random walk problem, the exponent characterizing probability distribution of areas of erased loops is superuniversal. In d-dimensions, the probability that the erased loop has an area A varies as A^{-2} for large A, independent of d, for 2 <= d <= 4. We estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d = 2 and 3 by large-scale Monte Carlo simulations. Our estimate of the fractal dimension z in two-dimensions is consistent with the known exact value 5/4. In three-dimensions, we get z = 1.6183 +- 0.0004. The exponent for the distribution of durations of avalanche in the three-dimensional abelian sandpile model is determined from this by using scaling relations.Comment: 25 pages, 1 table, 8 figure

Electronic structure of the cuprate superconducting and pseudogap phases from spectroscopic imaging STM

We survey the use of spectroscopic imaging scanning tunneling microscopy (SI-STM) to probe the electronic structure of underdoped cuprates. Two distinct classes of electronic states are observed in both the d-wave superconducting (dSC) and the pseudogap (PG) phases. The first class consists of the dispersive Bogoliubov quasiparticle excitations of a homogeneous d-wave superconductor, existing below a lower energy scale E = Delta(0). We find that the Bogoliubov quasiparticle interference (QPI) signatures of delocalized Cooper pairing are restricted to a k-space arc, which terminates near the lines connecting k = +/-(pi/a(0), 0) to k = +/-(0, pi/a(0)). This arc shrinks continuously with decreasing hole density such that Luttinger's theorem could be satisfied if it represents the front side of a hole-pocket that is bounded behind by the lines between k = +/-(pi/a(0), 0) and k = +/-(0, pi/a(0)). In both phases, the only broken symmetries detected for the vertical bar E vertical bar < Delta(0) states are those of a d-wave superconductor. The second class of states occurs proximate to the PG energy scale E = Delta(1). Here the non-dispersive electronic structure breaks the expected 90 degrees-rotational symmetry of electronic structure within each unit cell, at least down to 180 degrees-rotational symmetry. This electronic symmetry breaking was first detected as an electronic inequivalence at the two oxygen sites within each unit cell by using a measure of nematic (C-2) symmetry. Incommensurate non-dispersive conductance modulations, locally breaking both rotational and translational symmetries, coexist with this intra-unit-cell electronic symmetry breaking at E = Delta(1). Their characteristic wavevector Q is determined by the k-space points where Bogoliubov QPI terminates and therefore changes continuously with doping. The distinct broken electronic symmetry states (intra-unit-cell and finite Q) coexisting at E similar to Delta(1) are found to be indistinguishable in the dSC and PG phases. The next challenge for SI-STM studies is to determine the relationship of the E similar to Delta(1) broken symmetry electronic states with the PG phase, and with the E < Delta(0) states associated with Cooper pairing.Publisher PDFPeer reviewe

Probability distribution of the sizes of largest erased-loops in loop-erased random walks

We have studied the probability distribution of the perimeter and the area of the k-th largest erased-loop in loop-erased random walks in two-dimensions for k = 1 to 3. For a random walk of N steps, for large N, the average value of the k-th largest perimeter and area scales as N^{5/8} and N respectively. The behavior of the scaled distribution functions is determined for very large and very small arguments. We have used exact enumeration for N <= 20 to determine the probability that no loop of size greater than l (ell) is erased. We show that correlations between loops have to be taken into account to describe the average size of the k-th largest erased-loops. We propose a one-dimensional Levy walk model which takes care of these correlations. The simulations of this simpler model compare very well with the simulations of the original problem.Comment: 11 pages, 1 table, 10 included figures, revte

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

Fe I Oscillator Strengths for the Gaia-ESO Survey

The Gaia-ESO Public Spectroscopic Survey (GES) is conducting a large-scale study of multi-element chemical abundances of some 100 000 stars in the Milky Way with the ultimate aim of quantifying the formation history and evolution of young, mature and ancient Galactic populations. However, in preparing for the analysis of GES spectra, it has been noted that atomic oscillator strengths of important Fe I lines required to correctly model stellar line intensities are missing from the atomic database. Here, we present new experimental oscillator strengths derived from branching fractions and level lifetimes, for 142 transitions of Fe I between 3526 {\AA} and 10864 {\AA}, of which at least 38 are urgently needed by GES. We also assess the impact of these new data on solar spectral synthesis and demonstrate that for 36 lines that appear unblended in the Sun, Fe abundance measurements yield a small line-by-line scatter (0.08 dex) with a mean abundance of 7.44 dex in good agreement with recent publications.Comment: Accepted for publication in Mon. Not. R. Astron. So

How Unsplittable-Flow-Covering helps Scheduling with Job-Dependent Cost Functions

Generalizing many well-known and natural scheduling problems, scheduling with job-specific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4-approximation algorithm. This is true even for a special case that is equivalent to the covering version of the well-studied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasi-polynomial time $(1+\epsilon)$-approximation algorithm that yields an $(e+\epsilon)$-approximation for the above scheduling problem. Moreover, for the latter we devise the best possible resource augmentation result regarding speed: a polynomial time algorithm which computes a solution with \emph{optimal }cost at $1+\epsilon$ speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most $\log n$ many classes. This algorithm allows the jobs even to have up to $\log n$ many distinct release dates.Comment: 2 pages, 1 figur

Slow movement of a random walk on the range of a random walk in the presence of an external field

In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions (d \geq 5). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai's regime