High-Precision Entropy Values for Spanning Trees in Lattices

Shrock and Wu have given numerical values for the exponential growth rate of the number of spanning trees in Euclidean lattices. We give a new technique for numerical evaluation that gives much more precise values, together with rigorous bounds on the accuracy. In particular, the new values resolve one of their questions.Comment: 7 pages. Revision mentions alternative approach. Title changed slightly. 2nd revision corrects first displayed equatio

Critical percolation of free product of groups

In this article we study percolation on the Cayley graph of a free product of groups. The critical probability $p_c$ of a free product $G_1*G_2*...*G_n$ of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups $G_1,G_2,...,G_n$. For finite groups these equations are polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that $p_c$ for the Cayley graph of the modular group $\hbox{PSL}_2(\mathbb Z)$ (with the standard generators) is $.5199...$, the unique root of the polynomial $2p^5-6p^4+2p^3+4p^2-1$ in the interval $(0,1)$. In the case when groups $G_i$ can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of $G_1*G_2*...*G_n$ and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite. We show that the critical point, introduced by Schonmann, $p_{\mathrm{exp}}$ of the free product is just the minimum of $p_{\mathrm{exp}}$ for the factors

Is the Sun Lighter than the Earth? Isotopic CO in the Photosphere, Viewed through the Lens of 3D Spectrum Synthesis

We consider the formation of solar infrared (2-6 micron) rovibrational bands of carbon monoxide (CO) in CO5BOLD 3D convection models, with the aim to refine abundances of the heavy isotopes of carbon (13C) and oxygen (18O,17O), to compare with direct capture measurements of solar wind light ions by the Genesis Discovery Mission. We find that previous, mainly 1D, analyses were systematically biased toward lower isotopic ratios (e.g., R23= 12C/13C), suggesting an isotopically "heavy" Sun contrary to accepted fractionation processes thought to have operated in the primitive solar nebula. The new 3D ratios for 13C and 18O are: R23= 91.4 +/- 1.3 (Rsun= 89.2); and R68= 511 +/- 10 (Rsun= 499), where the uncertainties are 1 sigma and "optimistic." We also obtained R67= 2738 +/- 118 (Rsun= 2632), but we caution that the observed 12C17O features are extremely weak. The new solar ratios for the oxygen isotopes fall between the terrestrial values and those reported by Genesis (R68= 530, R6= 2798), although including both within 2 sigma error flags, and go in the direction favoring recent theories for the oxygen isotope composition of Ca-Al inclusions (CAI) in primitive meteorites. While not a major focus of this work, we derive an oxygen abundance of 603 +/- 9 ppm (relative to hydrogen; 8.78 on the logarithmic H= 12 scale). That the Sun likely is lighter than the Earth, isotopically speaking, removes the necessity to invoke exotic fractionation processes during the early construction of the inner solar system

Microwave properties of DyBa_2Cu_3O_(7-x) monodomains and related compounds in magnetic fields

We present a microwave characterization of a DyBa$_{2}$Cu$_{3}$O$_{7-x}$ single domain, grown by the top-seeded melt-textured technique. We report the (a,b) plane field-induced surface resistance, $\Delta R_s(H)$, at 48.3 GHz, measured by means of a cylindrical metal cavity in the end-wall-replacement configuration. Changes in the cavity quality factor Q against the applied magnetic field yield $\Delta R_s(H)$ at fixed temperatures. The temperature range [70 K ; T_c] was explored. The magnetic field $\mu_0 H <$ 0.8 T was applied along the c axis. The field dependence of $\Delta R_s(H)$ does not exhibit the steep, step-like increase at low fields typical of weak-links. This result indicates the single-domain character of the sample under investigation. $\Delta R_s(H)$ exhibits a nearly square-root dependence on H, as expected for fluxon motion. From the analysis of the data in terms of motion of Abrikosov vortices we estimate the temperature dependences of the London penetration depth $\lambda$ and the vortex viscosity $\eta$, and their zero-temperature values $\lambda(0)=$165 nm and $\eta(0)=$ 3 10$^{-7}$ Nsm$^{-2}$, which are found in excellent agreement with reported data in YBa$_{2}$Cu$_{3}$O$_{7-x}$ single crystals. Comparison of microwave properties with those of related samples indicate the need for reporting data as a function of T/T_c in order to obtain universal laws.Comment: 6 pages, 4 figures, LaTeX, submitted to Journal of Applied Physic

G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by GBrownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of introducing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this paper is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest

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