arXiv subjects

Markus Heydenreich

Publications and source records attributed to Markus Heydenreich.

39 records · Page 3Linked to original sources

arXiv2008

Functionals of Brownian bridges arising in the current mismatch in D/A-converters

Digital-to-analog converters (DAC) transform signals from the abstract digital domain to the real analog world. In many applications, DAC's play a crucial role. Due to variability in the production, various errors arise that influence the performance of the DAC. We focus on the current errors, which describe the fluctuations in the currents of the various unit current elements in the DAC. A key performance measure of the DAC is the Integrated Non-linearity (INL), which we study in this paper. There are several DAC architectures. The most widely used architectures are the thermometer, the binary and the segmented architectures. We study the two extreme architectures, namely, the thermometer and the binary architectures. We assume that the current errors are i.i.d. normally distributed, and reformulate the INL as a functional of a Brownian bridge. We then proceed by investigating these functionals. For the thermometer case, the functional is the maximal absolute value of the Brownian bridge, which has been investigated in the literature. For the binary case, we investigate properties of the functional, such as its mean, variance and density.

math.PR↗

arXiv2008

Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(α\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(α\wedge2)$ for percolation, where $d$ denotes the dimension and $α$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)

math.PR↗

arXiv2006

Random graph asymptotics on high-dimensional tori

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d>6 for sufficient spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times $V^{2/3}$ and below by a small constant times $V^{2/3}(log V)^{-4/3}$, where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on Z^d under which the lower bound can be improved to small constant times $V^{2/3}$, i.e., we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by Aizenman (1997), apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation. Our method is crucially based on the results by Borgs, Chayes, van der Hofstad, Slade and Spencer (2005a, 2005b), where the $V^{2/3}$ scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on Z^d. We also strongly rely on mean-field results for percolation on Z^d proved by Hara (1990, 2005), Hara and Slade (1990) and Hara, van der Hofstad and Slade (2003).

math.PR↗