Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where p(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.Comment: 6 pages, 4 figures, revtex

Professor C. N. Yang and Statistical Mechanics

Professor Chen Ning Yang has made seminal and influential contributions in many different areas in theoretical physics. This talk focuses on his contributions in statistical mechanics, a field in which Professor Yang has held a continual interest for over sixty years. His Master's thesis was on a theory of binary alloys with multi-site interactions, some 30 years before others studied the problem. Likewise, his other works opened the door and led to subsequent developments in many areas of modern day statistical mechanics and mathematical physics. He made seminal contributions in a wide array of topics, ranging from the fundamental theory of phase transitions, the Ising model, Heisenberg spin chains, lattice models, and the Yang-Baxter equation, to the emergence of Yangian in quantum groups. These topics and their ramifications will be discussed in this talk.Comment: Talk given at Symposium in honor of Professor C. N. Yang's 85th birthday, Nanyang Technological University, Singapore, November 200

Random walk generated by random permutations of {1,2,3, ..., n+1}

We study properties of a non-Markovian random walk $X^{(n)}_l$, $l =0,1,2, >...,n$, evolving in discrete time $l$ on a one-dimensional lattice of integers, whose moves to the right or to the left are prescribed by the \text{rise-and-descent} sequences characterizing random permutations $\pi$ of $[n+1] = \{1,2,3, ...,n+1\}$. We determine exactly the probability of finding the end-point $X_n = X^{(n)}_n$ of the trajectory of such a permutation-generated random walk (PGRW) at site $X$, and show that in the limit $n \to \infty$ it converges to a normal distribution with a smaller, compared to the conventional P\'olya random walk, diffusion coefficient. We formulate, as well, an auxiliary stochastic process whose distribution is identic to the distribution of the intermediate points $X^{(n)}_l$, $l < n$, which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of "turns" of the PGRW trajectories.Comment: text shortened, new results added, appearing in J. Phys.

A Search for Extraterrestrial Intelligence (SETI) toward the Galactic Anticenter with the Murchison Widefield Array

Following the results of the first systematic modern low-frequency search for extraterrestrial intelligence using the Murchison Widefield Array (MWA), which was directed toward a Galactic Center field, we report a second survey toward a Galactic Anticenter field. Using the MWA in the frequency range 99-122 MHz over a three-hour period, a 625 deg 2 field centered on Orion KL (in the general direction of the Galactic Anticenter) was observed with a frequency resolution of 10 kHz. Within this field, 22 exoplanets are known. At the positions of these exoplanets, we searched for narrowband signals consistent with radio transmissions from intelligent civilizations. No such signals were found with a 5σ detection threshold. Our sample is significantly different to the 45 exoplanets previously studied with the MWA toward the Galactic Center, since the Galactic Center sample is dominated by exoplanets detected using microlensing, and hence at much larger distances than the exoplanets toward the Anticenter, found via radial velocity and transit detection methods. Our average effective sensitivity to extraterrestrial transmitter power is therefore much improved for the Anticenter sample. Added to this, our data processing techniques have improved, reducing our observational errors, leading to our best detection limit being reduced by approximately a factor of four compared to our previously published results

Breakthrough Listen follow-up of the reported transient signal observed at the Arecibo Telescope in the direction of Ross 128

We undertook observations with the Green Bank Telescope, simultaneously with the 300 m telescope in Arecibo, as a follow-up of a possible flare of radio emission from Ross 128. We report here the non-detections from the GBT observations in C band (4–8 GHz), as well as non-detections in archival data at L band (1.1–1.9 GHz). We suggest that a likely scenario is that the emission comes from one or more satellites passing through the same region of the sky

Exact expressions for correlations in the ground state of the dense O(1) loop model

Conjectures for analytical expressions for correlations in the dense O$(1)$ loop model on semi infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries ''open''. We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to Fully Packed Loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O$(1)$ loop model and the XXZ spin chain for the different types of boundary conditions.Comment: 25 pages, version accepted by JSTA