A Simple Gamma Random Number Generator for Arbitrary Shape Parameters

This paper proposes an improved gamma random generator. In the past, a lot of gamma random number generators have been proposed, and depending on a shape parameter (say, alpha) they are roughly classified into two cases: (i) alpha lies on the interval (0,1) and (ii) alpha is greater than 1, where alpha=1 can be included in either case. In addition, Cheng and Feast (1980) extended the gamma random number generator in the case where alpha is greater than 1/n, where n denotes an arbitrary positive number. Taking n as a decreasing function of alpha, in this paper we propose a simple gamma random number generator with shape parameter alpha greater than zero. The proposed algorithm is very simple and shows quite good performance.Gamma Random Variable

Structure of Lefschetz thimbles in simple fermionic systems

The Picard-Lefschetz theory offers a promising tool to solve the sign problem in QCD and other field theories with complex path-integral weight. In this paper the Lefschetz-thimble approach is examined in simple fermionic models which share some features with QCD. In zero-dimensional versions of the Gross-Neveu model and the Nambu-Jona-Lasinio model, we study the structure of Lefschetz thimbles and its variation across the chiral phase transition. We map out a phase diagram in the complex four-fermion coupling plane using a thimble decomposition of the path integral, and demonstrate an interesting link between anti-Stokes lines and Lee-Yang zeros. In the case of nonzero mass, it is shown that the approach to the chiral limit is singular because of intricate cancellation between competing thimbles, which implies the necessity to sum up multiple thimbles related by symmetry. We also consider a Chern-Simons theory with fermions in $0+1$-dimension and show how Lefschetz thimbles solve the complex phase problem caused by a topological term. These prototypical examples would aid future application of this framework to bona fide QCD.Comment: 37 pages, 17 figures. v2: minor changes, the version to appear in JHE

Vacuum structure of bifundamental gauge theories at finite topological angles

We discuss possible vacuum structures of $SU(n)\times SU(n)$ gauge theories with bifundamental matters at finite $\theta$ angles. In order to give a precise constraint, a mixed 't Hooft anomaly is studied in detail by gauging the center $\mathbb{Z}_n$ one-form symmetry of the bifundamental gauge theory. We propose phase diagrams that are consistent with the constraints, and also give a heuristic explanation of the result based on the dual superconductor scenario of confinement.Comment: 28 pages, 6 figures; (v2) references adde