# The 35th International Symposium on Lattice Field Theory

18-24 June 2017
Palacio de Congresos
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# Contribution Parallel

Auditorio Manuel de Falla
Nonzero Temperature and Density

# Representations of complex probabilities on groups, Gibbs sampling, and local reweighting

## Speakers

• Prof. Lorenzo Luis SALCEDO

## Content

Lattice QCD, involving a large number of degrees of freedom, relies on the applicability of Monte Carlo methods. The presence of complex weights in the study of QCD at finite baryonic density, introduces the sign (or phase problem) in this context and is a mayor impediment for faster progress in this field. In addition to standard reweighting, several approaches have been tried to sort this problem, each of them with its own virtues and limitations, these include the complex Langevin algorithm and the Lefschetz thimbles approach, and variations of thereof.

Typically, the configuration space ${x}$ is extended into a complex manifold ${z}$ and observables are sampled through their analytical continuation, $A(x)\to A(z)$. In practice this means that a positive definite distribution $\rho(z)$ is constructed and sampled on the complex manifold in such a way that it reproduces the correct expectation values of the original complex weight $P(x)$ defined on the real configuration manifold, i.e., $\langle A(x)\rangle_P = \langle A(z)\rangle_\rho$. In other words, $\rho$ is a representation of $P$ [J.Math.Phys.38(1997)1710].

We will describe an approach based on a complex version of the heat bath method [Phys.Rev.D94(2016)074503]. The point is that, although the representation problem has been formally solved for weights defined on $R^n$ and arbitrary compact groups [J.Phys.A40(2007)9399], it is hard to actually construct direct representations of a given $P(x)$ in the many dimensional case. The idea of the Gibbs sampling is to sequentially update each degree of freedom by using a representation of its conditional probability. This only requires to construct a one-dimensional representation at each step, what is easily amenable, and has been done for Abelian groups and for non compact variables. We will discuss how the method extends to the non-Abelian case, in particular SU(n). Specific issues appear in the representation of these more general groups, both for one-dimensional and for higher dimensional complex weights, and we show how they are fixed.

Finally, we introduce the concept of local projection as dual of that of Gibbs sampling and show how this method can be used to enormously enlarge the range of applicability of the reweighting technique. Such an approach would allow to accommodate an (approximate) representation $\rho(z)$ to the given $P(x)$ in order to reduce the fluctuations introduced by standard reweighting. Its viability and efficiency is currently under study.

## Preferred track (if multiple tracks have been selected)

Nonzero Temperature and Density