No coincidence of center percolation and deconfinement in SU(4) lattice gauge theory

We study the behavior of center sectors in pure SU(4) lattice gauge theory at finite temperature. The center sectors are defined as spatial clusters of neighboring sites with values of their local Polyakov loops near the same center elements. We study the connectedness and percolation properties of the center clusters across the deconfinement transition. We show that for SU(4) gauge theory deconfinement cannot be described as a percolation transition of center clusters, a finding which is different from pure SU(2) or pure SU(3) Yang Mills theory, where the percolation description even allows for a continuum limit. Introduction and outline Understanding the high temperature transition of QCD to a phase of deconfined quarks and gluons is still an open problem. Finding a suitable description of the transition and the plasma phase is essential for understanding current and upcoming results from heavy ion experiments. An interesting approach is the idea of describing the deconfinement transition as a percolation phenomenon and to explore its phenomenological consequences. For the simpler case of pure gauge theory the idea that percolation of clusters related to the center of the gauge group can be used to describe deconfinement goes back to [1] for the case of SU(2) and to [2] for SU(N) with N > 2. The clusters were constructed from the local Polyakov loops, i.e., they reflect the properties of static quark sources under the transformation with the center group ZN of SU(N). More recently it was argued that for the cases of SU(2) and SU(3) even a continuum limit of the percolation description is possible for suitably defined clusters. First results for full lattice QCD were presented in [4]. With the encouraging results available for SU(2) and SU(3) one may ask the question whether the percolation description of deconfinement based on the center group is suitable for all gauge groups SU(N). This is not a priori clear: The number of center elements is N , such that if the center sectors are occupied uniformly the probability that a site belongs to a particular sector is 1/N . For N > 3, this probability 1/N is below the critical occupation probability pc ∼ 0.316 of random percolation on a three-dimensional simple cubic lattice. This implies that if the percolation picture of deconfinement were to apply also for SU(N) with N > 3, the deconfinement transition must be accompanied by the onset of very strong correlations between the center phases of neighboring sites. The main goal of the current letter (see also [5]) is to establish or disproof the existence of such strong correlations and the corresponding percolation picture for SU(N) Yang-Mills theory at N > 3. It should be kept in mind that rigorous results for a complete description of thermal transitions, i.e., the same transition temperatures of the percolation and the thermal transition and matching critical exponents, are only available for continuous transitions. For first order transitions only numerical simulations or results for simple models suggest that percolation may be used to effectively describe thermal transitions in some cases (see, e.g., [6, 7]). The current paper provides a counter example.