# Cildo Meireles and the Poisson-Clock transition

Cildo Meireles is a well-known Brazilian artist, associated with the art movement “Neo-concrete art”. He was recently given a full retrospective by the Tate Modern Museum in London, which I had the opportunity to see in December 2008.

Meireles’ intriguing and aesthetically seductive installations and sculptures were displayed in seven rooms [link]. I enjoyed most of the works, but one of them made a strong impression on me: Fontes 1992/2008 (Room 5). This work consists of a room with 6,000 rulers, 1,000 clocks and 500,000 vinyl numbers and “demonstrates the aesthetic accumulation, […] a feature of many of Meireles installations”. I was very intrigued by Meireles’s clocks and especially by their ticks:

With regard to this work, the Tate brochure explains that “the rulers and the clocks undermine their very purpose: the order of their numbers and the spacing of their measurements are illogical, so the ability of these almost-Readymade objects to measure either time or space is subverted”. However, Meireles’ clocks didn’t strike me as illogical, since in my research I often encounter mathematical plots with surprisingly similar features:

Eigenvalue Plots – pdf file

I will try to explain what these plots represent. In the joint paper with Rowan Killip “Eigenvalue Statistics for CMV Matrices: From Poisson to Clock via Random Matrix Ensembles” [link], we investigate the eigenvalue statistics for random CMV matrices. These matrices are unitary, hence their eigenvalues are random points on the circle of radius one and centered at the origin. We can also think of these eigenvalues as interacting particles lying on the unit circle. Depending on the type of randomness considered, we get three situations:

1. Plots where the points (particles) are randomly distributed on the unit circle, with no correlations between them. A typical eigenvalue plot will exhibit clumps and gaps, a situation described mathematically by the Poisson point process. Physically, this corresponds to a system with infinite temperature and no correlation between the particles.

2. Plots where the points (particles) are randomly distributed on the circle, but weakly repel each other. This situation described mathematically by the Circular Beta Ensemble ($C \beta E$), where $\beta$ is a positive parameter). Physically, this corresponds to a system with temperature $1/\beta$, where there is weak repulsion between the particles.

3. Plots where the points (particles) are randomly distributed on the circle and repel each other strongly. If you imagine many particles on the unit circle which do their best to stay away from each other, the best they can do is to distribute themselves like the ticks on a clock. This situation described mathematically by the Clock (or Picket Fence) point process. Physically, this corresponds to a system with zero temperature, where there is strong repulsion between the particles.

The number $\beta$, which appears in Case 2, is known as the “inverse temperature” coefficient. If we take $\beta \to 0$, we get Case 1 (Poisson, infinite temperature); if we take $\beta \to \infty$, we get Case 3 (Clock, zero temperature). Hence the transition from Poisson to Clock eigenvalue statistics passes through a continuum of intermediate processes.

I was stunned by the similarity between Meireles’ clocks and our eigenvalue plots for random CMV matrices. A question arises naturally: are we witnessing a pure coincidence or Meireles’ work and our paper on eigenvalue statistics are two snapshots of a bigger concept?

Note: Cildo Meireles’ exhibition will have its North American premiere at the Museum of Fine Arts, Houston in June 2009 [link].