Finite Temperature Lattice QCD in the Large N Limit

Our aim is to give a self-contained review of recent advances in the analytic description of the deconfinement transition and determination of the deconfinement temperature in lattice QCD at large N. We also include some new results, as for instance in the comparison of the analytic results with Montecarlo simulations. We first review the general set-up of finite temperature lattice gauge theories, using asymmetric lattices, and develop a consistent perturbative expansion in the coupling $\beta_s$ of the space-like plaquettes. We study in detail the effective models for the Polyakov loop obtained, in the zeroth order approximation in $\beta_s$, both from the Wilson action (symmetric lattice) and from the heat kernel action (completely asymmetric lattice). The distinctive feature of the heat kernel model is its relation with two-dimensional QCD on a cylinder; the Wilson model, on the other hand, can be exactly reduced to a twisted one-plaquette model via a procedure of the Eguchi-Kawai type. In the weak coupling regime both models can be related to exactly solvable Kazakov-Migdal matrix models. The instability of the weak coupling solution is due in both cases to a condensation of instantons; in the heat kernel case, it is directly related to the Douglas-Kazakov transition of QCD2. A detailed analysis of these results provides rather accurate predictions of the deconfinement temperature. In spite of the zeroth order approximation they are in good agreement with the Montecarlo simulations in 2+1 dimensions, while in 3+1 dimensions they only agree with the Montecarlo results away from the continuum limit.Comment: 66 pages, plain Latex, figures included by eps

Two dimensional QCD is a one dimensional Kazakov-Migdal model

We calculate the partition functions of QCD in two dimensions on a cylinder and on a torus in the gauge $\partial_{0} A_{0} = 0$ by integrating explicitly over the non zero modes of the Fourier expansion in the periodic time variable. The result is a one dimensional Kazakov-Migdal matrix model with eigenvalues on a circle rather than on a line. We prove that our result coincides with the standard expansion in representations of the gauge group. This involves a non trivial modular transformation from an expansion in exponentials of $g^2$ to one in exponentials of $1/g^2$. Finally we argue that the states of the $U(N)$ or $SU(N)$ partition function can be interpreted as a gas of N free fermions, and the grand canonical partition function of such ensemble is given explicitly as an infinite product.Comment: DFTT 15/93, 17 pages, Latex (Besides minor changes and comments added we note that for U(N) odd and even N have to be treated separately

Effective actions for finite temperature Lattice Gauge Theories

We consider a lattice gauge theory at finite temperature in ($d$+1) dimensions with the Wilson action and different couplings $\beta_t$ and $\beta_s$ for timelike and spacelike plaquettes. By using the character expansion and Schwinger-Dyson type equations we construct, order by order in $\beta_s$, an effective action for the Polyakov loops which is exact to all orders in $\beta_t$. As an example we construct the first non-trivial order in $\beta_s$ for the (3+1) dimensional SU(2) model and use this effective action to extract the deconfinement temperature of the model.Comment: Talk presented at LATTICE96(finite temperature

A theoretical model of neuronal population coding of stimuli with both continuous and discrete dimensions

In a recent study the initial rise of the mutual information between the firing rates of N neurons and a set of p discrete stimuli has been analytically evaluated, under the assumption that neurons fire independently of one another to each stimulus and that each conditional distribution of firing rates is gaussian. Yet real stimuli or behavioural correlates are high-dimensional, with both discrete and continuously varying features.Moreover, the gaussian approximation implies negative firing rates, which is biologically implausible. Here, we generalize the analysis to the case where the stimulus or behavioural correlate has both a discrete and a continuous dimension. In the case of large noise we evaluate the mutual information up to the quadratic approximation as a function of population size. Then we consider a more realistic distribution of firing rates, truncated at zero, and we prove that the resulting correction, with respect to the gaussian firing rates, can be expressed simply as a renormalization of the noise parameter. Finally, we demonstrate the effect of averaging the distribution across the discrete dimension, evaluating the mutual information only with respect to the continuously varying correlate.Comment: 20 pages, 10 figure

Uncertainty Quantification in Scale-Dependent Models of Flow in Porous Media

Equations governing flow and transport in randomly heterogeneous porous media are stochastic and scale dependent. In the moment equation (ME) method, exact deterministic equations for the leading moments of state variables are obtained at the same support scale as the governing equations. Computable approximations of the MEs can be derived via perturbation expansion in orders of the standard deviation of the random model parameters. As such, their convergence is guaranteed only for standard deviation smaller than one. Here, we consider steady-state saturated flow in a porous medium with random second-order stationary conductivity field. We show it is possible to identify a support scale, $\eta*$, where the typically employed approximate formulations of MEs yield accurate (statistical) moments of a target state variable. Therefore, at support scale $\eta*$ and larger, MEs present an attractive alternative to slowly convergent Monte Carlo (MC) methods whenever lead-order statistical moments of a target state variable are needed. We also demonstrate that a surrogate model for statistical moments can be constructed from MC simulations at larger support scales and be used to accurately estimate moments at smaller scales, where MC simulations are expensive and the ME method is not applicable

Synthesizing realistic neural population activity patterns using generative adversarial networks

The ability to synthesize realistic patterns of neural activity is crucial for studying neural information processing. Here we used the Generative Adversarial Networks (GANs) framework to simulate the concerted activity of a population of neurons. We adapted the Wasserstein-GAN variant to facilitate the generation of unconstrained neural population activity patterns while still benefiting from parameter sharing in the temporal domain. We demonstrate that our proposed GAN, which we termed Spike-GAN, generates spike trains that match accurately the first- and second-order statistics of datasets of tens of neurons and also approximates well their higher-order statistics. We applied Spike-GAN to a real dataset recorded from salamander retina and showed that it performs as well as state-of-the-art approaches based on the maximum entropy and the dichotomized Gaussian frameworks. Importantly, Spike-GAN does not require to specify a priori the statistics to be matched by the model, and so constitutes a more flexible method than these alternative approaches. Finally, we show how to exploit a trained Spike-GAN to construct’importance maps’ to detect the most relevant statistical structures present in a spike train. Spike-GAN provides a powerful, easy-to-use technique for generating realistic spiking neural activity and for describing the most relevant features of the large-scale neural population recordings studied in modern systems neuroscience

Deconfinement Transition in Large N Lattice Gauge Theory

We study analytically the phase diagram of the pure $SU(N)$ lattice gauge theory at finite temperature, and we attempt to estimate the critical deconfinement temperature. We apply large $N$ techniques to the Wilson and to the Heat Kernel action, and we study the resulting models both in the strong coupling and in the weak coupling limits. Using the Heat Kernel action, we establish an interesting connection between the Douglas-Kazakov phase transition of two-dimensional QCD and the deconfining transition in $d$ dimensions. The analytic results obtained for the critical temperature compare well with Montecarlo simulations of the full theory in $(2+1)$ and in $(3+1)$ dimensions.Comment: 39 pages (Latex) + 4 ps-figures (using EPSF), DFTT 30/9