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A Concentration of Measure Phenomenon in the Principal Chiral Model

Tamer Tlas

Abstract

We utilize the concentration of measure phenomenon to study the large $N$ limit of the $O(N)$ principal chiral model. The partition function in this limit is demonstrated to be that of a free massive theory.

A Concentration of Measure Phenomenon in the Principal Chiral Model

Abstract

We utilize the concentration of measure phenomenon to study the large limit of the principal chiral model. The partition function in this limit is demonstrated to be that of a free massive theory.

Paper Structure

This paper contains 6 sections, 1 theorem, 23 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Preliminary Analysis
  4. Concentration of Measure
  5. The Mean and the Variance
  6. Large $N$ Asymptotics

Key Result

Lemma 1

The function is Lipschitz with respect to the metric $d(O, O') = \int || O - O'||_{HS}$ where $|| \cdot ||_{HS}$ stands for the Hilbert-Schmidt norm.

Figures (6)

  • Figure 1: Summary of the graphical notation used below. Note that we've only chosen a particular representation of the Kronecker's delta. We could have drawn any other line. Also note that the last diagram is indeed equal to the displayed mathematical expression since the transpose interchanges the indices.
  • Figure 2: The graphical representation of the cases $k =1$ and $k=2$ of the asymptotic integration formula above. Note that in the $k=1$ case, the dominant asymptotic is the exact answer.
  • Figure 3: The dashed lines represent indices which are contracted with other terms. The triangle stands for the matrix $\hat{M}_{aa'}$.
  • Figure 4: The result of performing the $O$ integral in the case of non-coincident points. The bar denotes averaging with respect to the $\rho$ measure.
  • Figure 5: The result of doing the $O$ integrals when exactly two $x$'s are coincident. As above, the bar denotes averaging with respect to the $\rho$ measure. The starting expression is that obtained after integrating over the $O$'s at non-coincident points. The first and second equalities above follow from the fact that $O^t O = I$ and that the $dO$ is a probability measure.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma
  • proof