## Loop models, random matrices and planar algebras.(English)Zbl 1277.82013

A Temperley-Lieb element is a rectangle with an even number of points on the boundary and non-intersecting strings connecting the points pairwise. Each region, separated by nonintersecting strings, is colored black or white, and one point on the boundary is marked. Adjacent regions have different colors.
Let $$S_1,S_2,\dotsc, S_k$$ and $$S$$ be fixed Temperley-Lieb elements, and let $$t_1,t_2,\dotsc, t_k$$ be real numbers. Consider the loop model ${\mathcal J}_t(S)= \sum_{r_i\geq 0}\prod^k_{i=1} {t^{r_i}_i\over r_i!}\,\sum \delta^{\#\mathrm{loops}},$ where the sum is over all tangles which are non-intersecting strings on $S^2- \Biggl(\bigcup^{r_1}_{i=1} S_1\cup\dotsb\cup \bigcup^{r_k}_{i=1} S_k\cup S\Biggr),$ and the non-intersecting strings connect the boundary points on $$r_1$$ Temperley-Lieb elements of the kind $$S_1,\dotsc, r_ k$$ Temperley-Lieb elements of the kind $$S_k$$ and the Temperley-Lieb element $$S$$. ($$S^2$$ is the standard two-dimensional sphere.) The non-intersecting strings connect the boundary points so that the black and white colors are compatible, i.e., the black and white regions match. Then, the authors prove that the loop model $${\mathcal J}_t(S)$$ is equal to a matrix model provided that $\delta\in \Biggl\{2\cos\Biggl({\pi\over p}\Biggr)\Biggr\}_{p\geq 3}\cup [2,+\infty)$ and the numbers $$t_i$$ are small. The matrix model is the limit of the integral over a Cartesian product of rectangular matrices as the size of the matrices go to infinity. The integrand is explicitly written down as perturbations of independent Gaussian measures. The subtlety of this construction can be seen in that $$\delta$$ is an eigenvalue of the adjacency matrix of a certain bipartite graph. The corresponding eigenvector is used in the construction of this matrix model.
Consider the special case in which the Temperley-Lieb elements $$S_1$$ and $$S_2$$ are given as follows. $$S_1$$ has two non-intersecting strings connecting four boundary points. $$S_1$$ has one white region, and two black regions. $$S_2$$ has two non-intersecting strings connecting four boundary points. $$S_2$$ has one black region, and two white regions. Also, consider the Temperley-Lieb element $$B_n$$ given as follows. $$B_n$$ has $$n$$ non-nested, non-intersecting strings connecting $$2n$$ boundary points. There are $$n$$ black regions, and one white region. Let ${\mathcal J}_t(B_n)= \sum^\infty_{r_1,r_2= 0} {t^{r_1}_1 t^{r_2}_2\over r_1!r_2!} \sum \delta^{\#\text{loops}},$ where the sum is over all tangles which are non-intersecting strings on $S^2-\Biggl(\bigcup^{r_1}_{i=1} S_1\cup \bigcup^{r_2}_{i=1} S_2\cup B_n\Biggr),$ and the non-intersecting strings connect the boundary points on $$r_1$$ Temperley-Lieb elements $$S_1$$, $$r_2$$ Temperley-Lieb elements $$S_2$$ and one Temperley-Lieb element $$B_n$$. The non-intersecting strings connect the points on the boundary so that the black and white colors are compatible, i.e., the black and white regions match. They explicitly evaluate the matrix model corresponding to this loop model $${\mathcal J}_t(B_n)$$ using the Hubbard-Stratonovich transformation and the saddle-point technique in the large matrices limit. Consequently, the infinite series $C(\gamma,t)= \sum^\infty_{n=0} \gamma^n{\mathcal J}_t(B_n),$ $$\gamma\in\mathbb{C}$$, is shown to be analytic in $$\gamma$$ for small $$\gamma$$. Moreover, they provide an explicit description of this function.

### MSC:

 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 15B52 Random matrices (algebraic aspects) 60B20 Random matrices (probabilistic aspects) 46L53 Noncommutative probability and statistics
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