Random Unitary Representations of Surface Groups II: The large $n$ limit

Magee, Michael

doi:10.48550/arxiv.2101.03224

Cited by 3 publications

(8 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Besides, the recent results of [41] are furthermore coherent with the above statement as discussed in the next sub-section.…”

Section: Yang-mills Measure and Master Field Statement Of Resultssupporting

confidence: 84%

“…Another measure on connections is due to [4,23] when g ≥ 2. Recently, the limit of Wilson loops under this measure has been investigated by [40,41], we discuss the relation with our result.…”

Section: Atiyah-bott-goldman Measurementioning

confidence: 53%

“…In higher dimension, an analog 3 of this question for a lattice model has also been considered [10]. Very recently and independently from the current work, it was shown in [41,40] that under the Atiyah-Bott-Goldman measure, which can be understood as the weak limit of the Yang-Mills measure when the area of the surface vanishes, the expectation of Wilson loops converges and has a 1 N expansion when the group belongs to the series of special unitary matrices and the surface is closed, orientable and of genus g ≥ 2. For further details and references on the motivations of this problem, we refer to [16,Sec.…”

Section: Introductionmentioning

confidence: 81%

“…Thanks to the defining relation of M g , for any word w, the function W w depends only on the equivalence class of w. When γ ∈ Γ g is the evaluation of w in Γ g , denote this function by W γ . In [41], Magee obtained the following analog of asymptotic freeness of Haar unitary random matrices.…”

Section: Atiyah-bott-goldman Measurementioning

confidence: 99%

“…The paper is organised as follows. The first four following sections of the introduction give respectively an informal definition of the Yang-Mills measure and of the main results, a discussion on the relation with the Atiyah-Bott-Goldman measure and the work [40,41], a consequence of the result on the torus in non-commutative probability, and lastly, a sketch of the strategy of the main proofs. Section 2 recalls and adapts some combinatorial notions of discrete homotopy of loops in embedded graph instrumental to the proof.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Large N limit of the Yang-Mills measure on compact surfaces II: Makeenko-Migdal equations and planar master field

Dahlqvist¹,

Lemoine²

2022

Preprint

View full text Add to dashboard Cite

This paper considers the large N limit of Wilson loops for the two-dimensional Euclidean Yang-Mills measure on all orientable compact surfaces of genus larger or equal to 1, with a structure group given by a classical compact matrix Lie group. Our main theorem shows the convergence of all Wilson loops in probability, given that it holds true on a restricted class of loops, obtained as a modification of geodesic paths. Combined with the result of [16], a corollary is the convergence of all Wilson loops on the torus. Unlike the sphere case, we show that the limiting object is remarkably expressed thanks to the master field on the plane defined in [3,32] and we conjecture that this phenomenon is also valid for all surfaces of higher genus. We prove that this conjecture holds true whenever it does for the restricted class of loops of the main theorem. Our result on the torus justifies the introduction of an interpolation between free and classical convolution of probability measures, defined with the free unitary Brownian motion but differing from t-freeness of [5] that was defined in terms the liberation process of Voiculescu [55]. In contrast to [16], our main tool is a fine use of Makeenko-Migdal equations, proving their uniqueness under suitable assumptions, generalising the arguments of [17,29].

show abstract

“…Besides, the recent results of [41] are furthermore coherent with the above statement as discussed in the next sub-section.…”

Section: Yang-mills Measure and Master Field Statement Of Resultssupporting

confidence: 84%

Section: Atiyah-bott-goldman Measurementioning

confidence: 53%

Section: Introductionmentioning

confidence: 81%

Section: Atiyah-bott-goldman Measurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Large N limit of the Yang-Mills measure on compact surfaces II: Makeenko-Migdal equations and planar master field

Dahlqvist¹,

Lemoine²

2022

Preprint

View full text Add to dashboard Cite

show abstract

The Asymptotic Statistics of Random Covering Surfaces

Magee

Puder

2023

Forum of Mathematics, Pi

View full text Add to dashboard Cite

Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq 2$ . We develop a new method for integrating over the representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ , where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots ,n\}$ . Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g. Given $\phi \in \mathbb {X}_{g,n}$ and $\gamma \in \Gamma _{g}$ , we let $\mathsf {fix}_{\gamma }(\phi )$ be the number of fixed points of the permutation $\phi (\gamma )$ . The function $\mathsf {fix}_{\gamma }$ is a special case of a natural family of functions on $\mathbb {X}_{g,n}$ called Wilson loops. Our new methodology leads to an asymptotic formula, as $n\to \infty $ , for the expectation of $\mathsf {fix}_{\gamma }$ with respect to the uniform probability measure on $\mathbb {X}_{g,n}$ , which is denoted by $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ . We prove that if $\gamma \in \Gamma _{g}$ is not the identity and q is maximal such that $\gamma $ is a qth power in $\Gamma _{g}$ , then $$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{\gamma}\right]=d(q)+O(n^{-1}) \end{align*}$$ as $n\to \infty $ , where $d\left (q\right )$ is the number of divisors of q. Even the weaker corollary that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]=o(n)$ as $n\to \infty $ is a new result of this paper. We also prove that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$ .

show abstract

Local Statistics of Random Permutations from Free Products

Puder¹,

Zimhoni²

2022

Preprint

View full text Add to dashboard Cite

Let Γ = G 1 * . . . * G k be a free product of groups where each of G 1 , . . . , G k is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element γ ∈ Γ, a γ-random permutation in the symmetric group S N is the image of γ through a uniformly random homomorphism Γ → S N . In this paper we study local statistics of γ-random permutations and their asymptotics as N grows. We first consider E [fix γ (N )], the expected number of fixed points in a γ-random permutation in S N . We show that unless γ has finite order, the limit of E [fix γ (N )] as N → ∞ is an integer, and is equal to the number of subgroups H ≤ Γ containing γ such that H ∼ = Z or H ∼ = C 2 * C 2 . Equivalently, this is the number of subgroups H ≤ Γ containing γ and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for E [fix γ (N )] and determine the limit distribution of the number of fixed points as N → ∞. These results are then generalized to all statistics of cycles of fixed lengths.

show abstract

Random Unitary Representations of Surface Groups II: The large $n$ limit

Cited by 3 publications

References 25 publications

Large N limit of the Yang-Mills measure on compact surfaces II: Makeenko-Migdal equations and planar master field

Large N limit of the Yang-Mills measure on compact surfaces II: Makeenko-Migdal equations and planar master field

The Asymptotic Statistics of Random Covering Surfaces

Local Statistics of Random Permutations from Free Products

Contact Info

Product

Resources

About