#0: Tamás

Ref: [4]

Contents
Origin of invariant ensembles •
time-reversal invariance •
orthogonal ensemble: $T$-invariance $+$ even-spin or rotational symmetry •
symplectic ensemble $T$-invariance $+$ odd-spin •
unitary ensemble: "everything else".

#1: Feb 9 — Tamás

Ref: [1] Sec 2.1-2.3, [2] Lec 1 & 2

Contents
Three invariant ensembles (unitary, orthogonal, symplectic) •
definition •
invariance •
matrices with simple spectrum (open, dense, full measure).

#2: Feb 13 — Tivadar

Ref: [1] Sec 2.4, [2] Lec 3

Contents
Jacobian of $M\mapsto(\Lambda,U)$ •
$2\times 2$ example •
computing the Jacobian for UE,OE,SE.

#3: Feb 16 — Tivadar

Ref: [1] Sec 2.4-2.5, [2] Lec 3

Contents
Computing the Jacobian for UE,OE,SE.

#4: Feb 20 — Tivadar

Ref: [1] Sec 2.5, [2] Lec 3

Contents
Appearance of the Vandermonde-determinant •
integrating out variables other then eigenvalues.

#5: Feb 23 — Gyuri

Ref: [1] Sec 3.1-3.2, [2] Lec 3

Contents
Auxiliary fact from functional analysis •
$\det(\mathbb{1}+AB)=\det(\mathbb{1}+BA)$, first proof.

#6: Mar 5 — Gyuri

Ref: [1] Sec 3.2, [2] Lec 3

Contents
$\det(\mathbb{1}+AB)=\det(\mathbb{1}+BA)$, second proof.

#7: Mar 9 — Tivadar

Ref: [1] Sec 3.3, [2] Lec 3

Contents
Pfaffian $(\mathrm{Pf}(A))^2=\det(A)$ •
first approach: exisistence of $\mathrm{Pf}(A)$•
second approach: explicit formula for $\mathrm{Pf}(A)$ •
~~third approach~~.

#8: Mar 16 — Tamás

Ref: [1] Sec 3.4, [2] Lec 3, [3] Sec 5.4

Contents
Proof of three integral identities.

#9: Mar 27 — Gyuri

Ref: [1] Sec 4.1.1, [2] Lec 4, [3] Sec 5.4

Contents
Correlation kernel, gap probability, expectations for $\beta=2$.

#10: Apr 10 — Tamás

Ref: [1] Sec 4.2-4.4, [2] Lec 4, [3] Sec 5.4

Contents
Gap probability, "integrating out" lemma, (multiple) integrals of the correlation kernel,
$m$-point correlation function, $R_m$ is the $m$-th pricipal minor of $\det K_N(x_i,x_j)$,
occupational probabilities $A_m(\theta)$, two formulae for $A_m(\theta)$,
interpretation of $R_1$ and $R_2$.