The eigenvalue equations Regularized phase functions

4 Phase functions In this section we introduce the phase functions used to count the eigenvalues.

4.1 The eigenvalue equations

Let s j = p n − j − 12 and p j = p m − j − 12. Conjugating the matrix ˜ A n,m 6 with a 2n × 2n diagonal matrix D = D n with diagonal elements D 1,1 = 1, D 2i,2i = ˜ χ βm−i−1 p β p i i −1 Y ℓ=1 ˜ χ βm−ℓ χ βn−ℓ β p ℓ s ℓ , D 2i+1,2i+1 = i Y ℓ=1 ˜ χ βm−ℓ χ βn−ℓ β p ℓ s ℓ we get the tridiagonal matrix ˜ A D n,m = D −1 ˜ A n,m D: ˜ A D n,m =            p + X p 1 s + Y s 1 p 1 + X 1 ... ... ... p n −1 s n −2 + Y n −2 s n −1 p n −1 + X n −1 p n            29 where X ℓ = ˜ χ 2 βm−ℓ−1 β p ℓ+1 − p ℓ , ≤ ℓ ≤ n − 1, Y ℓ = χ 2 βn−ℓ−1 βs ℓ+1 − s ℓ , ≤ ℓ ≤ n − 2. The first couple of moments of these random variables are explicitly computable using the moment generating function of the χ 2 -distribution and we get the following asymptotics: EX ℓ = O m − ℓ −32 , EX 2 ℓ = 2β + O m − ℓ −1 , EX 4 ℓ = O 1, EY ℓ = O n − ℓ −32 , EY 2 ℓ = 2β + O n − ℓ −1 , EY 4 ℓ = O 1, 30 where the constants in the error terms only depend on β. We consider the eigenvalue equation for ˜ A D n,m with a given Λ ∈ R and denote a nontrivial solution of the first 2n − 1 components by u 1 , v 1 , u 2 , v 2 , . . . , u n , v n . Then we have s ℓ v ℓ + p ℓ + X ℓ v ℓ+1 = Λu ℓ+1 , ≤ ℓ ≤ n − 1, p ℓ+1 u ℓ+1 + s ℓ + Y ℓ u ℓ+2 = Λv ℓ+1 , ≤ ℓ ≤ n − 2, where we set v = 0 and we can assume u 1 = 1 by linearity. We set r ℓ = r ℓ,Λ = u ℓ+1 v ℓ , 0 ≤ ℓ ≤ n−1 and ˆr ℓ = ˆr ℓ,Λ = v ℓ u ℓ , 1 ≤ ℓ ≤ n. These are elements of R ∪ {∞} satisfying the recursion ˆr ℓ+1 = − 1 r ℓ · s ℓ p ℓ + Λ p ℓ 1 + X ℓ p ℓ −1 , ≤ ℓ ≤ n − 1 31 r ℓ+1 = − 1 ˆr ℓ+1 · p ℓ+1 s ℓ + Λ s ℓ 1 + Y ℓ s ℓ −1 , ≤ ℓ ≤ n − 2, 32 with initial condition r = ∞. We can set Y n = 0 and define r n via 32 with ℓ = n − 1, then Λ is an eigenvalue if and only if r n = 0. 324

4.2 The hyperbolic point of view

We use the hyperbolic geometric approach of [13] to study the evolution of r and ˆr. We will view R ∪ {∞} as the boundary of the hyperbolic plane H = {ℑz 0 : z ∈ C} in the Poincaré half-plane model. We denote the group of linear fractional transformations preserving H by PSL2, R. The recursions for both r and ˆr evolve by elements of this group of the form x 7→ b − ax with a 0. The Poincaré half-plane model is equivalent to the Poincaré disk model U = {|z| 1} via the con- formal bijection Uz = iz+1 z+i which is also a bijection between the boundaries ∂ H = R ∪ {∞} and ∂ U = {|z| = 1, z ∈ C }. Thus elements of PSL2, R also act naturally on the unit circle ∂ U. By lifting these maps to R, the universal cover of ∂ U, each element T in PSL2, R becomes an R → R function. The lifted versions are uniquely determined up to shifts by 2 π and will also form a group which we denote by UPSL2, R. For any T ∈ UPSL2, R we can look at T as a function acting on ∂ H , ∂ U or R. We will denote these actions by: ∂ H → ∂ H : z 7→ z.T, ∂ U → ∂ U : z 7→ z ◦

T, ∂ R → ∂ R : z 7→ z

∗ T. For every T ∈ UPSL2, R the function x 7→ f x = x ∗ T is monotone, analytic and quasiperiodic modulo 2 π: f x + 2π = f x + 2π. It is clear from the definitions that e i x ◦ T = e i f x and 2 tanx.T = 2 tan f x. Now we will introduce a couple of simple elements of UPSL2, R. For a given α ∈ R we will denote by Q α the rotation by α in U about 0. More precisely, ϕ ∗ Q α = ϕ + α. For a 0, b ∈ R we denote by Aa, b the affine map z → az + b in H . This is an element of PSL2, R which fixes ∞ in H and −1 in ∂ U. We specify its lifted version in UPSL2, R by making it fix π, this will uniquely determines it as a R → R function. Given T ∈ UPSL2, R, x, y ∈ R we define the angular shift ash T, x, y = y ∗ T − x ∗ T − y − x which gives the change in the signed distance of x, y under

T. This only depends on v = e

i x , w = e i y and the effect of T on ∂ U, so we can also view ashT, ·, · as a function on ∂ U ×∂ U and the following identity holds: ash

T, v, w = arg

[0,2π w ◦ T v ◦ T − arg [0,2π wv. The following lemma appeared as Lemma 16 in [13], it provides a useful estimate for the angular shift. Lemma 12. Suppose that for a T ∈ UPSL2, R we have i + z.T = i with |z| ≤ 1 3 . Then ash

T, v, w = ℜ

h ¯ w − ¯v −z − i2+¯ v+ ¯ w 4 z 2 i + ǫ 3 = −ℜ [ ¯ w − ¯vz] + ǫ 2 = ǫ 1 , 33 where for d = 1, 2, 3 and an absolute constant c we have |ǫ d | ≤ c|w − v||z| d ≤ 2c|z| d . 34 If v = −1 then the previous bounds hold even in the case |z| 1 3 . 325

4.3 Regularized phase functions

Because of the scaling in 11 we will set Λ = µ n + λ 4n 1 2 . We introduce the following operators J ℓ = QπAs ℓ p ℓ , µ n s ℓ , M ℓ = A1 + X ℓ p ℓ −1 , λ4n 1 2 p ℓ A p ℓ p ℓ+1 , 0, ˆ J ℓ = QπAp ℓ s ℓ , µ n p ℓ , ˆ M ℓ = A1 + Y ℓ s ℓ −1 , λ4n 1 2 s ℓ . Then 31 and 32 can be rewritten as r ℓ+1 = r ℓ . J ℓ M ℓ ˆ J ℓ ˆ M ℓ , r = ∞. We suppressed the λ dependence in r and the operators M, ˆ M. Lifting these recursions from ∂ H to R we get the evolution of the corresponding phase angle which we denote by φ ℓ = φ ℓ,λ . φ ℓ+1 = φ ℓ∗ J ℓ M ℓ ˆ J ℓ ˆ M ℓ , φ = −π. 35 Solving the recursion from the other end, with end condition 0 we get the target phase function φ ⊙ ℓ,λ : φ ⊙ ℓ = φ ⊙ ℓ+1∗ ˆ M −1 ℓ ˆ J −1 ℓ M −1 ℓ J −1 ℓ , φ ⊙ n = 0. 36 It is clear that φ ℓ,λ and φ ⊙ ℓ,λ are independent for a fixed ℓ as functions in λ, they are monotone and analytic in λ and we can count eigenvalues using the formula 21. In our case both M ℓ and ˆ M ℓ will be small perturbations of the identity so J ℓ ˆ J ℓ will be the main part of the evolution. This is a rotation in the hyperbolic plane if it only has one fixed point in H. The fixed point equation ρ ℓ = ρ ℓ . J ℓ ˆ J ℓ can be rewritten as ρ ℓ = p ℓ s ℓ    µ n p ℓ − p ℓ µ n s ℓ − 1 ρ ℓ    = ρ ℓ µ 2 n − p 2 ℓ − µ n s ℓ ρ ℓ µ n s ℓ − s 2 ℓ . This can be solved explicitly, and one gets the following unique solution in the upper half plane if ℓ n + 12: ρ ℓ = µ 2 n − m + n 2 µ n s ℓ + i s 1 − µ 2 n − m + n 2 4 µ 2 n s 2 ℓ . 37 One also needs to use the identity p 2 ℓ − s 2 ℓ = m − n and 12. This shows that if ℓ n then J ℓ ˆ J ℓ is a rotation in the hyperbolic plane. We can move the center of rotation to 0 in U by conjugating it with an appropriate affine transformation: J ℓ ˆ J ℓ = Q−2 argρ ℓ ˆ ρ ℓ T −1 ℓ . Here T ℓ = Aℑρ ℓ −1 , −ℜρ ℓ , X Y = Y −1 XY and ˆ ρ ℓ = µ 2 n + m − n 2 µ n p ℓ + i s 1 − µ 2 n + m − n 2 4 µ 2 n p 2 ℓ . 38 326 In order to regularize the evolution of the phase function we introduce ϕ ℓ,λ := φ ℓ,λ∗ T ℓ Q ℓ−1 , ≤ ℓ n where Q ℓ = Q2 argρ ˆ ρ · · · Q2 argρ ℓ ˆ ρ ℓ and Q −1 is the identity. It is easy to check that the initial condition remains ϕ 0, λ = π. Then ϕ ℓ+1 = ϕ ℓ∗ Q −1 ℓ−1 T −1 ℓ J ℓ M ℓ ˆ J ℓ ˆ M ℓ T ℓ+1 Q ℓ = ϕ ℓ∗ Q −1 ℓ−1 T −1 ℓ Q −2 argρ ℓ T −1 ℓ M ˆ J ℓ ℓ ˆ M ℓ T ℓ T −1 ℓ T ℓ+1 Q ℓ = ϕ ℓ∗  M ˆ J ℓ ℓ T ℓ ‹ Q ℓ € ˆ M ℓ T ℓ Š Q ℓ € T −1 ℓ T ℓ+1 Š Q ℓ Note that the evolution operator is now infinitesimal: M ℓ , ˆ M ℓ and T −1 ℓ T ℓ+1 will all be asymptotically small, and the various conjugations will not change this. We can also introduce the corresponding target phase function ϕ ⊙ ℓ,λ := φ ⊙ ℓ,λ∗ T ℓ Q ℓ−1 , ≤ ℓ n . 39 The new, regularized phase functions ϕ ℓ,λ and ϕ ⊙ ℓ,λ have the same properties as φ, φ ⊙ , i.e.: they are independent for a fixed ℓ as functions in λ, they are monotone and analytic in λ and we can count eigenvalues using the formula 24. See 22 and the discussion before it. We will further simplify the evolution using the following identities: − a r + b = ‚ b 2 + 1 a r − b Œ Q arg b − i b + i , r.ˆ J ℓ T ℓ = − 1 r p ℓ s ℓ ℑρ ℓ + µ n s ℓ ℑρ ℓ − ℜρ ℓ ℑρ ℓ . From this we get ˆ J ℓ T ℓ = ˆ T ℓ Q ℓ −2 arg ˆ ρ ℓ where r.ˆ T ℓ = ‚ s ℓ ℑρ ℓ p ℓ − 2 ℜρ ℓ ℑρ ℓ µ n + µ 2 n ℑρ ℓ p ℓ s ℓ Œ r − µ n s ℓ ℑρ ℓ + ℜρ ℓ ℑρ ℓ = 1 ℑ ˆ ρ ℓ r − ℜ ˆ ρ ℓ ℑ ˆ ρ ℓ . This allows us to write  M ˆ J ℓ ℓ T ℓ ‹ Q ℓ = M ˆ T ℓ ℓ Q −2 arg ˆ ρ ℓ Q ℓ = M ˆ T ℓ ℓ ˆ Q ℓ . 40 where ˆ Q ℓ = Q ℓ Q −2 arg ˆ ρ ℓ . Thus ϕ ℓ+1 = ϕ ℓ∗ M ˆ T ℓ ℓ ˆ Q ℓ ˆ M T ℓ ℓ Q ℓ € T −1 ℓ T ℓ+1 Š Q ℓ . 327 We will introduce the following operators to break up the evolution into smaller pieces: L ℓ,λ = A1, λ4n 1 2 p ℓ , ˆ L ℓ,λ = A1, λ4n 1 2 s ℓ , S ℓ,λ = L ˆ T ℓ ℓ,λ ˆ T −1 ℓ A p ℓ p ℓ+1 1 + X ℓ p ℓ −1 , 0 ˆ T ℓ , 41 ˆ S ℓ,λ = ˆ L T ℓ ℓ,λ € T ℓ −1 A1 + Y ℓ s ℓ −1 , 0 T ℓ+1 Š . Then ϕ ℓ+1 = ϕ ℓ∗ L ˆ T ℓ ℓ ˆ Q ℓ € S ℓ,0 Š ˆ Q ℓ ˆ L T ℓ ℓ Q ℓ €ˆ S ℓ,0 Š Q ℓ = ϕ ℓ∗ € S ℓ,λ Š ˆ Q ℓ €ˆ S ℓ,λ Š Q ℓ . 42 We also introduce the relative regularized phase function and target phase function: α ℓ,λ := ϕ ℓ,λ − ϕ ℓ,0 , α ⊙ ℓ,λ := ϕ ⊙ ℓ,λ − ϕ ⊙ ℓ,0 . 43 5 SDE limit for the phase function Let F ℓ denote the σ-field generated by ϕ j, λ , j ≤ ℓ − 1. Then ϕ ℓ,λ is a Markov chain in ℓ with respect to F ℓ . Indeed, the relation 42 shows that ϕ ℓ+1,λ = h ℓ,λ ϕ ℓ+1,λ , X ℓ , Y ℓ where h ℓ,λ is a deterministic function depending on ℓ and λ. Since X ℓ , Y ℓ are independent of F ℓ it follows that E ” ϕ ℓ+1,λ |F ℓ — = E ” ϕ ℓ+1,λ |ϕ ℓ,λ — . We will show that this Markov chain converges to a diffusion limit after proper normalization. In order to do this we will estimate E ” ϕ ℓ+1,λ − ϕ ℓ,λ |F ℓ — and E ” ϕ ℓ+1,λ − ϕ ℓ,λ ϕ ℓ+1,λ ′ − ϕ ℓ,λ ′ |F ℓ — using the angular shift lemma, Lemma 12. To simplify the computations we introduce ‘intermediate’ values for the process ϕ ℓ,λ by breaking the evolution operator in 42 into two parts: ϕ ℓ+ 1 2 , λ = ϕ ℓ∗ € S ℓ,λ Š ˆ Q ℓ , F ℓ+ 1 2 = σF ℓ ∪ {ϕ ℓ+ 1 2 , λ }. Note that ϕ ℓ,λ is still a Markov chain if we consider it as a process on the half integers. Remark 13. We would like to note that the ‘half-step’ evolution rules ϕ ℓ,λ → ϕ ℓ+ 1 2 , λ , ϕ ℓ+ 1 2 , λ → ϕ ℓ+1,λ are very similar to the one-step evolution of the phase function ϕ in [13]. Indeed, in [13], the evolution of ϕ is of the type ϕ ℓ+1 = ϕ ℓ∗ €˜ S ℓ,λ Š ˜ Q ℓ where ˜ S is an affine transformation and ˜ Q is a rotation similar to our S, ˆ S and Q, ˆ Q. In our case the evolution of ϕ ℓ+1,λ is the composition of two transformations with similar structure. The main difficulties in our computations are caused by the fact that Q and ˆ Q are rather different which makes the oscillating terms more complicated.

5.1 Single step estimates