The maximum deviation of the Sine$_β$ counting process
Diane Holcomb Elliot Paquette
math.PR 60B20


 


The maximum deviation of the Sine_β counting process
    Diane Holcomb and Elliot Paquette
    December 30, 2023
====================================================




In this paper, we consider the maximum of the 
Sine_β counting process from its expectation.  
We show the leading order behavior is consistent with the predictions of 
log–correlated Gaussian fields, also consistent with
work on the imaginary part of the log–characteristic polynomial of random 
matrices.
We do this by a direct analysis of the stochastic sine equation, which gives 
a description of the continuum limit of the Prüfer phases of a Gaussian β–ensemble matrix.  






The Sine_β point process (<cit.>), which arises as the local point process limit of the eigenvalues of β–ensembles, can be defined in terms of the SDE 

    dα_x,t= x β/4 e^- β/4 t dt+ [ ( e^-i α_x,t-1)dZ_t],     α_x,0=0.

Specifically, sending t →∞, α_x,t/(2π) converges for all x to an integer valued limit, which is the counting function of the Sine_β point process.    

We are interested in the question of whether this function is an example of a process that should satisfy log–correlated field predictions. For an overview on work related to log–correlated Gaussian and approximately Gaussian processes see <cit.>. This question follows naturally from the fact that the counting function of Sine_β is a scaling limit of the imaginary part of the logarithm of the characteristic polynomial of random matrices. Such Gaussian log–correlated field predictions have been proven for a variety of matrix models <cit.>. Similar work has been done for randomized models of the Riemann ζ function <cit.>, and also for the ζ function itself <cit.>. For further discussion of the connections between the ζ function and random matrix theory see <cit.>.

We consider the process 
N(x)=lim_t→∞α_x,t-α_-x,t/2π,
which counts the number of points in the Sine_β point process between [-x,x] for any x > 0.  This process exhibits a purer analogy with log–correlated fields (see Remark <ref> for details). 
We show that:

  
  
    max_0≤λ≤ x[N(λ)- λ/π]/log x[x]
      2/√(β)π.


Moreover, we do this by a direct argument for the Sine_β process. Another possible approach might be to use the recent <cit.>, which gives a coupling between the Sine_β and CβE point processes, to transfer estimates from the random matrix process to the continuum limit.



Observe that as the process N(λ) is almost surely non–decreasing, we may immediately replace this maximum over all 0 ≤λ≤ x by the maximum over any discrete net of [0,x] with maximum spacing o(log x).  Likewise, we may assume that x is an integer.  Going forward, we will take λ and x to be integers. The monotonicity of N(λ) may be seen from the SDE description by observing that the noise term vanishes at multiples of 2π and the drift is positive for λ>0 and negative for λ<0 (<cit.>). 

It should be noted there is another SDE description due to <cit.> (only recently proven to give rise to the same process by <cit.>, while another proof follows from <cit.>), which can be related to (<ref>) by a time–reversal.  This arises due to an order reversal of the Prüfer phases, for which reason the correlation structure is reversed from the previously studied CβE model. The processes α_x,t and α_y,t are strongly correlated for large times and weakly correlated for small times.  We elaborate upon the correlation structure in (<ref>).



 §.§ Heuristic



We will name the martingale part of α_λ, t-α_-λ,t diffusion:

    M_λ,t= ∫_0^t (e^-i α_λ,s-e^-iα_-λ,s)dZ_s.

As the process α_x,t converges for all x ∈ when t →∞, so does M_λ,t converge for all λ∈ when t →∞.  Moreover, 

    2π N(λ) - 2λ
      =
    ∫_0^∞ (e^-i α_λ,s-e^-iα_-λ,s)dZ_s
    =M_λ,∞.

Therefore we can reformulate Theorem <ref> as

    max_0≤λ≤ xM_λ,∞/log x[x] 
      4/√(β)
      .


Let T_λ = 4/βlogλ.  This is heuristically the length of time that M_λ,t needs to evolve so that it is within bounded distance of its limit.  Specifically, the variables M_λ,∞ - M_λ,T_λ have a uniform–in–λ exponential tail bound:

  There is a constant C = C_β so that for all λ,r ≥ 0,
  
    [ 
          M_λ,∞ - M_λ,T_λ≥
          C+
          r
        ]
        ≤ e^-r/C.

  

Using the monotonoicity of N(λ), we can also show that:

  
    max_0≤λ≤ x |M_λ,∞ - M_λ,T_λ | /log x[x] 0.



Hence we need only consider the process M_λ,t up to time t=T_λ. We delay the proofs of these propositions to Section <ref>.


Another representation for M_λ,t is given by, for all t ≥ 0 

    M_λ,t   = ∫_0^t (e^-i/2(α_λ,s-α_-λ,s)-e^-i/2(α_-λ,s-α_λ,s))e^-i/2(α_λ,s+α_-λ,s)dZ_s 
       =  ∫_0^t (e^-i/2(α_λ,s-α_-λ,s)-e^-i/2(α_-λ,s-α_λ,s)) (dV^(λ)_s+ i dW^(λ)_s) 
        = ∫_0^t 2sin( α_λ,s-α_-λ,s2)d W^(λ)_s.

where dV^(λ)_s+ i dW^(λ)_s=e^-i/2(α_λ,s+α_-λ,s)dZ_s is a standard complex Brownian motion.

Hence, the bracket process is given by

    [M_λ]_t = ∫_0^t 4sin( α_λ,s-α_-λ,s2)^2 ds.

Applying the trig identity 2sin(x)^2 = 1-cos(2x), and treating the oscillating the term as negligible, we can consider [M_λ]_t ≈ 2t, for t ≤ T_λ.  This allows us to roughly consider M_λ,T_λ, for the purpose of moderate deviations, as a centered Gaussian of variance 2T_λ.


As for the correlation structure, 

    [ M_λ, M_μ ]_t
         =
      ∫_0^t 
      (e^-i α_λ,s-e^-iα_-λ,s)
      (e^i α_μ,s-e^iα_-μ,s)
       ds

Approximating α_λ,t by its drift in the equation above, we are led to the heuristic that M_λ and M_μ behave approximately independently for t ≤4/βlog_+|λ-μ| and are maximally correlated for larger t.  

This leads to the cross variation heuristic:

    [ M_λ, M_μ ]_T_λ∧ T_μ≈ 2(T_λ∧ T_μ - 4βlog_+|λ-μ|).


We can define a Gaussian process that has the exact correlation structure suggested by the heuristics 
in (<ref>):

    G_λ,t = ∫_0^t (e^-i α_λ,s-e^-iα_-λ,s)dZ_s.

For this process, we have correlation given by

    [G_λ,G_μ]_t = 4∫_0^t 
      sin( λ (1-e^-β 4 s))
      sin( μ (1-e^-β 4 s))
       ds.

On the supposition that  the maximum of λ↦ M_λ,∞ is well modeled by the maximum of the field (G_λ, T_λ, 0 ≤λ≤ x), we are led to the following conjecture.

  
  There is a random variable ξ so that
  
    max_0≤λ≤ x(M_λ,∞)
          -4√(β)( log x  - 34 loglog x)
          [x] ξ.
 

Indeed by a theorem of <cit.>, full convergence could be proven for the G_λ,T_λ field. One might expect that the distribution of ξ is sensitive to the model and so could be different than in the Gaussian case.

 
  If we instead considered the one–sided problem, we would instead see
  
    max_0 ≤λ≤ x[ α_λ,∞ - λ]
        /log x[x]
        4/√(2β).

  We would be led to considering the martingale
  
    V_λ,t= ∫_0^t (e^-i α_λ,s-1)dZ_s.

  which has quadratic variation [V_λ]_t≈ 2t for t< T_λ and cross variation: 
  
    [ V_λ, V_μ ]_T_λ∧ T_μ   =
          ∫_0^t 
          (e^-i α_λ,s-1)
          (e^i α_μ,s-1)
           ds
          ≈ T_λ∧ T_μ  + 1/2 [ M_λ, M_μ ]_T_λ∧ T_μ.

  Thus, the process has an additional positive correlation, which is heuristically equivalent to adding a common standard normal of variance 4βlog x to every V_λ,∞ for δ x ≤λ≤ x.  In particular this is too small to change the behavior of the maximum.  As working with V_λ,t does not materially change the argument, we have not pursued it here.







§ BACKGROUND TOOLS


We begin with the proofs of Propositions <ref> and <ref>.
These rely heavily on basic properties of the diffusion established in <cit.>.


 §.§  Delayed proofs from introduction 


  Observe first by integrating the drift
  
    M_λ,∞ - M_λ,T_λ
        =α_λ,∞ - α_λ,T_λ
        -1.

  Consider the process v that satisies
  
    dv_t= λβ/4 e^- β/4 t[t ≤ T_λ] dt+ [ ( e^-i v_t-1)dZ_t],      v_0=0.

  Then α_λ,t and v_t are equal until T_λ.  After this time, v never crosses another multiple of 2π.  Moreover, it eventually converges to a multiple of 2π (<cit.>).  Hence we have
  
    |v_∞ - α_λ,T_λ| ≤ 2π.

  On the other hand α_λ,∞-v_∞ has the same law as α_1,∞.
  By <cit.>, this has an exponential tail bound.



  By (<ref>), it suffices to show the same for α_λ,∞ - α_λ,T_λ.  The diffusion α_λ,t can not cross below an integer multiple of 2π.  Hence if s ≤ t, for all λ≥ 0
  α_λ,s≤α_λ,t + 2π.
  This implies 
  
    min_0 < λ≤ x(α_λ,∞ - α_λ,T_λ) ≥ -2π,

  and it suffices to consider an upper bound.
For x/2 ≤λ≤ x, we can estimate

    α_λ,∞ - α_λ,T_λ≤α_λ,∞ - α_λ,T_x/2 + 2π

Let v_λ satisfy

    dv_λ,t= λβ/4 e^- β/4 t[t ≤ T_x/2] dt+ [ ( e^-i v_λ,t-1)dZ_t],      v_λ,0=0.

As v_λ can not cross multiples of 2π, for any λ∈, after T_x/2, we have

    α_λ,∞ - α_λ,T_x/2 + 2π≤α_λ,∞ - v_λ,∞ + 4π.

On the other hand α_λ,t - v_λ,t is monotone increasing in λ almost surely (as the difference for parameters λ_1>λ_2 satisfies an SDE that can not cross below 0, c.f. <cit.>).  
Combining the work so far, we have the bound

    max_x/2 ≤λ≤ x(
      α_λ,∞ - α_λ,T_λ)
      ≤α_x,∞ - v_x,∞ + 4π.

Using the equality in law given by

    (
      α_x,t+T_x/2 - v_x,t+T_x/2, t ≥ 0
      ) 
      (α_2,t, t ≥ 0),

and by <cit.>, α_2,∞ has an exponential tail bound depending only on β.  Applying the same argument for j∈ and x 2^-j-1≤λ≤ x 2^-j, we may use a union bound up to j on the order of log x to conclude that there is a constant C_β so that

    max_0 < λ≤ x(α_λ,∞ - α_λ,T_λ) ≤ C_βloglog x

with probability going to 1 as x →∞.




 §.§ Oscillatory integrals


For each λ∈, suppose that
A_λ,t is an adapted finite variation process so that |A_λ,t| ≤ξ∈ (0,∞) for all time almost surely 
and suppose that X_λ,t is a martingale satisfying d[X_λ]_t ≤ 2.
Suppose that

    du_λ, t
      =
      λβ4e^-β 4 tdt
      +A_λ,t dt
      +dX_λ,t,     u_λ,0 = 0.





Let u_λ,t satisfy (<ref>) and let (t) = β/4e^-β/4t, then for each fixed β>0 there exist constants R and γ uniform in T and λ, a∈ such that 

    [sup_0 ≤ t ≤ T|∫_0^t e^i a u_λ,s ds| ] ≤R(1+|ξ|)/|aλ| (T),

and for all C>0

    (sup_0≤ t ≤ T|∫_0^t e^i a u_λ,sds|- R(1+|ξ|)/|aλ| (T)≥ C ) ≤exp[ - γ C^2 a^2 λ^2 (T)^2].





    The theorem is vacuous if aλ = 0, so we may assume this is not the case.
  Writing u_t in its integrated form, we have
  
    u_t    = λ(1-4/β(t)) + ℛ_t 
    ℛ_t
       = ∫_0^t 
    { A_λ,sds + dX_λ,s}
    .

Let H(t) = 1-4/β(t) and Λ(t) = ∫_0^t e^i a λ H(s)ds , then we may use Itô integration by parts to get  

    ∫_0^t e^i aλ u_s ds 
    = ∫_0^t e^i a λ H(s) e^i aℛ_s ds 
    = e^i aℛ_tΛ(t)
    - ∫_0^t Λ(s) i a e^i aℛ_sdℛ_s 
    + a^2/2∫_0^t Λ(s) e^i aℛ_s d[ℛ]_s.

 Now observe that Λ(t) may be bounded in the following way:

    ∫_0^t e^i aλ  H(s)ds  
       = ∫_0^t1/i aλ(s)d/ds e^i a λ  H(s)ds
    =  4 e^β/4t/β i a λ{e^i aλ  H(t) - 1 }
    - 1/i aλ∫_0^t e^β/4s{e^i a λ  H(s)-1}ds.

This gives us |Λ(s)| ≤16/β |a λ| e^β/4t. Applying this to our integrated equation we get for the finite variation terms 

    | ∫_0^t Λ(s)e^i ℛ_s 
     aA_λ,s ds +  
    a^2/2∫_0^t Λ(s)e^i a ℛ_s d[ℛ]_s   | 
    ≤16/β a λ
    e^β/4t( |a|ξ + a^2 ).


By (<ref>) and the triangle inequality, it remains to show the desired tail bound and supremum bound for the martingale V_t given by

    V_t =  ∫_0^t Λ(s) i a e^i aℛ_s· dX_λ,s

Note we have an easy bracket bound, for σ∈{ 1, i } given by

    [ (σ V)]_t ≤∫_0^t 2Λ(s) a^2 ds ≤C_β/λ ^2 |a| e^β2 t

for some constant C_β.
Hence the desired bounds follow immediately from the Dambis–Dubins–Schwarz theorem (<cit.> or <cit.>) and Doob's inequality.





 §.§ Tilting

We now want to look at the measure tilted so that W^(λ) (see (<ref>)) has a drift. In particular 
for deterministic η∈,
we consider the measure Q_ξ,λ so that  

    dX_s = dW^(λ)_s - ηsin( α_λ,s-α_-λ,s2) ds

is a standard Brownian motion up to time T under Q_η,λ. By Girsanov (see e.g. <cit.>) we get that 

    dQ_η,λ/d = 
      ℰ(η M_λ)
      =
      exp( η M_λ,T - η^22[M_λ]_T)

Since sin^2(x) ≤ 1 we have that the bracket process of [M_λ]_t ≤ T almost surely for all t ≥ 0.  In particular, the exponential martingale is uniformly integrable by Novikov's condition for all η∈.

Under Q_η, λ the law of α_λ,t - α_-λ,t changes; it can be succinctly described as the solution to

    du_λ,η,t
      =
      2λβ4e^-β 4 tdt
      +2ηsin(  u_λ,η,t2)^2dt
      +2sin(u_λ,η,t2)dX_t,     u_0 = 0

for a Brownian motion dX, which we call the accelerated stochastic sine equation with acceleration η.  Let M_λ,η,t be the martingale part of u_λ,η,t. 






 §.§ Martingale bounds


Using the Girsanov transformation, we now give a nearly sharp tail bound for M_λ.

  For any η∈, there is an R > 0 so that for all λ > 0,  all T ≤ T_λ

    (sup_0≤ t ≤ T M_λ,η,t≥ C) 
    ≤exp[-C^2/4(T+R)( 1- C^2R/2(T+R)^3) ∧-C^4/3/4T^1/3].

and

    (inf_0≤ t ≤ T M_λ,η,t≤ -C) 
    ≤exp[-C^2/4(T+R)( 1- C^2R/2(T+R)^3) ∧-C^4/3/4T^1/3]




  For C up to the order of magnitude of T^3/2 the Gaussian tail majorizes the martingale tail.  For larger C, the second term majorizes the martingale tail.  For much much larger C (on the order T^2) a small change in the proof gives decay of order e^-cC^4/3.  A large deviations principle for N_λ is proven in <cit.> which suggests a stronger tail bound ought to be true.




  Let X_t be a standard Brownian motion, and let w solve (<ref>) the accelerated stochastic sine equation with acceleration η.  Let M be the martingale part of w.
Let ξ∈,
and apply Doob's inequality to the submartingale e^ξ M_t to get 

    ( sup_0≤ t ≤ T M_t≥ C) ≤ e^-ξ C(e^ξ M_T).
 
Applying (<ref>), we have that

    (e^ξ M_T)
     =
     (ℰ(ξ M_T) e^ξ^22 [M]_T)
     =
     Q̂_E(e^ξ^22 [M]_T),

with Q̂_E(·) the expectation under the probability measure Q̂ defined by

    dQ̂/d 
      = ℰ(ξ M_T).

By the Girsanov theorem,

    dY_s = dX_s - 
      ξsin(  w_t2) ds

is a Q̂–Brownian motion.  Hence, 

    M_t 
    = ∫_0^t 2sin( w_s2)dY_s
    + ∫_0^t 2ξsin( w_s2)^2 ds.

Further, the law of w_s changes under Q̂, as we have that

    dw_t
      =
      2λβ4e^-β 4 tdt
      +2(ξ+η)sin(  w_t2)^2dt
      +2sin(w_t2)dY_t,     w_0 = 0.

Hence, under Q̂, w is a solution of the accelerated stochastic sine equation with acceleration ξ+η. 

As for the bracket, we have that for t ≤ T

    [M_λ]_t
      = 
      ∫_0^t 4sin( w_s2)^2 ds
      = 
      2t - 
      ∫_0^t 2cos( w_s) ds.

Using Proposition <ref>, we have that for T ≤ T_λ, there is an R independent of ξ and η so that for all C>0

    Q̂(∫_0^T -2cos( w_s) ds ≥ R(1+|ξ+η|) + C) ≤ e^-C^2/R.

Therefore, we have that for T ≤ T_λ 

    Q̂_E(e^ξ^22 [M_λ]_T)
     =e^ξ^2 TQ̂_E( exp(∫_0^T -ξ^2cos( w_s) ds))
     ≤ 
     e^ξ^2(T+S) + Sξ^4

for some constant S>0 independent of ξ,λ or T but depending on η.  

There remains to optimize in ξ.  From the work so far, we have

    ( sup_0≤ t ≤ T M_t≥ C) 
      ≤ e^-ξ C(e^ξ M_T)
      ≤
     e^-ξ C + ξ^2(T+S) + S|ξ|^3.

Taking ξ = C2(T+S),

    ( sup_0≤ t ≤ T M_t≥ C) 
      ≤exp[
        -C^2/4(T+S) + S C^4/8(T+S)^4],

and taking ξ = (C/(4T+4S))^1/3 gives

    ( sup_0≤ t ≤ T M_t≥ C) 
      ≤exp[
        -3C^4/3/4(4(T+S))^1/3 + C^2/3(T+S)^1/3/4^2/3].

Hence the desired bound holds
by taking the second bound for C > P(T+S) and P sufficiently large, and the first 
bound for C ≤ P(T+S).

The statement about the infimum may be proved in an identical fashion by reformulating it as an equivalent bound on the supremum of -M_λ. We would then use the submartingale e^-ξ M_λ and use [M_λ]_t = [-M_λ]_t. 




§ MAIN THEOREM




 §.§ The one–point upper bound


Using Proposition <ref> with η=0, we can give the upper bound in (<ref>).  

  For any δ > 0
  
    lim_x →∞(
        max_0≤λ≤ xM_λ,T_λ >  (4/√(β) + δ)log x) = 0



  As commented, it suffices to bound the probability for natural numbers λ and x.
  By Proposition <ref> for any δ>0 sufficiently small
  there is an ϵ > 0 
  and an x_0 sufficiently large so that for all x > x_0

  and all x > λ > exp( (log x)^3/4) 
  
    (
        M_λ,T_λ >  (4/√(β) + δ)log x)
        ≤exp(-(log x)^2(4/√(β) + 2δ)^216βlogλ)
        ≤exp( -(log x)(1+ϵ)).

  For smaller λ, we have, taking the 4/3–power bound in Proposition <ref>, that for some C_β,δ
  
    (
        M_λ,T_λ >  (4/√(β) + δ)log x)
        ≤exp(-(log x)^13/12C_β,δ)

  Hence, taking a union bound over all natural numbers λ less than x gives the desired bound.


  In fact, the proof is easily modified to give 
  
    lim sup_λ→∞(
        M_λ,T_λ/logλ)
        ≤4/√(β),   a.s.





 §.§ The tube event and the lower bound


Let x be a natural number,
and let R be a large parameter to be chosen later.
Let T_λ' = T_λ - R^2√(logλ).
Define an event 𝒜_λ given by 

    𝒜_λ
      ={   |M_λ,t - √(β)t| ≤ R √(log x),  ∀ 0 ≤ t ≤ T_x'; 
       |[M_λ]_t - 2t| ≤ R,  ∀ 0 ≤ t ≤ T_x'  
      }.


Let x be a natural number, and define 

    S_x = ∑_λ=x^2xℰ( √(β) M_λ,T_x') [ 𝒜_λ]

Notice that with this definition of S_x we will have that S_x>0 if and only if the event 𝒜_λ occurs for some integer λ≤ x. Using the Cauchy-Schwarz inequality for non-negative random variables, we arrive at the Paley-Zygmund inequality

    ( S_x > 0) S_x^2 ≥ ( S_x)^2.

We wish to show that this has probability going to 1 as λ→∞ for any δ > 0.
Hence, we need to produce a lower bound of the form 

    [ℰ( √(β) M_λ,T_x') [ 𝒜_λ]]
      =Q_√(β),λ(𝒜_λ) ≥  1 - C_β e^-R^4/3/C_β,

and we need to produce a similar upper bound on

    [
        ℰ( √(β) M_λ_1,T_x') [ 𝒜_λ_1]
        ℰ( √(β) M_λ_2,T_x') [ 𝒜_λ_2]
      ].

From these bounds we will be able to show that as x→∞

    ( S_x^2)/x^2 → 0      and      S_x ≥ x(1 - C_β e^-R^4/3/C_β).

Hence, we conclude (<ref>) that for any ϵ >0 there is an R sufficiently large and an x_0 sufficiently large so that for all x > x_0 

    ( S_x > 0) ≥ ( S_x)^2/ S_x^2≥ 1-ϵ.

We have therefore shown that by letting R_x tend arbitrarily slowly to infinity

    max_x ≤λ≤ 2x{
        M_λ,T_x'}≥√(β)T_x' - R_x√(log x),

with probability going to 1 as x →∞.




 §.§ One point lower bound

We need to find a lower bound on  

    [ℰ( √(β) M_λ,T_x') [ 𝒜_λ]]
      =Q_√(β),λ(𝒜_λ),

which is on the order of unity.  Recall that under Q_√(β),λ the process α_λ,· - α_-λ,· follows the accelerated stochastic sine equation (<ref>) with ξ=√(β).  The process M_λ,t referenced in the event 𝒜_λ can be expressed as 

    M_λ,t = u_λ,ξ,t - 2λ(1- 4β(t)).

Meanwhile, the performing the Doob decomposition on u_λ,ξ,t, we have

    M_λ,ξ,t = u_λ,ξ,t 
      - 2λ(1- 4β(t))
      - ∫_0^t 2ξsin(  u_λ,ξ,s2)^2ds

The bracket process [M_λ,ξ]_t is given as before by

    [M_λ,ξ]_t
      = 
      ∫_0^t 4sin( u_λ,ξ,s2)^2 ds
      = 
      2t - 
      ∫_0^t 2cos( u_λ,ξ,s) ds.

Hence we can write

    Q_ξ,λ(𝒜_λ)
      ≥
      1
      -
         Q_ξ,λ( 
      sup_0 ≤ t ≤ T_x'|M_λ,ξ,t + ∫_0^t ξcos(u_λ,ξ,s) ds| > R√(log x)) 
      
    
      -   Q_ξ,λ( 
      sup_0 ≤ t ≤ T_x'|∫_0^t 2cos( u_λ,ξ,s) ds| > R
      ).

By Propositions <ref> and <ref>, we conclude that

    Q_ξ,λ(𝒜_λ)
      ≥ 1 - C_β e^-R^4/3/C_β

for some C_β sufficiently large and all λ sufficiently large.





 §.§ Two point bound


Following the heuristic (<ref>), we treat M_λ_1,t and M_λ_2,t as uncorrelated until T_* = 4βlog_+ | λ_1 - λ_2 |. 
Without loss of generality, suppose that λ_2 ≥λ_1.
On the event 𝒜_λ_2, we can estimate 

    ℰ( √(β) M_λ_2,T_x')
           
        =
        ℰ( √(β) M_λ_2,T_*)
        exp(
        √(β) ( M_λ_2,T_x' - M_λ_2,T_*)
        -β2 ([M_λ_2]_T_x' - [M_λ_2]_T_*)
        ) 
       ≤ℰ( √(β) M_λ_2,T_*)
        exp(
        2√(β) R √(log x) + β R
        ).

Hence, we have the estimate

    [
        ℰ( √(β) M_λ_1,T_x') [ 𝒜_λ_1]
        ℰ( √(β) M_λ_2,T_x') [ 𝒜_λ_2]
      ] 
    ≤   [
        ℰ( √(β) M_λ_1,T_x') 
        ℰ( √(β) M_λ_2,T_*)
        exp(
        2√(β) R √(log x) + β R
        )
      ].


We now observe that

    ℰ( √(β) M_λ_1,T_x') 
        ℰ( √(β) M_λ_2,T_*)
        =
        ℰ( √(β) (M_λ_1,T_x' + M_λ_2,T_*))
        exp( 
        β
        [M_λ_1,M_λ_2]_T_* ∧ T_x').

By the Girsanov theorem, under the measure 𝕊 with Radon–Nikodym derivative

    d𝕊/d = ℰ( √(β) (M_λ_1,T_x' + M_λ_2,T_*)),

we have that there is a finite variation process A_t bounded almost surely by an absolute constant so that

    dU_t = dZ_t - √(β)A_t dt

is a standard complex 𝕊–Brownian motion. Here Z_t is the standard complex Brownian motion used in equation (<ref>) under the measure ℙ. Meanwhile (<ref>) (also c.f. (<ref>)) shows that  
[M_λ_1,M_λ_2]_t
is a sum of integrals of e^iσ_1(σ_1α_σ_2 λ_1,t + σ_3 α_σ_4 λ_2,t) with σ_j ∈{ 1,-1 }. 
Applying Proposition <ref> to each of these integrals, we can conclude 

    (
      [M_λ_1,M_λ_2]_T_* ∧ T_x' > t + C
      )
      ≤ e^-t^2/C

for sufficiently large C.  Hence we conclude using (<ref>) and (<ref>) that there is some constant C_β so that for any R>0

    [
        ℰ( √(β) M_λ_1,T_x') [ 𝒜_λ_1]
        ℰ( √(β) M_λ_2,T_x') [ 𝒜_λ_2]
      ]
      ≤
      e^C_β + 2R√(βlog x) + β R.




 §.§ Fine estimate


We also need an estimate that improves when λ_1 and λ_2 are well separated.  Once more, we estimate by dropping the indicators and writing

    [
        ℰ( √(β) M_λ_1,T_x') [ 𝒜_λ_1]
        ℰ( √(β) M_λ_2,T_x') [ 𝒜_λ_2]
      ]
      ≤𝕊( 
      exp( 
        β
        [M_λ_1,M_λ_2]_T_x')
      ),

where 

    d𝕊/d = ℰ( √(β) (M_λ_1,T_x' + M_λ_2,T_x')).

Now, on applying Proposition <ref>, we have a tail bound of the form

    (
      [M_λ_1,M_λ_2]_T_x' > t + C_β/Δ)
      ≤ e^-t^2Δ^2/C_β

where Δ = |λ_1-λ_2|(T_x') and C_β > 0 is a constant.  This leads to an estimate of the form

    [
        ℰ( √(β) M_λ_1,T_x') [ 𝒜_λ_1]
        ℰ( √(β) M_λ_2,T_x') [ 𝒜_λ_2]
      ]
      ≤exp( 
      C_β/Δ).

for some other C_β and all Δ≥ 1. 



 §.§ The second moment


Here we estimate S_x^2.  
Recalling (<ref>), we can write

    S_x^2 = 
      ∑_λ_1=x^2x∑_λ_2=x^2x[
        ℰ( √(β) M_λ_1,T_λ_1) [ 𝒜_λ_1]
          ℰ( √(β) M_λ_2,T_λ_2) [ 𝒜_λ_2]
      ].

We partition this sum according to the magnitude of |λ_1-λ_2|.  Let S_0 be all those pairs (λ_1,λ_2) so that |λ_1-λ_2| ≥ x e^- 12 R^2 √(log x).  Let S_1 be the remaining pairs.  Observe that the cardinality of S_1 is at most 2x^2e^- 12 R^2 √(log x).









For terms in S_0, we apply the fine bound (<ref>).  The term Δ that appears for such terms can be estimated uniformly by

    Δ≥ xe^-12 R^2 √(log x)·β4 e^-log x + R^2 √(log x),

which tends to ∞ with x.  In particular, we can estimate 

    ∑_S_0[
        ℰ( √(β) M_λ_1,T_x') [ 𝒜_λ_1]
          ℰ( √(β) M_λ_2,T_x') [ 𝒜_λ_2]
      ]
      ≤ x^2· (1+ O(e^-12 R^2√(log x))).


For the remaining terms, we apply the coarse bound (<ref>), using which we conclude that

    ∑_S_1[
        ℰ( √(β) M_λ_1,T_x') [ 𝒜_λ_1]
          ℰ( √(β) M_λ_2,T_x') [ 𝒜_λ_2]
      ]
      ≤ x^2e^C_β-12 R^2√(log x) + 2R√(βlog x)+β R.

Hence picking R sufficiently large (anything larger than 4√(β) will do), we have combining (<ref>), (<ref>) and (<ref>) that

    ( S_x^2)/x^2 → 0

as x →∞.



 §.§ Proof of main theorem


As in the proofs of Propositions <ref> and <ref>, we have that
( α_λ,t - α_λ,T_x' - 4π : t ≥ T_x', λ > 0 )
is stochastically dominated by 
( α_λ 4 β(T_x'),t : t ≥ 0, λ > 0 ).
Therefore we have by Proposition <ref> that there is a γ > 0 so that for all C>0,

    max_x ≤λ≤ 2x( 
      α_λ,T_x - α_λ,T_x' - 2λ(  4 β )( (T_x) - (T_x')) ≤ -C + 4π) ≤ e^-γ C^2/(T_x - T_x').

In particular we conclude that 

    max_x ≤λ≤ 2x{
        -
        M_λ,T_x
        +
        M_λ,T_x'}≤ C_β R_x (log x)^3/4

with probability going to 1.

Finally, we observe that for 0 ≤λ≤ 2x,

    0
      ≤α_λ,∞
      -
      α_λ,T_x
      =
      M_λ,∞
      -
      M_λ,T_x
      +2λ( 4β(T_x))
      ≤
      M_λ,∞
      -
      M_λ,T_x
      +16β.

Therefore, we conclude that 

    max_x ≤λ≤ 2x{ 
        M_λ,∞}≥max_x ≤λ≤ 2x{ 
        M_λ,T_x}
      -16/β

Combining (<ref>), (<ref>) and (<ref>), we conclude that

    max_x ≤λ≤ 2x{ 
        M_λ,∞}≥4/√(β)log(x) - C_β R_x (log x)^3/4 - (R_x^2+R_x)√(log x) - 16/β

with probability going to 1 as x →∞. 

abbrv



