Lower Deviations in β -ensembles and Law of Iterated Logarithm in Last Passage Percolation
Riddhipratim Basu, Shirshendu Ganguly, Milind Hegde, Manjunath Krishnapur
LLOWER DEVIATIONS IN β -ENSEMBLES AND LAW OF ITERATEDLOGARITHM IN LAST PASSAGE PERCOLATION RIDDHIPRATIM BASU, SHIRSHENDU GANGULY, MILIND HEGDE, AND MANJUNATH KRISHNAPUR
Abstract.
For the last passage percolation (LPP) on Z with exponential passage times, let T n denote the passage time from (1 ,
1) to ( n, n ). We investigate the law of iterated logarithmof the sequence { T n } n ≥ ; we show that lim inf n →∞ T n − nn / (log log n ) / almost surely converges to adeterministic negative constant and obtain some estimates on the same. This settles a conjectureof Ledoux [14] where a related lower bound and similar results for the corresponding upper tailwere proved. Our proof relies on a slight shift in perspective from point-to-point passage timesto considering point-to-line passage times instead, and exploiting the correspondence of the latterto the largest eigenvalue of the Laguerre Orthogonal Ensemble (LOE). A key technical ingredient,which is of independent interest, is a new lower bound of lower tail deviation probability of thelargest eigenvalue of β -Laguerre ensembles, which extends the results proved in the context of the β -Hermite ensembles by Ledoux and Rider [15]. Contents
1. Introduction and statement of main results 12. The Law of Iterated Logarithm: Proof of Theorem 1 53. Lower Deviations in β -Laguerre ensemble: Proof of Theorem 2 11References 201. Introduction and statement of main results
Last passage percolation on Z , where one puts i.i.d. weights on the vertices of Z and studies themaximum weight of an oriented path between two vertices, is a canonical model believed to be in the(1+1)-dimensional Kardar-Parisi-Zhang (KPZ) universality class. A handful of such models, the so-called exactly solvable models, have been rigorously analysed using some remarkable bijections andconnections to random matrix theory, leading to an explosion of activities in the field of integrableprobability in recent decades. We shall consider the exponential LPP model on Z where the fieldof vertex weights { ξ v } v ∈ Z is a family of i.i.d. rate one exponentially distributed random variables. Definition 1.1.
For any up/right path γ in Z , define the weight of γ as (cid:96) ( γ ) := (cid:80) v ∈ γ ξ v , and for u, v ∈ Z , with u (cid:22) v in the usual partial order, the last passage time T u,v = T v,u from u to v isdefined by T u,v := max γ : u → v (cid:96) ( γ ) where the maximum is taken over all oriented paths from u to v .For n ≥ , we shall denote by T n the passage time from to n ( r will denote the point ( r, r ) for r ∈ Z ). Our primary object of interest will be the family of coupled random variables { T n } n ≥ . It is a fact,by now classical, [22] that T n ∼ n and it was shown by Johansson in [12] that Z n := n − / ( T n − n )is a tight sequence of random variables and in particular converges to a scalar multiple of the GUETracy-Widom distribution from random matrix theory. Indeed, [12] established the remarkabledistributional equality: T n d = λ n (LUE n ) (1) a r X i v : . [ m a t h . P R ] S e p R. BASU, S. GANGULY, M. HEGDE, AND M. KRISHNAPUR where λ n (LUE n ) is the largest eigenvalue of the Laguerre Unitary Ensemble (LUE), i.e. the matrix X ∗ X where X is an n × n matrix of i.i.d. standard complex Gaussian random variables.Inspired by a result of Paquette and Zeitouni [20] (see Section 1.1 for details), Ledoux [14]considered the law of iterated logarithm for the sequence { T n : n ≥ } , and showed that there exist0 < C < C < ∞ such that almost surely C ≤ lim sup n →∞ Z n (log log n ) / ≤ C . (2)Note that the lim sup above and the lim inf below are almost sure constants by a 0-1 law (seeLemma 2.1). For the lim inf, it was shown in [14] thatlim inf n →∞ Z n (log log n ) / > − C , (3)almost surely for some C < ∞ , and it was conjectured that (log log n ) / is indeed the right scaleof fluctuation for the lower deviations. The first main result of this paper completes the picture byestablishing this conjecture. Theorem 1.
There exists C > such that, almost surely lim inf n →∞ Z n (log log n ) / = − C . (4)A comparison with the classical law of iterated logarithm for the simple random walk and theresults in [20], will be presented in Section 1.1. While we have stated Theorem 1 in the simplestpossible form here, a more detailed discussion on the settings considered in [14], the conjecturedlim inf value and some possible extensions of Theorem 1 is presented in Section 2.1.Ledoux’s proof for the upper tail [14] is reminiscent of the classical law of iterated logarithm forrandom walk and uses sub-additivity of T n and the moderate deviation estimates for the largesteigenvalue of LUE from [15]. The standard sub-additivity is less useful for the lim inf and hencethe weaker result in [14]. The starting point in this paper is the observation that the above issuecan be circumvented by considering point-to-line LPP and using stochastic ordering between thesame and point-to-point LPP.The main technical ingredient we rely on then is a new lower bound of the lower tail moderatedeviation probabilities for the point-to-line last passage time in Exponential LPP. Formally, for anyvertex v and a line L , define the point-to-line last passage time T v, L := sup w ∈ L T v,w . A particularlycanonical case is when v = and L n = { x + y = 2 n } , in which case we define T ∗ n := sup v ∈ Z : v + v =2 n T ,v . (5)Similar to how n − / ( T n − n ) converges to a scalar multiple of the GUE Tracy-Widom distribu-tion, it remarkably turns out and is well-known that n − / ( T ∗ n − n ) converges to a scalar multipleof the Gaussian Orthogonal Ensemble (GOE) Tracy-Widom distribution [4]. We shall prove thefollowing new left tail moderate deviation lower bound with optimal exponent for T ∗ n , which willbe the key ingredient in the proof of Theorem 1. Theorem 1.2.
There exist c > and n ∈ N such that for all n > n and x ∈ (1 , n / ) we have P ( T ∗ n ≤ n − xn / ) ≥ e − c x . To prove Theorem 1.2, we shall use the following correspondence from integrable probabilitybetween the point-to-line passage time and the largest eigenvalue of the Laguerre Orthogonal En-semble with certain parameters. Though implicit in the results of [3, 4, 1], we were not able to findan explicit quotable statement in the literature and for completeness provide a proof later in thearticle using results from [11] and [18].
OWER DEVIATIONS IN β -ENSEMBLES AND LAW OF ITERATED LOGARITHM IN LPP 3 Proposition 1.3.
As defined above, T ∗ n has the same distribution as λ n − , where λ n − is thelargest eigenvalue of LOE n − (i.e., the largest eigenvalue of X T X where X is a n × (2 n − matrix of i.i.d. N (0 , variables). Using Proposition 1.3, Theorem 1.2 will follow from a new general lower deviation tail inequalityfor Laguerre β -ensembles, which is our second main result and is of independent interest (seeTheorem 2). In the next section we define β - ensembles, and review in some detail the relevantliterature on them and connections to last passage percolation.1.1. β -Ensembles, Background and Related Results. Spectra of classical random matrixensembles are special cases of a wide class of point processes termed as β -ensembles which are definedthrough a family of Gibbs measures, with β playing the classical role of inverse temperature. Inthis framework, the well known Hermite, Laguerre and Jacobi ensembles for parameters β = 1 , , β = 2 caseis special as it admits a determinantal structure, for which the Hermite ensemble ( H n ) correspondsto the eigenvalues of a GUE n matrix, i.e., a hermitian matrix of size n with i.i.d. standard complexGaussian entries above the diagonal and independent i.i.d. real Gaussian entries on the diagonal;while for m ≥ n ≥
1, the Laguerre ensemble LUE m,n (for m = n this will simply be denoted byLUE n ) corresponds to the eigenvalues of a complex Wishart matrix, i.e., X ∗ X where X is an m × n matrix of i.i.d. standard complex Gaussians.For the purposes of this paper we need to define the general β > Definition 1.4.
The Laguerre β -ensemble LE βm,n , with parameters m ≥ n ≥ is a point processon R + whose ordered points λ ≤ λ ≤ · · · ≤ λ n have joint density proportional to (cid:89) ≤ i 1, 0 < ε ≤ m ≥ n we have for some absolute constants C, c > P ( λ n ≥ ( √ m + √ n ) (1 + ε )) ≤ Ce − cβε / ( mn ) / ( √ ε ∧ ( mn ) / ) (9) P ( λ n ≤ ( √ m + √ n ) (1 − ε )) ≤ C β e − cβε mn ( ε ∧ ( mn ) / ) (10)In particular, observe that, when m = n , and ε ≈ n − / , this gives the optimal exponents aspredicted from the tails of the Tracy-Widom distribution.For the Hermite case, [15] also proved lower bounds of the the deviation probabilities withmatching exponents, see [15, Theorem 4] for the precise statement. It was remarked there that thelower bound for the upper tail deviation probability in the Laguerre case can be proved using theirmethods but the lower bound for the lower tail would require a different argument. Our secondmain result in this article is to complement the results of [15], by providing the corresponding lowerbound with matching exponents for the lower tail probability in the Laguerre case. Theorem 2. There exists absolute constants C , c, c (cid:48) , c (cid:48)(cid:48) > such that for any < ε < c (cid:48) , and forall integers m ≥ n ≥ and β ≥ , we have P { λ n ≤ ( √ m + √ n ) (1 − ε ) } ≥ (cid:40) exp {− cβ ( ε √ mn ) } if ε ≥ c (cid:48)(cid:48) √ n √ m , ( C ) β · exp {− cβ ( ε m n ) } if < ε ≤ c (cid:48)(cid:48) √ n √ m . (11) OWER DEVIATIONS IN β -ENSEMBLES AND LAW OF ITERATED LOGARITHM IN LPP 5 In the square case, (by taking c (cid:48) = c (cid:48)(cid:48) ) we have the simpler looking P { λ n ≤ n (1 − ε ) } ≥ ( C ) β · exp {− cβ ( nε ) } . (12) Remark 1.5. Observe that the exponents in Theorem 2 are optimal as they match the correspondingupper bounds in (10) . Typically λ n fluctuates on the scale σ m,n = n − / m / , and Theorem 2(together with (10) ) shows that if mn is bounded, then P ( λ n ≤ ( √ m + √ n ) − xσ m,n ) decays like e − cx for x large, as expected. It is worthwhile to notice the following interesting transition in theregime mn → ∞ . If n is bounded one can observe that P ( λ n ≤ ( √ m + √ n ) − xσ m,n ) decays as e − cx as x large. On the other hand if n → ∞ then for each large but fixed x , P ( λ n ≤ ( √ m + √ n ) − xσ m,n ) decays as e − cx . This transition from Gaussian to Tracy-Widom tail behaviour isnot surprising and is understood at the level of Wishart matrices ( β = 1 , ). For β = 2 , there isalso an interpretation in terms of the fluctuation of last passage times across a thin rectangle inexponential LPP, which, via a coupling (or an invariance principle) can also be extended to moregeneral LPP models [7, 5, 24] . The proof of Theorem 2, at a high level, follows the general program of [15]. To obtain the upperbounds for tails in Laguerre and Hermite β -ensembles [15] used the the bi-diagonal and tri-diagonalmodels respectively for these ensembles. For the Hermite case, the proof of lower bound of the lefttail in [15] relied on the independence in the tridiagonal model and Gaussianity of the diagonalentries in a crucial way, which unfortunately is not available for the bi-diagonal models, and hencecould not be extended to the Laguerre case, as pointed out in [15]. We circumvent this issue byusing the idea of linearisation, elaborated in Section 3, which lets us write λ n = s n where s n isthe largest eigenvalue of a certain 2 n × n tridiagonal matrix with independent entries. Note thatTheorem 2 covers ε only up to some small constant. When nm is bounded away from 0, one canprove similar tail bounds for c (cid:48) ≤ ε < 1, by a much simpler argument presented in Section 3.4.1.2. Organization of the article. The rest of this paper is organised as follows. In Section 2 weprovide a proof of Proposition 1.3, prove Theorem 1.2 using Theorem 2, and complete the proof ofTheorem 1 using Theorem 1.2. In Section 3, we provide the proof of Theorem 2. Acknowledgements. The authors thank Ofer Zeitouni for bringing the law of iterated logarithmquestion to their attention and Ivan Corwin for pointing out the LOE connection. RB is partiallysupported by a Ramanujan Fellowship (SB/S2/RJN-097/2017) from the Government of India andan ICTS-Simons Junior Faculty Fellowship. SG is partially supported by a Sloan Research Fellow-ship in Mathematics and NSF Award DMS-1855688. MH is supported by a summer grant of theUC Berkeley Mathematics department. MK is partially supported by UGC Centre for AdvancedStudy and the SERB-MATRICS grant MTR2017/000292.2. The Law of Iterated Logarithm: Proof of Theorem 1 We start by proving Proposition 1.3. As explained before, this result is implicitly known, follow-ing works of Baik and Rains in early 2000s, but we could not find a precise reference in the literatureand hence for completeness provide a short proof using the recent works [18, 11], borrowing theirnotations where convenient. Proof of Proposition 1.3. Theorem 1.2 of [18] says that λ n − d = 4 (cid:32) sup t ∈ [0 , B n − ( t ) (cid:33) , where B < . . . < B n − is a collection of 2 n − , 1] and λ < . . . < λ n − are the eigenvalues of a LOE matrix X T X , where X is a 2 n × (2 n − 1) matrixwith i.i.d. N (0 , 1) random variables. R. BASU, S. GANGULY, M. HEGDE, AND M. KRISHNAPUR Now the calculation immediately preceding equation (5) in [11] shows that (cid:32) sup t ∈ [0 , B n − ( t ) (cid:33) d = sup t ≥ λ max ( H ( t ) − tI ) , where H ( t ) is a (2 n − × (2 n − 1) Hermitian Brownian motion, i.e. a (2 n − × (2 n − 1) Hermitianmatrix with i.i.d. standard complex Brownian motions below the diagonal and i.i.d. standard realBrownain motions along the diagonal. Finally, Theorem 1 of [11] says thatsup t ≥ λ max ( H ( t ) − tI ) d = max π ∈ Π flat n (cid:88) ( i,j ) ∈ π ξ (cid:48) ij , where ξ (cid:48) ij are i.i.d. rate 2 exponential random variables, and Π flat n is the collection of up-right pathsfrom (1 , 1) to the line i + j = 2 n . Combining these and using the scaling property of exponentialsthen yields that T ∗ n = max π ∈ Π flat n (cid:88) ( i,j ) ∈ π ξ ij d = 12 λ n − , where ξ ij are i.i.d. rate 1 exponential random variables and the first equality is by definition (5). (cid:3) We next prove Theorem 1.2, which is an almost immediate consequence of Proposition 1.3 andTheorem 2. Proof of Theorem 1.2. Using Proposition 1.3 and setting the parameters ( m, n ) = (2 n, n − β = 1, and ε = xn − / it follows that P (cid:16) T ∗ n ≤ n − xn / (cid:17) = P (cid:16) λ n − ≤ n − xn / (cid:17) ≥ P (cid:16) λ n − ≤ ( √ n + √ n − (1 − ε ) (cid:17) and the proof is completed by invoking Theorem 2. (cid:3) Note that Theorem 1 states that lim inf Z n (log log n ) / is almost surely a constant. This is proved inthe next lemma, which then reduces proving Theorem 1 to showing that it is bounded below from0 with positive probability. Lemma 2.1. lim inf n →∞ Z n (log log n ) / is a constant, almost surely.Proof. This is a straightforward consequence of the Kolmogorov 0-1 law since the random variablein question is a tail random variable. To see this, fix r and let L r be the line x + y = 2 r . Then,since all the variables ξ v are non-negative, by definition, for n ≥ r ,sup v ∈ L r T v, n ≤ T n ≤ sup v ∈ L r T v, n + sup v ∈ L r T ,v . Now clearly sup v ∈ L r T v,n is a function of only the independent field including and above the line L r , while lim inf n →∞ sup v ∈ L r T ,v − rn / (log log n ) / = 0 , as sup v ∈ L r T ,v is finite a.s. Thus, for every r ,lim inf n →∞ T n − nn / (log log n ) / = lim inf n →∞ sup v ∈ L r T v, n − n − r ) n / (log log n ) / , and hence it is a tail random variable. (cid:3) Observe that the same argument would also show that lim sup n →∞ Z n / (log log n ) / is constantalmost surely.We are now ready to prove the following intermediate result, which in conjunction with Lemma 2.1finishes the proof of Theorem 1. OWER DEVIATIONS IN β -ENSEMBLES AND LAW OF ITERATED LOGARITHM IN LPP 7 Proposition 2.2. For c as in Theorem 1.2, there exists δ > such that P (cid:18) lim inf n →∞ Z n (log log n ) / ≤ − (2 c ) − / (cid:19) ≥ δ. The choice of parameters in the proof can be slightly tweaked allowing us to replace (2 c ) − / by c − / , as pointed out in Remark 2.3. Proof. Define n j = 2 j and fix k > r ∈ Z + , L r denotes the line x + y = 2 r . Our objective is to establish that for ˜ c = (2 c ) − / > 0, as in the statement of theproposition, and for every k sufficiently large, with probability δ > j such that k/ ≤ j ≤ k and T n j ≤ n j − ˜ cn / j (log log n j ) / . Clearly this will suffice.To achieve this, we will divide up the region from (1 , 1) to ( n k , n k ) in dyadic scales(so the j thregion is between L n j − and L n j ), and proceed by examining these regions sequentially, startingfrom the k th region and decreasing the index, till we find the first j such that T n j , L nj − (weight ofthe largest weight path from the line L n j − to ( n j , n j )) is sufficiently low. Suppose this region is theone between L n j − and L n j . Since, we know that the regions are disjoint and hence independent,with high probability the point-to-line weight from (1 , 1) to L n j − does not have too high a weightand hence on the intersection of the above events, one can conclude that T n j is low, finishing theproof (see Figure 1 for an illustration). The rigorous argument will require some minor tweaks tothe above high level description to ensure the necessary independence.Moving now to make the above argument precise, let us, for notational convenience, denoteby T ∗ ( j ) , the weight of the line-to-point polymer from the anti-diagonal line L n j − to the point( n j − , n j − n j − = n j , by the symmetry of the random environment weget T ∗ ( j ) d = T ∗ n j − , and that T ∗ ( j ) are independent as j varies. Also throughout the proof, c is as in Theorem 1.2.Let us fix ε > C ε := (1 − ε ) / c − / . We need to define a number ofevents. For j ∈ N let us define A j := (cid:110) T ∗ ( j ) ≤ n j − − C ε n / j − (log log n j − ) / (cid:111) , and for a fixed k sufficiently large let us set A := (cid:83) kj = k/ A j . Now define τ = max { j ≤ k : A j occurs } so that on A , τ > k/ 2. For v = ( x, y ) ∈ Z we shall denote the random variable ξ v by ξ x,y for notational convenience. Define the σ -algebras F j := σ ( { ξ x,y : x + y < n j − } ) and F cj := σ ( { ξ x,y : x + y ≥ n j − } ) . Notice that the event { τ = j } is F cj -measurable. Define the events B j = (cid:110) (cid:101) T ∗ n j − < n j − (cid:111) and B τ = (cid:110) (cid:101) T ∗ n τ − < n τ − (cid:111) , where (cid:101) T ∗ n is the weight of the best path from (1 , 1) to the line x + y = 2 n with the weight of thefinal vertex excluded, i.e., (cid:101) T ∗ n = sup v ∈ L n ( T ,v − ξ v ) . R. BASU, S. GANGULY, M. HEGDE, AND M. KRISHNAPUR ( n k , n k )... L n τ − L n τ ( n τ − ,n τ − Figure 1. The argument of Proposition 2.2. The dotted lines to the right of( n τ − , n τ − 1) are the boundaries of regions where the event A j did not oc-cur, i.e. the line-to-point weight was not sufficiently low. The region boundedby black lines is the first region, when traveling southwest, that A j occurs, andthe corresponding line-to-point polymer is marked in red. Since the red point-to-line polymer from (1 , 1) to L n τ − has weight less than 4 n τ − , the weight of thepoint-to-point polymer from (1 , 1) to ( n τ − , n τ − 1) must have weight less than4 n τ − (1 − ε ) / c − / n / τ − (log log n τ − ) / . The red points are endpoints of polymerswhose weight is included in the weight of the polymer, while the blue point is theendpoint whose weight is not included.We want to show that A ∩ B τ occurs with uniformly positive probability. To this end, notethat using Theorem 1.2, we have P (cid:16) T ∗ ( j ) ≤ n j − − xn / j − (cid:17) ≥ e − c x for x ∈ (1 , n / ) and setting x = C ε (log log n j − ) / we get P ( A j ) ≥ e − (1 − ε ) log log n j − ≥ j − ε , since n j = 2 j and log 2 < 1. Then, using independence of the A j ’s we have for any δ > P ( A ) ≥ − (cid:18) − k − ε (cid:19) k/ ≥ − δ (13)for k > k ( ε, δ ).Note that T ∗ n is F j measurable. Hence we have P ( B τ | τ = j ) = P ( B j ) , (14)since B j ∈ F j and { τ = j } ∈ F cj , which are independent σ -algebras (it is for this independencethat we removed the weight of the last vertex in the definition of (cid:101) T ∗ n ). Since (cid:101) T ∗ n j ≤ T ∗ n j for every OWER DEVIATIONS IN β -ENSEMBLES AND LAW OF ITERATED LOGARITHM IN LPP 9 j , from Theorem 1.2 there exists δ > P ( B j ) ≥ δ for all k/ ≤ j ≤ k and all k largeenough , and so (13) and (14) together imply that, P ( A ∩ B τ ) ≥ δ, for large enough k .Now on A ∩ B τ , we have k/ ≤ τ ≤ k and T n τ − ≤ (cid:101) T ∗ n τ − + T ∗ ( τ ) < n τ − + 4( n τ − n τ − ) − C ε n / τ − (log log n τ − ) / = 4 n τ − C ε n / τ − (log log n τ − ) / . Since n τ − = n τ / 2, replacing ε by 2 ε, for all large enough k , we get, P k (cid:91) j = k/ (cid:26) T n j − − n j − n j − / (log log( n j − / < − − / C ε (cid:27) ≥ P ( A ∩ B τ ) ≥ δ, (15)This shows that with probability at least δ ,lim inf n →∞ T n − nn / (log log n ) / < − − / C ε = (1 − ε ) / (2 c ) − / . Indeed, we have P (cid:18) lim inf n →∞ T n − nn / (log log n ) / < − − / C ε (cid:19) = P ∞ (cid:92) k =0 ∞ (cid:91) j = k (cid:26) T j − jj / (log log j ) / < − − / C ε (cid:27) = lim k →∞ P ∞ (cid:91) j = k (cid:26) T j − jj / (log log j ) / < − − / C ε (cid:27) ≥ lim k →∞ P k − (cid:91) j =2 k/ − (cid:26) T j − jj / (log log j ) / < − − / C ε (cid:27) ≥ δ. where the last inequality follows from (15). As this is true for every ε > 0, sending ε to 0 we getthat P (cid:18) lim inf n →∞ Z n (log log n ) / ≤ − (2 c ) − / (cid:19) = P (cid:18) lim inf n →∞ T n − nn / (log log n ) / ≤ − (2 c ) − / (cid:19) ≥ δ, completing the proof. (cid:3) Remark 2.3. If instead of a dyadic breakup with n j = 2 j as in the proof, choosing n j = (cid:100) (1+ η ) j (cid:101) forsome η > , allows us to replace (2 c ) − / in the statement of the proposition by (cid:0) (1 + η − ) c (cid:1) − / ,which, on taking η → ∞ , converges to c − / . Sharpness and possible extensions. We wrap up this section with a discussion on thesharpness of our argument and several possible extensions. The first natural question to determinethe limiting value of lim inf n →∞ Z n (log log n ) / . It is widely believed that for T n , a stronger moderatedeviation estimate than is given by the results of [15] holds. As is well known, it was shown in[12] that 2 − / n − / ( T n − n ) converges weakly to the GUE Tracy-Widom distribution ( β = 2).Comparing with the tails of Tracy-Widom distribution from [21], one can make predictions aboutthe optimal constants for the tail estimates in (9) and (10) which are indeed conjectured to be true.In particular, it is believed thatlog P ( T n ≥ n + xn / ) = − 43 (2 − / x ) / + O ( x n − / ) + O (log x ) (16) This in fact is just a straightforward consequence of the weak convergence result alluded to right after (5). for all large x ≤ δn / and for all n sufficiently large. Similarly, for the lower tail, it is believed thatlog P ( T n ≤ n − xn / ) = − 112 (2 − / x ) + O ( x n − / ) + O (log x ) (17)for all large x ≤ δn / and for all n sufficiently large. Such sharp results have indeed been provedin cases of Poissonian and Geometric last passage percolation using the Riemann-Hilbert approachin [16, 17, 2, 9]. Under the stronger hypothesis (16), Ledoux in [14] showed thatlim sup n →∞ Z n (log log n ) / = 3 / , almost surely . He also conjectured based on the believed bound (17) thatlim inf n →∞ Z n (log log n ) / = − / . (18)(The statistic considered in [14] is 2 − / n − / ( T n − n ) and so the numerical values there differfrom the ones above by a factor of 2 / .) Furthermore, the Borel-Cantelli lemma based argumentin [14] in conjunction with the conjectured bound (17), does yield a lower bound of − / for theLHS in (18).On the other hand in the point-to-line case, as indicated before, it is known (see [4, 23, 8]) that2 − / n − / ( T ∗ n − n ) converges weakly to GOE Tracy-Widom distribution ( β = 1). In analogy with(17), comparing with the left tail of GOE Tracy-Widom distribution, the optimal estimate in thiscase is predicted to be P ( T ∗ n ≤ n − xn / ) = − x 96 (1 + o (1)) , (19)for x (cid:28) n / . Even though we are unaware of such a sharp estimate in the literature, (19), alongwith our arguments will indeed imply thatlim inf n →∞ Z n (log log n ) / ≤ − / almost surely (see Remark 2.3), which is still far from the conjectured value in [14], indicating, notsurprisingly, that dominating the point-to-point passage times by point-to-line counterparts incursa loss in the constant.Going beyond Exponential LPP, Ledoux points out in [14] that his results hold also for LPPon Z with geometrically distributed weights. This is because, the upper bounds for the moderatedeviation probabilities (for both the left and right tails), i.e., analogues of (9), (10) are availablealso for the geometric case (see e.g. [2, 9]) which are the only inputs needed for the argument of[14]. On the other hand, for our argument, we rely on the lower tail bounds for the point-to-linelast passage times and while there does exist an explicit distributional formula for the latter forthe geometric case as well (see [3]), the random matrix connection, as far as we understand, existsonly in the Laguerre limit. While it is possible that using the formula of [3] one can obtain a resultanalogous to Theorem 1.2 for geometric LPP, we are unaware of any such result, rendering ourcurrent arguments inapplicable in the geometric case.Finally, passage times in more general non-axial directions other than along the diagonal werealso considered in [14]. That is, for any fixed γ ∈ (0 , ∞ ), a similar law of iterated logarithm wasproved for the (properly centered) sequence T γn := T , ( n, (cid:98) γn (cid:99) ) . Since our proof relies on point-to-lineestimates, it is not hard to see that the same proof verbatim also yields Theorem 1 in this moregeneral case. However we do not attempt to provide any details.All that is left to be done now is prove Theorem 2 which is accomplished in the following section. OWER DEVIATIONS IN β -ENSEMBLES AND LAW OF ITERATED LOGARITHM IN LPP 11 Lower Deviations in β -Laguerre ensemble: Proof of Theorem 2 As mentioned earlier, for the proof of Theorem 2, we shall rely on a tridiagonal matrix modelfor the β -Laguerre ensemble [10]. To define the tridiagonal matrix, we start by introducing somenotation. We write χ r for the Chi-square distribution with parameter r , and by an abuse of notation,also for a random variable having this distribution. Its density is proportional to x r/ − e − x/ on R + and it has expectation r . Similarly, we write χ r for the random variable (and the distribution)which is the positive square root of a χ r variable. It has density proportional to x r − e − x / and itsexpectation is equal to 2 / Γ( r/ / r/ . (20)We also recall the well-known facts (see e.g. [15] for a reference) that E χ r is increasing in r forall r > E χ r ≤ r / (Jensen’s inequality) and E χ r ≥ (cid:112) r − / r ≥ m ≥ n and β ≥ 1, let X k − = a k , for 1 ≤ k ≤ n, and X k = b k , for 1 ≤ k ≤ n − βa k ∼ χ β ( m +1 − k ) and βb k ∼ χ β ( n − k ) .Given the above, we define the following matrices.(1) Let B = B β be a n × n bi-diagonal matrix with B k,k = a k and B k +1 ,k = b k .(2) Let L = L β = BB T , an n × n positive semi-definite matrix, where B T as usual denotes thetranspose of B .(3) Let M = (cid:20) B T B (cid:21) , a 2 n × n symmetric matrix.(4) Let T be a 2 n × n symmetric tridiagonal matrix with zeros on the diagonal and X =( X , . . . , X n − ) on the super-diagonal and sub-diagonal, i.e. for each i = 1 , , . . . , n − ,T i,i +1 = T i +1 ,i = X i . (22) Fact: The joint density of eigenvalues of L is given by (6), the β -Laguerre ensemble, LE βm,n . Thisis by now well known (see for example [10, Theorem 3.1]). We shall, however, not be relying onthis joint density.We next state and prove a simple lemma relating the eigenvalues of L and M and T. Lemma 3.1. If L has eigenvalues s ≤ . . . ≤ s n (since L is positive semi-definite), then M and T have eigenvalues ± s , ± s , . . . , ± s n .Proof. The characteristic polynomial of M is det( zI n ) det( zI n − z BB T ) = det( z I n − L ). Thisshows that M has eigenvalues ± s k . One can check easily that if we permute the rows and columnsof M in the order n + 1 , , n + 2 , , n + 3 , , . . . , then we get the matrix T . Thus T has the sameeigenvalues as M . (cid:3) Now note that using Lemma 3.1, our Theorem 2 reduces to proving the following result. Forsome c, c (cid:48) , c (cid:48)(cid:48) > < ε < c (cid:48) and all β ≥ P { s n ≤ ( √ m + √ n )(1 − ε ) } ≥ (cid:40) exp {− cβ ( ε √ mn ) } if ε ≥ c (cid:48)(cid:48) √ n √ m ,C β · exp {− cβ ( ε m n ) } if 0 < ε ≤ c (cid:48)(cid:48) √ n √ m . (23)Note that Theorem 2 is a statement about s n , while (23) concerns s n . There is no issue in makingthis change, except that ε changes by a factor of 2; this is safely absorbed in the constant c . As mentioned earlier, the lower bound for the lower tail in the Laguerre case was not addressed in[15]. The main reason why we are able to analyze it is that we do not use BB T (in which the entriesare not independent and are sums of products of χ random variables with different parameters) butthe matrix T which has independent entries. Further, the matrix T is very similar in appearanceto the tridiagonal matrix for the Hermite model. However the proof in [15] for the Hermite modeluses in an essential way the Gaussians on the diagonal, while T has zeros on the diagonal. Hencesome modification is needed in the proof of the lower bound. This idea of linearization is oftenuseful when working with the Laguerre ensembles, and has been used before (see for example, theappendix to [25]).As the following proof is rather technical, before delving into it we provide a brief high leveloverview. The most natural idea would be to condition X k s to be small, and indeed, that togetherwith a simple approximation for the largest eigenvalue works for ε bounded away from 0 (see Section3.4). To treat all values of ε down do the fluctuation scale, one needs to estimate the eigenvaluemore accurately.We achieve this by an appropriate tilting argument. We do a change of measure changing X k sto Y k , s (defined in (32)), so that for the tridiagonal matrix obtained in (22) by replacing the X k sby the Y k s, the largest eigenvalue is typically smaller than ( √ m + √ n )(1 − ε ).The sought lower bound is then obtained by lower bounding the Radon-Nikodym derivative of X k s with respect to Y k s. To achieve the first step, recalling λ n = max (cid:107) w (cid:107) =1 Q ( w ), we shall define aquadratic form Q b (that arises naturally by approximating E Q and completing squares, see below)with Q b ≥ Q and estimate max (cid:107) w (cid:107) =1 Q b ( w ) instead. It turns out (Lemma 3.6) there exists a change ofmeasure from X k s to Y k s which is simply scaling a number of X k s by a factor of √ − ε that givesthe lower bound with the right exponents. We now start with the details.3.1. The quadratic forms and their comparison. The quadratic form corresponding to T − ( √ m + √ n ) I n is Q ( w ) = 2 n − (cid:88) k =1 X k w k w k +1 − ( √ m + √ n ) n (cid:88) k =1 w k . (24)where w = ( w , w , . . . , w n ) . Let ˆ X k = X k − E [ X k ], and for b > Q b ( w ) = 2 n − (cid:88) k =1 ˆ X k w k w k +1 − b √ n n (cid:88) k =0 ( w k − w k +1 ) − b √ m n (cid:88) k =1 ( w k − − w k ) − b √ n n (cid:88) k =1 kw k , (25)where w = w n +1 = 0 by convention. If Z k are centered random variables, define Q b ( w ; Z ) by thesame expression as Q b , except that ˆ X k is replaced by Z k . In particular, Q b ( w ) = Q b ( w, ˆ X ).We briefly describe the motivation for defining Q b ( w ) as above, which naturally arises fromcompleting squares in E Q ( w ). Observe that we have E Q ( w ) := 2 n (cid:88) k =0 E X k − w k − w k + 2 n − (cid:88) k =1 E X k w k w k +1 − ( √ m + √ n ) n (cid:88) k =1 w k . Now, using the approximation (coming from (20)) E X k − ≈ √ m − k ≈ √ m − k √ m and E X k ≈√ n − k ≈ √ n − k √ n and the identities 2 w k w k +1 = w k + w k +1 − ( w k − w k +1 ) we get E Q ( w ) ≈ −√ n n (cid:88) k =0 ( w k − w k +1 ) − √ m n (cid:88) k =1 ( w k − − w k ) − (cid:18) √ n + 12 √ m (cid:19) n (cid:88) k =1 kw k . OWER DEVIATIONS IN β -ENSEMBLES AND LAW OF ITERATED LOGARITHM IN LPP 13 It is now easy to see that (at least for m = n ), Q b ( w ) ≈ ( Q ( w ) − E Q ( w )) + b E Q ( w )and for m > n , Q b ( w ) is even larger. As the approximant of E Q ( w ) is negative, one could reasonablyexpect that for small b , Q b ( w ) would be larger than Q ( w ). This is the content of the next lemma. Lemma 3.2. If b is sufficiently small ( b < suffices), then Q ≤ Q b .Proof. Observe that Q b ( w ) − Q ( w ) = n (cid:88) k =1 V k,k w k + 2 n − (cid:88) k =1 V k,k +1 w k w k +1 where V k,k = √ m + √ n − b √ m − b √ n − bk √ n and V k,k +1 = − E [ X k ] + b √ m for k odd and V k,k +1 = − E [ X k ] + b √ n for k even.That is, if we form the symmetric tridiagonal matrix V with these entries, then( Q b − Q )( w ) = (cid:104) V w, w (cid:105) . Thus, to show that Q b ≥ Q , it suffices to show that V k,k ≥ | V k,k − | + | V k,k +1 | for all 1 ≤ k ≤ n (with the interpretation that V , = 0 and V n, n +1 = 0). In fact, consideringthe different combinations of signs of V k,k − and V k,k +1 , it is sufficient if we have √ m + √ n − b √ m − b √ n − bk √ n ≥ b √ m + b √ n − E [ X k − ] − E [ X k ]; E [ X k − ] + E [ X k ] − b √ m − b √ n ; E [ X k − ] − E [ X k ] − b √ m + b √ n if k is even E [ X k − ] − E [ X k ] − b √ n + b √ m if k is odd . − E [ X k − ] + E [ X k ] + b √ m − b √ n if k is even − E [ X k − ] + E [ X k ] + b √ n − b √ m if k is odd . Notice that the left hand side is at least √ m (1 − b ) + √ n (1 − b ). By using the fact that theexpectation of χ variables increases with its parameter, it follows that the contribution of theexpectation terms is negative in the right hand side of the first, fourth and the fifth inequality.If we ignore these negative terms, what remains is at most b √ m + b √ n . Therefore, all threeinequalities now follow by choosing b ≤ . Using E χ α ≤ √ α it also follows that the right handsides of the third and the sixth inequalities are at most (1 − b ) √ m + b √ n , and these inequalitiesalso follow by taking b ≤ .It remains to prove the second inequality. For this, assume that k = 2 (cid:96) (similar reasoning worksfor odd k ) in which case, invoking the facts following (20) again, the right hand side is at most √ m + 1 − (cid:96) + √ n − (cid:96) − b √ m − b √ n . Since √ m + 1 − (cid:96) ≤ √ m , all we need is that2 (cid:96)b √ n ≤ √ n − √ n − (cid:96) = (cid:96) √ n + √ n − (cid:96) . The right hand side is at least (cid:96) √ n , hence the desired inequality is valid if b < . (cid:3) Exponential tail bound to the right. We now prove a deviation inequality for the uppertail of Q b ( · , · ), which, via Lemma 3.2 also provides a bound on the upper deviation of Q ( · , · ). Asimilar result was proved in [15] for a different but related quadratic form on the way to prove anupper tail deviation inequality for the largest eigenvalue for Laguerre β -ensemble (see [15, Section3.2]). Lemma 3.3. Assume that Z k are independent random variables with zero mean and satisfying E [ e λZ k ] ≤ e cλ for all k ≤ n and some c > and all λ ∈ R . Then for any < ε < , P (cid:26) max (cid:107) w (cid:107) =1 Q b ( w ; Z ) ≥ ε √ m (cid:27) ≤ Ce − c (cid:48) ε / √ mn ( √ ε ∧ ( mn ) / ) (26) where c (cid:48) , C > depends only on c and b .Proof. Define S k = Z + . . . + Z k for 1 ≤ k ≤ n (and S = 0 and S k = S n for k > n ). For p ≥ p ( k ) = max {| S k + j − S k | : 1 ≤ j ≤ p } . By the summation by parts formula ( [15, Lemma 8]), we have for any unit vector w , n − (cid:88) k =1 Z k w k w k +1 = n − (cid:88) k =0 p [ S k + p − S k ] w k +1 w k +2 + n − (cid:88) k =0 (cid:32) p k + p − (cid:88) (cid:96) = k [ S (cid:96) − S k ] (cid:33) w k +1 ( w k +2 − w k ) ≤ p n − (cid:88) k =0 ∆ p ( k )( w k +1 + w k +2 ) + 2 b √ n n − (cid:88) k =0 ∆ p ( k ) w k +1 + b √ n n (cid:88) k =0 ( w k +2 − w k ) ≤ p n − (cid:88) k =0 ∆ p ( k )( w k +1 + w k +2 ) + 2 b √ n n − (cid:88) k =0 ∆ p ( k ) w k +1 + 12 b √ n n (cid:88) k =0 ( w k +1 − w k ) (27)where for the first inequality we bound all S (cid:96) − S k terms by ∆ p ( k ) and then used Cauchy-Schwarzin the form | ∆ p ( k ) w k +1 ( w k +2 − w k +1 ) | ≤ λ ∆ p ( k ) w k +1 + λ ( w k +2 − w k ) , with λ = 18 b √ n. (28)To see the second inequality in (27), write( w k +2 − w k ) ≤ w k +1 − w k ) + 2( w k +2 − w k +1 ) , to see that the last term is at most b √ n (cid:80) nk =0 ( w k +1 − w k ) .Now recalling the definition Q b ( w ; Z ) = 2 n − (cid:88) k =1 Z k w k w k +1 − b √ n n (cid:88) k =0 ( w k − w k +1 ) − b √ m n (cid:88) k =1 ( w k − − w k ) − b √ n n (cid:88) k =1 kw k , and plugging in the above upper bound for (cid:80) n − k =1 Z k w k w k +1 , together with m ≥ n we obtain Q b ( w ; Z ) ≤ p n − (cid:88) k =0 ∆ p ( k )( w k +1 + w k +2 ) + 4 b √ n n − (cid:88) k =0 ∆ p ( k ) w k +1 − b √ n n (cid:88) k =1 kw k = n − (cid:88) k =0 w k +1 (cid:20) p (∆ p ( k ) + ∆ p ( k − b √ n ∆ p ( k ) − b ( k + 1) √ n (cid:21) ≤ max ≤ k ≤ n − (cid:20) p (∆ p ( k ) + ∆ p ( k − b √ n ∆ p ( k ) − bk √ n (cid:21) since (cid:80) k w k = 1 and we define ∆ p ( − 1) = 0. Now, for k ∈ [( j − p + 1 , jp ] and any i , since | S k + i − S k | ≤ | S k − S ( j − p | + | S k + i − S ( j − p | as in [15] we can write, ∆ p ( k ) ∨ ∆ p ( k − ≤ p (( j − p ) , OWER DEVIATIONS IN β -ENSEMBLES AND LAW OF ITERATED LOGARITHM IN LPP 15 and it can separately be verified that the above inequality also holds for the case k = 0 and j = 1.Therefore, Q b ( w ; Z ) ≤ max ≤ j ≤(cid:100) n/p (cid:101) (cid:20) p ∆ p (( j − p ) + 16 b √ n ∆ p (( j − p ) − b ( j − p √ n (cid:21) . (29)Thus it follows that, P (cid:18) max (cid:107) w (cid:107) =1 Q b ( w ; Z ) ≥ ε √ m (cid:19) ≤ n/p (cid:88) j =1 P (cid:18) p ∆ p (( j − p ) − b ( j − p √ n ≥ ε √ m (cid:19) + n/p (cid:88) j =1 P (cid:18) b √ n ∆ p (( j − p ) − b ( j − p √ n ≥ ε √ m (cid:19) , ≤ n/p (cid:88) j =1 P (cid:18) ∆ p (( j − p ) ≥ εp √ m + b ( j − p √ n (cid:19) + n/p (cid:88) j =1 P (cid:18) ∆ p (( j − p ) ≥ bε √ mn + b ( j − p (cid:19) . Using the assumption about the exponential moments of the Z k , and applying Doob’s maximalinequality, we see that for any k ≥ t > 0, by choosing λ = tp P { ∆ p ( k ) ≥ t } ≤ e − c (cid:48) t /p . Thus we get P (cid:18) ∆ p (( j − p ) ≥ εp √ m + b ( j − p √ n (cid:19) ≤ exp (cid:18) − c (cid:48) ε pm − c (cid:48) b ( j − p n (cid:19) and P (cid:18) ∆ p (( j − p ) ≥ bε √ mn + b ( j − p (cid:19) ≤ exp (cid:18) − c (cid:48) bε √ mnp − c (cid:48) b ( j − (cid:19) . The sum of the first bound over j = 1 , . . . , (cid:100) n/p (cid:101) is upper bounded by C n / p / e − c (cid:48) ε pm , while the sum of the second bound is upper bounded by Ce − c (cid:48) bε √ mn/p , using that exp (cid:0) − c (cid:48) b ( j − (cid:1) is summable and bounded independent of m, n and p . We now set p = max( (cid:98) ε − / m − / n / (cid:99) , p > 1, this yields an overall bound of C ( ε / m / n / ) / e − c (cid:48) ε / m / n / + Ce − c (cid:48) ε / m / n / ≤ Ce − c (cid:48) ε / m / n / by increasing the constant C and reducing the constant c (cid:48) suitably. On the other hand, if p = 1,i.e., if ε ≥ √ n √ m , we get, using ε m ≥ n , an overall bound of Ce − c (cid:48) ε / m / n / + Ce − c (cid:48) ε √ mn . Combining the two cases we get P (cid:18) max (cid:107) w (cid:107) =1 Q b ( w ; Z ) ≥ ε √ m (cid:19) ≤ Ce − c (cid:48) ε / √ mn ( √ ε ∧ ( mn ) / ) ;as desired. (cid:3) Remark 3.4. For later purposes, we note that if Y ∼ χ ν and ˆ Y = Y − E [ Y ] then E [ e λ ˆ Y ] ≤ e λ for all λ > and for all ν > . This is the content of [15, Lemma 9] . Hence, if Z k s are independentcentred χ -random variables (not necessarily identically distributed), then Lemma 3.3 is applicable. Remark 3.5. It can be checked using Remark 3.4 and definition of X k s that Z k = ˆ X k we have E [ e λZ k ] ≤ e cλ /β for all λ ∈ R and all k . Tracking the dependence of β throughout the calculations,it follows that P (cid:18) max (cid:107) w (cid:107) =1 Q b ( w ; Z ) ≥ ε √ m (cid:19) ≤ Ce − c (cid:48) βε / √ mn ( √ ε ∧ ( mn ) / ) ; where C, c (cid:48) do not depend on β . Clearly, since β ≥ the β -term can simply be dropped fromthe exponent to get a uniform upper bound. Using Lemma 3.2, one also gets an upper bound for P ( λ n ≥ (1 + ε )( √ m + √ n ) ) of the form Ce − c (cid:48) βε / √ mn ( √ ε ∧ ( mn ) / ) . This recovers the first item of [15, Theorem 2] with possibly different absolute constants. Lower bound for the left tail. We are now in a position to prove (23). Since s n − ( √ m + √ n ) = max (cid:107) w (cid:107) =1 Q ( w ) and Q ≤ Q b for small enough b (by Lemma 3.2), we have that P (cid:16) s n < ( √ m + √ n )(1 − ε ) (cid:17) ≥ P (cid:18) max (cid:107) w (cid:107) =1 Q b ( w ) ≤ − ε ( √ m + √ n ) (cid:19) , (30)and so the following result implies (23). Note that we can replace ε ( √ m + √ n ) by ε √ m as ourprobability estimates are not sharp enough to differentiate ε from 2 ε . Lemma 3.6. Fix < b < / . Then there are constants constant c, C > and m ∈ N such thatfor all m ≥ m ∨ n and all β ≥ P (cid:26) max (cid:107) w (cid:107) =1 Q b ( w ) ≤ − ε √ m (cid:27) ≥ (cid:40) exp {− cβ ( ε √ mn ) } if b √ n √ m ≤ ε ≤ b ,C β · exp {− cβ ( ε m n ) } if < ε ≤ b √ n √ m ∧ b . (31)Observe that Lemma 3.6 completes the proof of Theorem 2 except for the case m ≤ m . However,for m ≤ m , s n < √ m + √ n − ε √ m can simply be ensured by making all (non-zero) entries of thematrix T sufficiently small. Considering the density of χ random variables, it is easy to check thatthe probability of such and event is lower bounded by C βmn ≥ C βm for some (possibly different)constant C (depending on b and m ). This takes care of the remaining case and completes theproof of Theorem 2. Proof of Lemma 3.6. We first assume ε ≥ ε := C m − / n − / , for a C which will be taken to bea sufficiently large absolute constant, chosen appropriately later. We shall choose m sufficientlylarge so that ε will be much smaller that b/ 2. We further divide into two cases depending on therange of ε . Case 1: ε ≤ b √ n √ m ∧ b . In this case we fix K = (cid:100) ε √ mn/ b (cid:101) ≤ n +12 . Let Y k be independent randomvariables such that Y k − d = (1 − ε ) X k − ∼ Gam (cid:18) β ( m + 1 − k )2 , β − ε ) (cid:19) (32)for 1 ≤ k ≤ K , while Y k d = X k for all other k ≤ n − 1. The constraint on ε was used to ensure that K < n (otherwise the definition of Y k does not make sense). In fact, we have that K ≤ ( n + 1) / χ β ( m +1 − k ) distributions of X k are all at least βm/ k ≤ K . Gam( α, κ ) denotes the Gamma distribution with density proportional to x α − e − κx . OWER DEVIATIONS IN β -ENSEMBLES AND LAW OF ITERATED LOGARITHM IN LPP 17 We adopt the following perturbative strategy: Using Y k in place of X k in the definition of Q b ,implies that Q b ( w ) ≤ − ε √ m for all unit vectors w , with probability close to 1. This is because thefirst K of the Y k − random variables have a slightly reduced mean (by about ε √ m ), as comparedto X k − . Then we compute the Radon-Nikodym derivatives of X with respect to Y to get a lowerbound for the probability of the same event under the X k .So replacing X by Y , in (25), consider (cid:101) Q b ( w, Y ) := 2 n − (cid:88) k =1 ( Y k − E [ X k ]) w k w k +1 (33) − b √ n n (cid:88) k =0 ( w k − w k +1 ) − b √ m n (cid:88) k =1 ( w k − − w k ) − b √ n n (cid:88) k =1 kw k = Q b/ ( w, Z ) + 2 K (cid:88) k =1 ( E [ Y k − ] − E [ X k − ]) w k − w k (34) − b √ n n (cid:88) k =0 ( w k − w k +1 ) − b √ m n (cid:88) k =1 ( w k − − w k ) − b √ n n (cid:88) k =1 kw k where Z k = Y k − E [ Y k ].Using Remark 3.5 (it is applicable since Y k s are all scalar multiples of X k s in distribution witha multiplication factor uniformly bounded from above and below), recalling the upper bound of e − c (cid:48) ε / √ mn ( √ ε ∧ ( mn ) / ) from there, and our choice of ε := C m − / n − / , setting C large enough(independent of β ) allows us to ensure that for all ε ≥ ε , P ( Q b/ ( w, Z ) ≤ ε √ m ) ≥ P ( Q b/ ( w, Z ) ≤ ε √ m ) ≥ . . (35)To estimate (cid:101) Q b ( Y ) , in (34) we will use2 w k − w k = w k + w k − − ( w k − − w k ) (36)Furthermore, using the distributional equality in (32), it follows from (20) that − ε √ m ≤ E [ Y k − ] − E [ X k − ] ≤ − ε √ m (37)for all 1 ≤ k ≤ K . Putting these together, we have that with probability 0 . w , (cid:101) Q b ( w, Y ) ≤ ε √ m − ε √ m K (cid:88) k =1 w k − (cid:18) b − ε (cid:19) √ m n (cid:88) k =1 ( w k − − w k ) − bK √ n n (cid:88) k =2 K +1 w k (38) ≤ ε √ m − ε √ m K (cid:88) k =1 w k − ε √ m n (cid:88) k =2 K +1 w k = − ε √ m. (39)The first inequality follows by considering (34), and:(1) Using (35) to bound Q b/ ( w, Z ) . (2) Using (36), and (37) to the term 2 (cid:80) Kk =1 ( E [ Y k − ] − E [ X k − ]) w k − w k to obtain the term − ε √ m (cid:80) Kk =1 w k . (3) Reduce the coefficient in the third term to b − ε for all k. (4) Dropping the term − b √ n (cid:80) nk =0 ( w k − w k +1 ) . (5) Dropping the terms with k ≤ K in the last sum on the RHS and lower bounded k ≥ K +1by 2 K. For the second inequality (39), we have dropped the third term (which can be done since 2 ε < b by hypothesis on ε ). In the final term we use that by choice, bK √ n ≥ ε √ m . Finally using (cid:80) j w j = 1,yields the final equality. Bounding the Radon-Nikodym derivative. For notational convenience we start by definingthe following nice set. A = (cid:40) t ∈ R n − : ∀ w with (cid:107) w (cid:107) = 1 , n − (cid:88) k =1 ( √ t k − E [ X k ]) w k w k +1 − b √ n n (cid:88) k =0 ( w k − w k +1 ) − b √ m n (cid:88) k =1 ( w k − − w k ) x − b √ n n (cid:88) k =1 kw k < − ε √ m (cid:41) . Through the discussion so far leading to (39), we have shown that P { Y ∈ A} ≥ . , where Y := ( Y , . . . , Y n − ) . Similarly we will use X to denote ( X , . . . , X n − ) . Now let g k denote the density of X k and let f k denote the density of Y k . Of course f k = g k except for k = 1 , , . . . , K − 1. Recalling from (32), for convenience, we list here the exact formsof f k − and g k − , for t ≥ ≤ k ≤ K : f k − ( t ) = t β ( m +1 − k ) − exp (cid:16) − βt − ε ) (cid:17) Γ (cid:16) β ( m +1 − k )2 (cid:17) (2(1 − ε )) β ( m +1 − k ) ,g k − ( t ) = t β ( m +1 − k ) − exp (cid:16) − βt (cid:17) Γ (cid:16) β ( m +1 − k )2 (cid:17) β ( m +1 − k ) . Next we substitute these expressions to get, P { X ∈ A} = (cid:90) A K (cid:89) k =1 g k − ( t k − ) f k − ( t k − ) n − (cid:89) k =1 f k ( t k ) dt . . . dt n − = (cid:90) A exp (cid:40) βε − ε ) K (cid:88) k =1 t k − (cid:41) (1 − ε ) β (cid:80) Kk =1 ( m +1 − k ) 2 n − (cid:89) k =1 f k ( t k ) dt . . . dt n − . Now let B = (cid:110) t ∈ R n − : (cid:80) Kk =1 t k − > (1 − ε ) (cid:80) Kk =1 ( m + 1 − k ) (cid:111) . Since E [ Y k − ] = (1 − ε )( m +1 − k ), it follows that P { Y ∈ B} ≥ . . (40)A quick way to see this is to use the normal approximation for Gamma variables. More preciselyby (32): For any K as above, K (cid:88) k =1 Y k − ∼ Gam (cid:32) (cid:80) Kk =1 β ( m + 1 − k )2 , β − ε ) (cid:33) . Now as ε is bounded away from 1 and β is a given constant, uniformly for all K ≤ m , as m → ∞ since (cid:80) Kk =1 β ( m +1 − k )2 goes to infinity, and β − ε ) stays fixed, by the Central Limit Theorem for OWER DEVIATIONS IN β -ENSEMBLES AND LAW OF ITERATED LOGARITHM IN LPP 19 gamma variables with increasing shape parameters and a given scale parameter, we have, P (cid:34) Gam (cid:32) (cid:80) Kk =1 β ( m + 1 − k )2 , β − ε ) (cid:33) ≥ E (cid:32) Gam (cid:32) (cid:80) Kk =1 β ( m + 1 − k )2 , β − ε ) (cid:33)(cid:33)(cid:35) → , and hence in particular (using β ≥ 1) for any large enough m ≥ m , (40) is satisfied. Thus, P { X ∈ A} ≥ (cid:90) A∩B exp (cid:40) βε − ε ) K (cid:88) k =1 t k − (cid:41) (1 − ε ) β (cid:80) Kk =1 ( m +1 − k ) 2 n − (cid:89) k =1 f k ( t k ) dt . . . dt n − , ≥ exp (cid:40) βε − ε ) (1 − ε ) K (cid:88) k =1 ( m + 1 − k ) (cid:41) (1 − ε ) β (cid:80) Kk =1 ( m +1 − k ) P { Y ∈ A ∩ B}≥ 14 exp (cid:32) β K (cid:88) k =1 ( m + 1 − k ))( ε + log(1 − ε )) (cid:33) ≥ e − ε βKm . (41)In the last line we used P { Y ∈ A ∩ B} ≥ . . − > and (cid:80) Kk =1 ( m + 1 − k ) ≤ Km togetherwith ε + log(1 − ε ) ∈ ( − ε , 0) (if 0 < ε < ). By our choice of K we get that this is further lowerbounded by e − cβε m / n / .This is exactly what was asked for in Lemma 3.6, for the case when ε ≤ c (cid:48)(cid:48) (cid:112) n/m for a smallenough c (cid:48)(cid:48) . Case 2: b √ n √ m < ε < b . In this case, we take K = n and define Y k as in (32). Thus all the odd Y k are different from all the odd X k . Proceeding exactly as before, in the analysis of (38) (thefirst place where the definition of K was used) the last term is an empty sum and hence droppedyielding the following instead of (39). (cid:101) Q b ( w, Y ) ≤ ε √ m − ε √ m K (cid:88) k =1 w k − (cid:18) b − ε (cid:19) √ m n (cid:88) k =1 ( w k − − w k ) ≤ − ε √ m. (42)Continuing, we get the same probability bound (41) as before, except that K = n . Thus wearrive at P { X ∈ A} ≥ e − cε βmn ≥ e − cε βmn by increasing the constant c , and using β ≥ ε mn ≥ b n ≥ b . This is what thelemma claims. Finally we address the case: ε ≤ ε = C m − / n − / . P (cid:26) max (cid:107) w (cid:107) =1 Q b ( w ) ≤ − ε √ m (cid:27) ≥ P (cid:26) max (cid:107) w (cid:107) =1 Q b ( w ) ≤ − ε √ m (cid:27) ≥ e − cβC ≥ ( C ) β e − cβε m / n / , for a suitably increased constant c (depending on b ) where C = e − cC . (cid:3) Deviation bounds in the large deviation regime. We finish with a discussion of lowerbounds of the large deviation probabilities in the left tail: i.e., P ( λ n ≤ ( √ m + √ n ) (1 − ε )) where c (cid:48) ≤ ε < 1. Notice that this case is not covered by Theorem 2. For β = 2 case, Johansson [12]obtained that if m = γn , then − n − log P ( λ n ≤ ( √ m + √ n ) (1 − ε )) → J γ ( ε ) Since Gam( α, κ ) = Gam( (cid:98) α (cid:99) , κ ) ∗ Gam( α − (cid:98) α (cid:99) , κ ) , and the first term is a sum of (cid:98) α (cid:99) many i.i.d. Gam(1 , κ ) andGam( α − (cid:98) α (cid:99) , κ ) is a tight random variable. for a large deviation rate function J γ ( · ). We shall briefly describe below how to show a correspondinglower bound for finite n in this regime with a rather simple argument (the upper bound was coveredin [15]).While for this discussion we shall restrict ourselves to the case m = n sufficiently large, and alsoto β = 1, one can use the same argument for all β ≥ m, n with mn is bounded away frominfinity.Recall the 2 n × n tridiagonal matrix T with largest eigenvalue s n where s n = λ n . We use thefollowing well known and easy to prove bound (Gershgorin theorem) s n ≤ max ≤ i ≤ n − ( X i + X i +1 )where X i s are the independent χ variables defined in (22) with the convention that X = X n = 0.It now follows easily that P ( s n ≤ √ n (1 − ε )) ≥ n − (cid:89) i =1 P ( X i ≤ (1 − ε ) √ n ) . Now each term in the product can be lower bounded by a constant (say, ) if E X i ≤ (1 − ε ) n .Using the left tail of χ distribution one can lower bound each of the other terms by e − c ( ε ) n , leadingto a lower bound of the form e − c (cid:48) ( ε ) n for n sufficiently large. A slightly more careful version of theabove calculation yields c (cid:48) ( ε ) ≈ ε if ε is sufficiently small (but still bounded away from 0) thusmatching the lower bound in Theorem 2.Note however, that this approach cannot be carried over to get optimal tails bounds all the wayup to the moderate deviation tails, i.e., ε ≈ n − / . In this case, observe that at least the firstΘ( n / ) many of the terms in the product (cid:81) n − i =1 P ( X i ≤ (1 − ε ) √ n ) are bounded away from 1.Hence this approach gives us a lower bound that decays to 0 at least as fast as e − cn / , much worsethan the constant order lower bound obtained in Theorem 2. References [1] Jinho Baik. 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