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Optimal Tail Estimates for Directed Last Passage Site Percolation with Geometric Random Variables

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Optimal Tail Estimates for Directed Last Passage Site Percolation with Geometric Random Variables © 2001 International Press Adv. Theor. Math. Phys. 5 (2001) 1207-1250 Optimal tail estimates for directed last passage site percolation with geometric random variables Jinho Baik, Percy Deift, Ken T-R McLaughlin, Peter Miller, and Xin Zhou Deparment of Mathematics, Princeton University, Princeton, NJ 08544, USA Institute for Advanced Study, Princeton, NJ 08540, USA [email protected] Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA deift @ cims. ny u. edu Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA [email protected] Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA [email protected] Department of Mathematics, Duke University, Durham, NC 27708, USA [email protected] Abstract e-print archive: http://xxx.lanl.gov/math.PR/0112162 1208 BAIK, DEIFT, MCLAUGHLIN, MILLER, AND ZHOU In this paper, we obtain optimal uniform lower tail estimates for the prob- ability distribution of the properly scaled length of the longest up/right path of the last passage site percolation model considered by Johansson in [12]. The estimates are used to prove a lower tail moderate deviation result for the model. The estimates also imply the convergence of moments, and also provide a verification of the universal scaling law relating the longitudinal and the transversal fluctuations of the model. 1 Introduction In [12], Johansson considered directed last passage site percolation on Z^_ = {(m, n) : m,n G N} with geometric random variables. More precisely, for (i,j) € Z+, let w(i,j) be independent, identically distributed geometric random variables with lP(t«(t,j) = fe) = (l-*2)(*2)*, fc = 0,l,2,---, (1.1) and 0 < t < 1. An up/right path TT from (1,1) to (M,N) is, by definition, a collection of sites {(ik,jk) : fc = 1,2, • • • ,n}, n := M + N — 1 such that (ii,ji) = 1 (1,1), (in,jn) = (M,A0 and (ik+ujk+i) - {hjk) = (1,0) or (0,1). Let (1,1) Z (M,iV) be the (finite) set of all such up/right paths from (1,1) to (M,N). Now define the maximal 'length', G(M,N) := max{ £ w(i,j) : TT e (1,1) / (M,N)}. (1.2) (*,i)G7r Fix 7 > 1 and set M = [yN], the integer part of 7iV. The main result in [12] is the following: for any x e M, T /G([7iV],iV)-f N \ ^ { LN'» s.)-rw. 0.3) where ao (1 4) = t((7 + 1^+2/7)' - and 2 3 2/3 b0 = *^I^l(t + >/7) / (l + tV7) . (1-5) and where F(x) is the Tracy-Widom distribution [23] for the largest eigenvalue of a random matrix chosen from the Gaussian unitary ensemble (GUE). In addition to (1.3), Johansson also proved large deviation results, lim -llogP(G([7iV],iV) < N(--y)) = -l(y), (1.6) lim -J-logP(G([7iV],iV) > N(-+y)) = -i(y), (1.7) OPTIMAL TAIL ESTIMATES... 1209 for some explicit positive functions £(y) and i(y), y > 0. The result (1.3) parallels an earlier result in [1] on the length of the longest increasing subsequence £N(^) of a random permutation a of N letters. The main result in [1] is the following: for any x £ M, where again F(x) is the Tracy-Widom distribution appearing in (1.3). The authors in [1] also proved the convergence of moments: if Xiv(cr) :— i^^i/QN and X is a random variable with distribution function F(x), then for ra = 0,1,2, • • •, Km E0$) = E(xm). (1.9) In earlier work other authors proved large deviation results for £/v, lim ^ logP(^ < VN(2 - y)) = -H(y), (1.10) JV-»oo iv lim -L]ogV{l > VN(2 + y)) = -I(y), (1.11) N-¥oo YiV N for y > 0, where H(y),I(y) are certain explicit positive functions. The result (1.10) is due to Deuschel and Zeitouni [9] and the result (1.11) is essentially due to Seppalainen [21]. In two recent papers, [18] [17], the authors have considered £N in the moderate deviation regime. More precisely, for 0 < a < |, they showed [18] that for y > 0 7V->oo y3j\fl-3a 12 V ' and [17] 1. logF(£N>(2 + yN-«)VN) _ 4 iV^o yS/2N(l-3a)/2 " 3' ^-^J These moderate deviation results can be motivated by noting that F(x) - ex3/12 as x -> -00, (1.14) and l-^)-6^^ as x -^+00. (1.15) Thus from (1.8), one anticipates that as N -> 00, a 1 3- 1 P(^Ar < (2 - yN~ )y/N) = P(lN < 2VN - (yTV / ")^ ^) 1 3 1 16 ~ log^-J/TV / -") ~ -l2/3ivl-3a) ( - ) 1210 BAIK, DEIFT, MCLAUGHLIN, MILLER, AND ZHOU and a 1 6 P(^v > (2 + yN- )VN) = P(tN > 2y/N + (yN^^N ^ ) ~ log(l - FiyN1'3-")) ~ -tyWNl1-3""2. (1'17) Of course, when a = 0, we are in the large deviation regime, and when a = |, we are in the GUE central limit theorem regime. The above moderate deviation results can also be motivated by estimating the functions I(y) and H(y) for the large-deviation regime. In [17], the authors proved (1.13) by refining certain estimates in [1] and using a careful summation argument. In [18], the authors utilized an analogous summation argument together with the estimate Lemma 6.3 (ii) in [1], Calculations similar to (1.16), (1.17), motivate the following moderate deviation results for G{[jN],N): for 0 < a < f, logP(G([7iV], AT) < (i - yboN'a)N) 1 lim —ri^V ^- = -7^ (1-18) and logV{G{[>yN],N)>{±+yboN-°)N) 4 jfe, yB/ajyi-sa/a = -3- (L19) One of the principal goals in this paper is to prove (1.18). Relation (1.19) is slightly simpler and can also be approached using the techniques in this paper. We hope to return to this problem in the future. Relation (1.18) is a consequence of the following result. Theorem 1.1. Fix 0 < t < 1 and 70 > 1. Then there exist a (large) constant L > 0 and a (small) constant S > 0, such that for large N, logP(G([7iV], JV) < —N - xboN1/3) = -^-x3 + 0(x4iV-2/3) + O(logz) (1.20) QJQ ±2, uniformly for all L < x < 5N2/3 and 1 < 7 < 7o- In particular, for the variables £,7 in the same range, F(G([7iV], N) < —N - xboNl/3) < e-c^3 (1.21) do for some constant c > 0. Setting x = yN2/3~a in (1.20), we immediately obtain (1.18) together with error estimates. Corollary 1.2 (Estimate for lower moderate deviation). ForO < a < | and y > 0, as N -> 00, logF(G([7JV],iV)<(^-i/boJV-)iV) __!_,„, „_„. , n(log{yN*"-°U (1.22) OPTIMAL TAIL ESTIMATES... 1211 Theorem 1.1 can also be used for other applications. Corollary 1.3 (Convergence of moments). Forj > 1, setOw := b ^1/3°°— and let x be the random variable with distribution function F(x) as above. Then for m = 0,1,2, • • •, lim E((9^) = E(xm). (1.23) Proof By Remark 2.5 of [12], (1.23) follows from the estimate (1.21). □ In particular, setting m = 2, we see that the fluctuation y/Var(G([yN], N)) of G is of order N11 where 77 = |. It is believed (see e.g., [16]) that the transversal fluc- tuations of G have order iV^ where £ and 7] are related by a dimension-independent universal scaling law 2£ = 77 + 1. In other words, it is expected that in our case f = f • In [13], Johansson considered transversal fluctuations for the Poissonized version of the longest increasing subsequence problem and showed in that case that the scaling law 2£ = 77 + 1 is satisfied. By [1], 77 is again 1/3, and Johansson [13] verified the scaling law by showing directly that £ = 2/3. A key role in his analysis was again played by Lemma 6.3 (ii) of [1]. This Poissonized problem can be viewed as a continuum version of the above site percolation problem and in Remark 1.2 of [13] Johansson notes that the scaling law 2£ = 77 + 1 for the site percolation problem would follow from an estimate of type (1.21) above. The modifications in the argument in [13] that are needed for the site percolation problem are detailed in [14]. We thus have Corollary 1.4 (Transversal fluctuations). For any 7 > 1, the above coeffi- cients 77 and £ for longitudinal and transversal fluctuations of the site percolation model obey the scaling law 2f = fj + l. (1.24) In order to prove Corollary 1.3, 1.4, weaker bounds than (1.21) suffice. Indeed, using an observation of Harold Widom [24] (see in particular Lemma 2), it is possible to prove the bound 1 3 c, 3/2 IP(G([7iV]5 N) < —N - xboN ' ) < e- l*l (1.25) ao for £,7 in the range of Theorem 1.1, for some constant c' > 0. As opposed to the proof of (1.21), which requires a steepest-descent Riemann-Hilbert analysis (see below), the proof of (1.25) uses only classical steepest-descent methods. A key role in [24] is played by a special case of a beautiful conjecture of Widom for the spectral properties of a class of singular integral operators (see [24]). Although the general conjecture is still unproved, in our case, as in the case considered by Widom in [24], the conjecture can be verified by using an elegant formula of Borodin and Okounkov [4] (see identity (4.9)).
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