One-sided reflected Brownian motions and the KPZ fixed point
OONE-SIDED REFLECTED BROWNIAN MOTIONS AND THE KPZ FIXED POINT
MIHAI NICA, JEREMY QUASTEL, AND DANIEL REMENIKA
BSTRACT . We consider the system of one-sided reflected Brownian motions which is in variationalduality with Brownian last passage percolation. We show that it has integrable transition probabilities,expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that itconverges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling invariant Markov process definedin [MQR17] and believed to govern the long time large scale fluctuations for all models in the KPZuniversality class. Brownian last passage percolation was shown recently in [DOV18] to converge to theAiry sheet (or directed landscape), defined there as a strong limit of a functional of the Airy line ensemble.This establishes the variational formula for the KPZ fixed point in terms of the Airy sheet.
1. RBM
AND THE
KPZ
UNIVERSALITY CLASS
The model of one-sided reflected Brownian motions (RBM for short) is a system of reflectedBrownian motions X t ( N ) ≤ X t ( N − ≤ · · · ≤ X t (1) on R . They start from an ordered initialcondition X ( N ) < X ( n − < . . . < X (1) , perform Brownian motions, and interact with eachother by one-sided reflections: X t ( k + 1) is reflected to the left off X t ( k ) , so that the particles alwaysremain ordered. This process and its variational dual, Brownian last passage percolation, have beenstudied intensively from many perspectives, see [OY01; O’C03; War07] and references therein forsome early work, and [GS15; WFS17; DOV18; Ham19; AOW19] for more recent results. RBM canbe defined formally in several equivalent ways. The easiest is to start with N standard Brownianmotions B t (1) , . . . , B t ( N ) initially at X (1) , . . . , X ( N ) , let X t (1) = B t (1) , and then, recursivelyfor k = 1 , , . . . , construct X t ( k ) by reflecting B t ( k + 1) off X t ( k ) , the reflection R f B t of a Brownianmotion B t off any continuous function f t with f ≥ B being easy to define, e.g. by the Skorokhodrepresentation R f B t = min { inf ≤ s ≤ t ( f s + B t − B s ) , B t } . Since the definition is recursive, it is notdifficult to have N = ∞ ; the first n < N particles don’t even know the other N − n are there. Notealso that using the Skorokhod representation one can let some of the initial positions coincide. Thesystem is alternatively defined by a system of stochastic differential equations involving the joint localtimes (see [AOW19, Sec. 4] and references therein). One can define other types of reflections, orpoint interactions for Brownian motions, but as far as is understood at this point, only this one has theintegrability described in this article and we will not consider other models here.From the Skorokhod representation it is not hard to see (see [WFS17, Eqn. (2.1.4)], but note inthat book the reflections go the opposite way) that RBM are in variational duality with Brownian lastpassage percolation (BLPP): Given a family of independent, standard two-sided Brownian motions ( W k ) k ≥ and t ≤ t (cid:48) in R , m ≤ m (cid:48) in Z , if we define the last passage time G [( t, m ) → ( t (cid:48) , m (cid:48) )] = sup t = t m < ···
2. T
RANSITION PROBABILITIES OF
RBMIn [WFS17], the m (spatial) point distributions of RBM and their asymptotics are computed for a fewspecial initial conditions: Packed (converging to the Airy process), periodic (or flat, Airy ), stationary(Airy stat ), half-periodic (Airy → ), half-Poisson (Airy → BM ), and periodic-Poisson (Airy → BM ). Notealso that, for special initial data, there has been a recent breakthrough in which two time distributionshave been computed [Joh17]. We are now going to give a formula for the m (spatial) point distributionsof RBM for general right finite initial data, at a fixed later time t . These generate the transitionprobabilities, in the same way that finite dimensional distributions define the Wiener measure.Let ∂ denote the derivative operator ∂f = f (cid:48) and ∂ − its formal inverse ∂ − f ( x ) = (cid:90) x −∞ d y f ( y ) , (2.1)which can be thought of as an integral operator with kernel ∂ − ( x, y ) = x>y . Also, for a fixed vector a ∈ R m and indices n < . . . < n m we introduce the operators χ a ( n j , x ) = x>a j , ¯ χ a ( n j , x ) = x ≤ a j (we will use the same notation if a is a scalar, writing χ a ( x ) = 1 − ¯ χ a ( x ) = x>a ). Theorem 2.1.
Consider RBM with initial condition { X ( i ) } ∞ i =1 . For any indices ≤ n < n <. . . < n m , any locations a , . . . , a m ∈ R and any t > , we have P (cid:0) X t ( n j ) ≥ a j , j = 1 , . . . , m (cid:1) = det (cid:0) I − ¯ χ a K RBM t ¯ χ a (cid:1) L ( { n ,...,n m }× R ) , where det is the Fredholm determinant, with K RBM t ( n i , z i ; n j , z j ) = − ∂ − ( n j − n i ) ( z i , z j ) n i Note that the formulas from [WFS17] for special deterministic initial conditions can be recoveredby computing explicitly the hitting law. This is straightforward in the case of packed initial data: here X ( k ) = 0 for each k ≥ , so inside the expectation in (2.4) we have τ = 0 if z ≥ , in whichcase B τ = z , and otherwise τ = ∞ ; using this together with (3.6) and (3.7) in (2.3) leads directly totheir formula after removing the conjugation e z j − z i . When m = 1 this is the classic Hermite kernel,and the distribution is the same as that of the largest eigenvalue of an n × n GUE random matrix[GTW01; Bar01], a remarkable fact establishing the key link between random matrix theory andrandom growth models, and providing much of the motivation for the study of this particular model.For the half-periodic case X ( k ) = − k , k ≥ (which after a limit leads also to the full periodic initialcondition) it turns out to be simpler to use the biorthogonal representation of the kernel, (5.1) below,together with Prop. 5.6; see [MQR17, Ex. 2.10] for the analogous computation in the case of TASEP.3. F ROM RBM TO THE KPZ FIXED POINT The KPZ fixed point is a Markov process on the space UC of upper semi-continuous functions h : R −→ R ∪ {−∞} satisfying h ( x ) ≤ A | x | + B for some A, B < ∞ , with the topology oflocal Hausdorff convergence of hypographs. It is shown in [MQR17] that it is the limit of the 1:2:3rescaled TASEP height functions, h ( t , x ; h ) = lim ε → ε / (cid:2) h ε − / t (2 ε − x ) + ε − / t (cid:3) , as long as h ( x ) = lim ε → ε / h (2 ε − x ) , all in the sense of the topology of UC, in probability; here we areusing h ( t , x ; h ) to denote the state of the Markov process at time t given initial state h . The KPZ fixedpoint is conjectured to be the universal limit under such scalings for models in the KPZ universalityclass (see [MQR17] and references therein for more background on the KPZ fixed point). Our nexttheorem proves this for RBM.Before stating the result we present the precise definition of the KPZ fixed point through its transitionprobabilities (for simplicity we present here the formula which uses K lim from [MQR17, Eqn. (3.21)]rather than the one appearing in the main results of that paper). For x ∈ R , t > let S − t , x ( u ) = t − / e x t + z xt Ai ( t − / u + t − / x ) , which is the integral kernel of the operator e x ∂ − t ∂ . For h ∈ UC with h ( x ) = −∞ for x > , let S epi ( − h − ) − t , x ( v, u ) = E B (0)= v (cid:2) S − t , x − τ ( B ( τ ) , u ) τ < ∞ (cid:3) , where τ is now defined as the hitting time of the epigraph of − h − by B ( x ) , a Brownian motion withdiffusion coefficient , with h − ( x ) = h ( − x ) . For such initial data, the KPZ fixed point kernel reads K FP t ( n i , · ; n j , · ) = − e ( x j − x i ) ∂ x i > x j + ( S − t , x i ) ∗ S epi ( − h − ) − t , − x j , (3.1)and the transition probabilities for the KPZ fixed point h ( t , x ) are defined through their finite dimen-sional distributions, which are given by Fredholm determinants, P h ( h ( t , x ) ≤ a , . . . , h ( t , x m ) ≤ a m ) = det (cid:0) I − ¯ χ − a K FP t ¯ χ − a (cid:1) L ( { x ,..., x m }× R ) . (3.2)The process is statistically spatially invariant, so the corresponding formula for h ∈ UC with h ( x ) = −∞ for x > x are easily recovered. Such data are dense in UC, and it is shown in [MQR17] theprobabilities are continuous functions of h ∈ UC. So the general formula can be obtained from (3.2)by approximation (see [MQR17, Thm. 3.8]).Consider a family of initial conditions X ( ε )0 for RBM satisfying, for some h ∈ UC, − ε / ( X ( ε )0 ( − ε − x ) − ε − x ) −−−→ ε → h ( x ) (3.3)in distribution in UC, where the left hand side is interpreted as a linear interpolation to make it acontinuous function of x ∈ R . The left hand side of (3.3) has the interpretation as a kind of inversefunction of the height function (which we have not defined here). The scaling limits look at perturbationsof height functions from flat, so the limit of the inverse function naturally picks up a minus sign. Notethat the convergence (3.3) requires a far more restrictive lower bound on the growth of X ( − (cid:96) ) , (cid:96) > than the one used after (1.1) to show that the minimum is achieved; it is being assumed to grow linearly. NE-SIDED REFLECTED BROWNIAN MOTIONS AND THE KPZ FIXED POINT 5 These conditions correspond to upper bounds needed on the initial data for the KPZ fixed point toprevent blowup: Once the initial data grows quadratically a blowup occurs in finite time. We assumelinear upper bounds on UC because it is a nice class where the solution stays and exists for all time.Let X ( ε ) t denote RBM with this initial data and define the 1:2:3 rescaled RBM, X ( ε ) t ( x ) = ε / (cid:0) X ( ε ) ε − / t ( ε − / t − ε − x ) + 2 ε − / t − ε − x (cid:1) . Theorem 3.1. Assume that the initial data satisfies (3.3) . Then, for each t > , in UC in distribution, − X ( ε ) t ( x ) −−−→ ε → h ( t , x ; h ) . (3.4)At the level of convergence of kernels, the proof is analogous to the proof of the convergence of theTASEP kernels in [MQR17], but in this case uses the standard convergence of Hermite polynomials toAiry functions. Our goal is to study the limit of the kernel ε − / (cid:101) K RBM t ( n i , z i ; n j , z j ) z i ≤− ˜ a i , u j ≤− ˜ a j defined in terms of (cid:101) K RBM t from (2.3) with t = ε − / t , n i = ε − / t − ε − x i , z i = − ε − / t + 2 ε − x i + ε − / u i and η = ε − / v and ˜ a i = − ε − / t + 2 ε − x i − ε − / a i (the ε − / in front of the kernel comes from the z i changeof variables). In view of (3.2), our goal is to show that this kernel converges in a suitable way to K FP t ( x i , u i ; x j , u j ) u i ≤− a i , u j ≤− a j Note that the change of variables transforms the indicator functionsin the desired manner.Recall first that e y − x ∂ − ( x, y ) is the transition probability for the exponential random walk B k .Thus, under this scaling and for x i > x j , the first term on the right hand side of (2.3) becomes − ε − / times the probability density for the walk B k to go from ε − x i + ε − / u i to ε − x j + ε − / u j intime ε − ( x i − x j ) which, as needed, converges to − e ( x i − x j ) ∂ ( u i , u j ) by the Central Limit Theorem.Define S ε − t , x ( v, u ) = ε − / e − t S − t, − n ( η, z ) = ε − / e − t e η − z ψ n ( t, η − z ) , ¯ S ε − t , − x ( v, u ) = ε − / e t ¯ S − t,n ( η, z ) = ε − / e t e z − η ¯ ψ n ( t, η − z ) so that, after scaling, the second term on the right hand side of (2.3) reads ( S ε − t , x i ) ∗ ¯ S ε, epi ( − h − ) − t , − x j with ¯ S ε, epi ( − h − ) − t , − x j ( v, u ) = ε − / E ε / B = v (cid:104) ¯ S ε − t , − x j − ετ ( ε / B τ , u ) τ Let h = d ba and let X ( ε )0 be their approximations prescribed above. Then (3.4) holds.Proof. We need to prove the convergence of the scaled RBM kernel in trace norm. For simplicity wewill only prove this for the kernel corresponding to one-point distributions, i.e. the kernel (cid:101) K ( n ) , RBM t coming from (2.3) (after scaling) with n i = n j = n . The only difference in the general case is that, inorder for the first term on the right hand side of (2.2) to converge in trace class, an additional conjugationby a multiplication operator is needed, but this conjugation does not affect the convergence of the otherterm; see [MQR17, Rem. B.5], where the same conjugation is employed in the proof of the convergenceof the TASEP kernels. Additionally, for notational simplicity we will take all b k ’s to be ; the extensionto general b k ’s is straightforward, as will be clear from the proof. We will write l k = (cid:100)− ε − a k (cid:101) .Consider first the single narrow wedge case, (cid:96) = 1 , for which X ( ε )0 ( i ) = ∞ for ≤ i < − ε − a and X ( ε )0 ( i ) = 2 ε − a for i ≥ − ε − a . Since the walk B k takes strictly negative steps, τ < n if andonly n > l and B l > − ε − a , in which case τ = l . Then, recalling again that the transition matrixof the walk B k is Q exp ( x, y ) := e y − x ∂ − ( x, y ) , we have ¯ S epi ( X ) − t,n = ( Q exp ) l ¯ S − t,n − l l 4. F ROM RBM TO THE A IRY SHEET VARIATIONAL FORMULA By coupling copies of TASEP starting with different initial conditions, and using compactness, the Airy sheet A ( x , y ) = h (1 , y ; d x ) + ( x − y ) is defined in [MQR17] as a two parameter process, where we have started the KPZ fixed point with theUC function d x ( x ) = 0 , d x ( u ) = −∞ for u (cid:54) = x ; the narrow wedge at x . This leads to the variationalformula h ( t , x ; h ) dist = sup y ∈ R (cid:8) t / A ( t − / x , t − / y ) − t ( x − y ) + h ( y ) (cid:9) . (4.1)The equality is in distribution, as functions of x , for fixed t .More generally, one can consider the space-time Airy sheet or directed landscape [CQR15; DOV18], A ( s , x , t , y ) = h ( t , y ; s , d x ) + ( x − y ) t − s , where h ( t , y ; s , ¯ h ) denotes the KPZ fixed point at time t starting at ¯ h at time s . This gives, for any s ≥ and any ¯ h ∈ UC, h ( t , x ; s , ¯ h ) dist = sup y ∈ R (cid:8) A ( s , x ; t , y ) − ( x − y ) t − s + ¯ h ( y ) (cid:9) , (4.2)with equality in distribution in the space of continuous functions of t ∈ [ s , ∞ ) into UC (or, if oneprefers, in the space of continuous functions of t ∈ ( s , ∞ ) into Hölder − functions.)The disadvantage of resorting to compactness arguments is that they cannot tell one that the limitingobject is defined uniquely and all one can conclude is that the variational expressions (4.1) and (4.2) NE-SIDED REFLECTED BROWNIAN MOTIONS AND THE KPZ FIXED POINT 8 hold for any limit point. In [DOV18] this is overcome by showing that the sheet is a (non-explicit)functional of the Airy line ensemble [CH14]. Since the Airy line ensemble is defined uniquely as adeterminantal point process, the uniqueness of the Airy sheet follows as a consequence. The followingresult follows from the construction of the Airy sheet in [DOV18] together with the relation betweenRBMs and Brownian last passage percolation. Proposition 4.1. In the setting of Thm. 3.1, assume now that ε / ( X ( ε )0 (2 ε − x ) + 2 ε − x ) → − h ( − x ) for some finitely supported h ∈ UC , meaning that h ( x ) = −∞ for x outside some finite interval.Then for each t > , − X ( ε ) t ( x ) −−−→ ε → sup y ∈ R (cid:8) A (0 , x ; t , y ) − ( x − y ) t + h ( y ) (cid:9) (4.3) in distribution (locally) in UC .Proof. For simplicity we only consider the case t = 1 , the general case follows easily from the usual1:2:3 scaling arguments. Additionally, by spatial invariance we may assume without loss of generalitythat the support of h is contained in ( −∞ , , so that we can work in the setting of RBM with a firstparticle X t (1) as in the previous section. Consider the scaling t = ε − / , n = ε − / − ε − x , so that X ( ε )1 ( x ) = ε / (cid:0) X ( ε ) t ( n ) − t − n ) and (1.1) yield − X ( ε )1 ( x ) = max ≤ (cid:96) ≤ n (cid:110) ε / (cid:0) − X ( ε )0 ( (cid:96) ) + (cid:96) (cid:1) + ε / (cid:0) G [(0 , (cid:96) ) → ( t, n )] + t + ( n − (cid:96) ) (cid:1)(cid:111) . Changing variables (cid:96) (cid:55)→ − ε − y this becomesmax y ∈ [ x − ε − / , ε ] (cid:110) − X ( ε )0 ( − y ) + ε / G [(0 , − ε − y ) → ( ε − / , ε − / − ε − x )]+ 2 ε − − ε − / ( x − y ) (cid:111) . X ( ε )0 ( − y ) converges in distribution, locally in UC, to − h ( y ) , and since h ( y ) = −∞ for y outsidesome finite interval [ a , b ] , then as ε → we obtain the same thing by optimising over the set [ a − , b + 1] . Let s = − ε / y and s (cid:48) = 1 − ε / x , to getmax y ∈ [ a − , b +1] (cid:110) − X ( ε )0 ( − y ) + ε / G [( ε − / s + 2 ε − y , ε − / s ) → ( ε − / s (cid:48) + 2 ε − x , ε − / s (cid:48) )]+ 2 ε − − ε − / ( x − y ) (cid:111) . The three variables s , s (cid:48) , y vary over a compact set as ε → , so assuming that x also does, using thethat the convergence in [DOV18, Thm. 1.5] is uniform, we deduce (4.3) in distribution, locally inUC. (cid:3) We can now fill in the gap between [DOV18] and [MQR17] that results from the limiting objects in[MQR17] having been defined as limits from TASEP and those in [DOV18] having been derived fromBLPP. Since the KPZ fixed point is defined through its transition probabilities, there is no ambiguityin its definition. So Prop. 3.2 shows that the right hand side of (4.3) is given by the KPZ fixed point h ( t , x ; h ) for multiple narrow wedge data (defined before Prop. 3.2). Since such data are dense in UC,and both sides of (4.2) are continuous on UC, we have that (4.1) holds for all h ∈ UC. At the sametime, a second group [DNV+] are filling the gap from the other side, proving that the result in [DOV18]can also be obtained from exponential last passage percolation, which is in variational duality withTASEP. Since the Airy sheet is obtained as the same functional of the Airy line ensemble, which isunique, again there is no ambigiuty, and TASEP converges to the unique Airy sheet of [DOV18]. Eitherroute leads to the main result: Corollary 4.2. The KPZ fixed point Markov process constructed in [MQR17] and the (unique) Airysheet/directed landscape constructed in [DOV18] are related by the variational formula (4.2) . NE-SIDED REFLECTED BROWNIAN MOTIONS AND THE KPZ FIXED POINT 9 5. F ROM TASEP TO RBM5.1. Biorthogonal ensemble for RBM. Recall (2.1) the operator ∂ − f ( x ) = (cid:82) x −∞ d y f ( y ) , the nota-tion being consistent with the fact that, when restricted to a suitable domain, ∂ − is the inverse of thederivative operator ∂ . Recall also that ∂ − can be regarded as an integral operator with integral kernel ∂ − ( x, y ) = x>y ; more generally, it is easy to check that ∂ − m := ( ∂ − ) m has integral kernel ∂ − m ( x, y ) = ( x − y ) m − ( m − x>y . The next result can be derived by following the biorthogonalization approach introduced in [Sas05;BFPS07] for TASEP in the case of RBM. The proof is also contained implicitly in the proof of [FSW15,Prop. 4.2], see also [WFS17, Lem. 3.5]. Or it can alternately be derived by taking the low density limitof the result for TASEP from [Sas05; BFPS07]. Theorem 5.1. Consider RBM with initial condition { X ( i ) } ∞ i =1 . For any indices ≤ n < n <. . . < n m , any locations a , . . . , a m ∈ R and any t > , P (cid:0) X t ( n j ) > a j , j = 1 , . . . , m (cid:1) = det (cid:0) I − ¯ χ a K RBM t ¯ χ a (cid:1) L ( { n ,...,n m }× R ) with K RBM t ( n i , x i ; n j , x j ) = − ∂ − ( n j − n i ) ( x i , x j ) n i By (5.11) we have Ψ nk ( x ) = ( − n t n/ h n ( t, x − X ( n − k )) w t ( x − X ( n − k )) with h n ( t, · ) the scaled Hermite polynomials h n ( t, x ) = H n ( x/ √ t ) , which are orthogonal with respect tothe Gaussian weight w t ( x ) = (2 πt ) − / e − t x . Hence (ignoring the prefactor ( − n t n/ ) the problemof finding the Φ nk ’s can be rephrased as follows:For fixed n > , and given a family of shifted Hermite functions ( f k ) k =0 ,...,n − , f k ( x ) = h n ( t, x − X ( n − k )) w t ( x − X ( n − k )) , find a family of polynomials ( g k ) k =0 ,...,n − , with g k of degree k , which is biorthogonal to ( f k ) k =0 ,...,n − .The challenge in such problems is to actually find the biorthogonal functions Φ nk in a form whichis useful. Our strategy here is to compute them formally as a limit of the corresponding biorthogonalfunctions found for TASEP in [MQR17] and then simply check that the result satisfies (1) and (2)above.5.2. TASEP. N -particle TASEP was solved by Schütz [Sch97] using the coordinate Bethe ansatz,which leads to a formula for the transition probabilities given as the determinant of an explicit N × N matrix (an analogous formula can be written for RBM, see [War07, Prop. 8], and is the starting pointin the derivation of Thm. 5.1). However, and as in Thm. 5.1, one is usually interested in studying m -point distributions of the process for arbitrary m ≤ N , and moreover in obtaining formulas which aresuitable for taking N → ∞ . [Sas05; BFPS07] realized that this can be achieved by rewriting Schütz’sformula in terms of a (signed) determinantal point process on a space of Gelfand-Tsetlin patterns andemploying the Eynard-Mehta technology [EM98] to derive a Fredholm determinant formula for the m -point distributions. The result is precisely the TASEP version of Thm. 5.1, but we will not need tostate it explicitly. Instead, we will state a version of this result which follows from [MQR17], where thebiorthogonalization is performed explicitly. NE-SIDED REFLECTED BROWNIAN MOTIONS AND THE KPZ FIXED POINT 10 In order to state the result we need to introduce some additional notation. Define kernels Q ( x, y ) = x>y , Q − ( x, y ) = x = y − − x = y . They can be regarded as operators acting on suitable functions f : Z −→ R , so for example Qf ( x ) = (cid:80) y We have stated the last result in a slightly different way than (but equivalent to) [MQR17].First, while in that paper the TASEP kernels were conjugated by x in order to connect them directlyto certain probabilistic objects (basically a random walk with Geom [ ] steps), here we will omit thatconjugation; this will allow us to state formulas in terms of slightly simpler kernels which are availablein the continuous space setting of RBM. Second, the TASEP biorthogonal functions Φ nk , which solve adiscrete space version of (1) and (2) of Thm. 5.1, were expressed in [MQR17] in terms of the solutionof an initial–boundary value problem for a discrete backwards heat equation, while here we write downthis solution explicitly; this allows us to compute the limiting Φ nk ’s very easily.5.3. Brownian scaling limit of the TASEP biorthogonal functions. We compute now the limit ofthe TASEP formulas under the scaling (1.2). The limits can be proved rigorously but since we don’tneed it, we will just state them and explain how they arise (see comments just after Thm. 5.1.)Since Q is a discrete integration operator, it is not surprising that after scaling it converges to ∂ − .In fact, we have for any t ∈ R and m ∈ Z ≥ that κ − ( m − / Q m ( √ κ x + κt, √ κ y + κt ) −−−→ κ →∞ ∂ − m ( x, y ) , as can be checked for instance by using the explicit formula Q m ( x, y ) = (cid:0) x − y − m − (cid:1) x ≥ y + m . The limitcan be extended suitably to all m ≤ to get that, after scaling, Q − m converges to ∂ m . Consider nextthe Poisson semigroup ( e − t ∇ − ) t ≥ . We are interested in the scaling √ κ e − κt ∇ − ( √ κ x, √ κ y + κt ) ,which is simply √ κ times the probability that a Poisson random variable with parameter κt equals √ κ ( y − x ) + κt . By the Central Limit Theorem, √ κ e − κt ∇ − ( √ κ x, √ κ y + κt ) −−−→ κ →∞ e t∂ ( x, y ) . (5.4) NE-SIDED REFLECTED BROWNIAN MOTIONS AND THE KPZ FIXED POINT 11 Combining the above two facts leads directly to the following: For k < n , and replacing t by κt andtaking X ( n − k ) = √ κ X ( n − k ) in (5.2), κ k/ Ψ nk ( √ κ x + κt ) −−−→ κ →∞ ∂ k e t∂ δ X ( n − k ) ( x ) = Ψ nk ( x ) . We turn now to the Φ nk ’s. For the functions h nk which are used to construct them (see (5.3)) we have κ − ( k − (cid:96) ) / h nk ( (cid:96), √ κx ) −−−→ κ →∞ h nk ( (cid:96), x ) (5.5)where h nk ( (cid:96), x ) is given by h nk ( k, z ) = 1 and h nk ( (cid:96), z ) = (cid:90) X ( n − (cid:96) ) x d y h nk ( (cid:96) + 1 , y ) for (cid:96) < k. (5.6)Each h nk ( (cid:96), · ) is a polynomial of degree k − (cid:96) . We want to use (5.5) to write a limit for Φ nk . This functionis defined by applying e t ∇ − to h nk (0 , · ) , and from (5.4) we have, formally, that under the scaling we areinterested in, e t ∇ − should converge to e − t∂ . Hence κ − ( k/ Φ nk ( √ κ x + κt ) −−−→ κ →∞ e − t∂ h nk (0 , · ) . Remark 5.5. The backwards heat kernel appearing above does not make sense in general, but in thissetting it is applied to the polynomial h nk (0 , · ) , in which case its action can be defined by expanding itas a (finite) power series. Furthermore, one can check that the group property e s∂ e t∂ p = e ( s + t ) ∂ p holds for any s, t ∈ R and any polynomial p .The preceeding computations suggest the following result, which we prove directly. It is worthnoting how simple the proofs are using this method once one has the candidate biorthogonal functions. Proposition 5.6. The functions Φ nk defined through (1) and (2) of Thm. 5.1 are given explicitly by Φ nk = e − t∂ h nk (0 , · ) . Proof. The fact that Φ nk is a polynomial of degree at most n − follows from the above discussion andthe same fact for h nk (0 , · ) . For the biorthogonality, we note first that by (5.6) we have the simple identity ∂ k h n(cid:96) (0 , X ( n − k )) = ( − k k = (cid:96) . Using this and the comment made in Rem. 5.5 we compute (cid:104) Ψ nk , Φ n(cid:96) (cid:105) L ( R ) = (cid:104) ∂ k e t∂ δ X ( n − k ) , e − t∂ h n(cid:96) (0 , · ) (cid:105) L ( R ) = (cid:104) ∂ k δ X ( n − k ) , h n(cid:96) (0 , · ) (cid:105) L ( R ) = ( − k ∂ k h n(cid:96) (0 , X ( n − k )) = k = (cid:96) as desired, where we have used the formula (cid:104) δ ( k ) x , f (cid:105) L ( R ) = ( − k f ( k ) ( x ) for δ ( k ) x the k -th distribu-tional derivative of δ x . (cid:3) Representation as hitting probabilities. The next step is to represent K RBM t in terms of hittingtimes, thus producing a formula which is nicely set up for the 1:2:3 KPZ scaling limit.Let ( B ∗ k ) k ≥ denote a random walk with Exp [1] steps to the right; its transition matrix is Q ∗ exp with Q exp ( x, y ) = e y − x ∂ − ( x, y ) the transition matrix of the walk B k introduced earlier. We claim thatbelow the “curve” ( X ( n − (cid:96) )) (cid:96) =0 ,...,n − defined by the initial data, h nk ( (cid:96), · ) can be represented as ahitting probability, h nk ( (cid:96), x ) = e X ( n − k ) − x P B ∗ (cid:96) − = x (cid:16) τ (cid:96),n = k (cid:17) , x < X ( n − (cid:96) ) , (5.7)with τ (cid:96),n = min { (cid:96) ≤ k ≤ n, B ∗ k ≥ X ( n − k ) } . This can be proved easily using (5.6) and the formula P B ∗ (cid:96) − = z ( τ (cid:96),n = k ) = (cid:82) X ( n − (cid:96) ) z d y e − ( y − z ) P B ∗ (cid:96) = y ( τ (cid:96),n = k ) , valid for (cid:96) < k . Next define G ,n ( x , x ) = n − (cid:88) k =0 ∂ − ( n − k ) δ X ( n − k ) ( x ) h nk (0 , x ) , NE-SIDED REFLECTED BROWNIAN MOTIONS AND THE KPZ FIXED POINT 12 so that K RBM t ( n i , · ; n j , · ) = − ∂ − ( n j − n i ) n i Let ( B k ) k ≥ be a random walk taking Exp [1] steps to the left and τ = min { k ≥ B k ≥ X ( k + 1) } . For any x , x , G ,n ( x , x ) = E B = x (cid:104) e x − B τ ¯ ∂ ( − n + τ ) ( B τ , x ) τ We have G ,n ( x , x ) = (cid:80) n − k =0 P B ∗− = x ( τ ,n = k ) P B ∗ = X ( n − k ) ( B ∗ n − k = x ) e x − x for x < X ( n ) by (5.7) and the definition of B ∗ k . On the other hand, P B ∗ = X ( n − k ) ( B ∗ n − k = x ) = (cid:82) ∞ X ( n − k ) d η e X ( n − k ) − η P B ∗ k = η ( B ∗ n − = x ) while, by the memoryless property of the exponential, P B ∗− = x ( τ ,n = k, B ∗ k ∈ d η ) = P B ∗− = x ( τ ,n = k ) e X ( n − k ) − η d η for η ≥ X ( n − k ) . Thus G ,n ( x , x ) = e x − x E B ∗− = x (cid:104) ( Q ∗ exp ) n − τ ,n ( B ∗ τ ,n , x ) τ ,n MN and JQ were supported by the Natural Sciences and Engineering ResearchCouncil of Canada. DR was supported by Programa Iniciativa Científica Milenio grant numberNC120062 through Nucleus Millenium Stochastic Models of Complex and Disordered Systems, and byConicyt through the Fondecyt program and through Basal-CMM Proyecto/Grant PAI AFB-170001. 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Probab. Reflected Brownian motions in the KPZ universalityclass . Vol. 18. SpringerBriefs in Mathematical Physics. Springer, Cham, 2017, pp. vii+118. (M. Nica) D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF T ORONTO , 40 S T . G EORGE S TREET , T ORONTO ,O NTARIO , C ANADA M5S 2E4 E-mail address : [email protected] (J. Quastel) D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF T ORONTO , 40 S T . G EORGE S TREET , T ORONTO ,O NTARIO , C ANADA M5S 2E4 E-mail address : [email protected] (D. Remenik) D EPARTAMENTO DE I NGENIERÍA M ATEMÁTICA AND C ENTRO DE M ODELAMIENTO M ATEMÁTICO (UMI-CNRS 2807), U NIVERSIDAD DE C HILE , A V . B EAUCHEF ORRE N ORTE , P ISO 5, S ANTIAGO , C HILE E-mail address ::