Stationary Gap Distributions for Infinite Systems of Competing Brownian Particles

Consider the infinite Atlas model: a semi-infinite collection of particles driven by independent standard Brownian motions with zero drifts, except for the bottom-ranked particle which receives unit drift. We derive a continuum one-parameter family of product-of-exponentials stationary gap distributions, with exponentially growing density at infinity. This result shows that there are infinitely many stationary gap distributions for the Atlas model, and hence resolves a conjecture of Pal and Pitman (2008) in the negative. This result is further generalized for infinite systems of competing Brownian particles with generic rank-based drifts.


Introduction and main results
Consider a system of infinitely many Brownian particles on the real line: X i (t), i = 1, 2, . . ., t ≥ 0. Assume we can rank them from bottom upward at any time t ≥ 0: X (1) (t) ≤ X (2) (t) ≤ . . ., and they satisfy the following system of SDEs: dX i (t) = 1 X i (t) = X (1) (t) dt + dW i (t), i = 1, 2, . . . , where W 1 , W 2 , . . . denote independent Brownian motions. In plain English, the bottom particle moves as a Brownian motion with drift one, and all other particles move as driftless Brownian motions. This system of Brownian particles is called the infinite Atlas model, for the bottom particle supporting all other particles "on its shoulders", as the ancient Atlas hero.

Infinite systems of competing Brownian particles
Although the main interest of our work is the infinite Atlas model (1.1), our result can be naturally generalized to more general systems of competing Brownian particles. In this subsection, we rigorously define these infinite systems. Finite systems of competing Brownian particles are defined very similarly in Section 2.
Hereafter, standard Brownian motion refers to a one-dimensional Brownian motion with zero drift and unit diffusion coefficient. Throughout this paper, we operate on a filtered probability space (Ω, F, (F t ) t≥0 , P) with the filtration satisfying the usual conditions, and fix independent standard Brownian motions W 1 , W 2 , . . . with respect to the filtration (F t ) t≥0 . Definition 1.1. Assume X = (X(t), t ≥ 0) is an R ∞ -valued adapted process such that X(t) = (X i (t)) i≥1 is rankable for every t ≥ 0, each coordinate X i = (X i (t), t ≥ 0) is a.s. continuous, and dX i (t) = ∞ k=1 1 p X(t) (k) = i g k dt + dW i (t), i = 1, 2, . . .
Remark 1.4. The gap process Z = Z(t) is invariant under adding a drift g ∞ dt to each named particle. Therefore, the conditions (1.5) is readily generalized to Similarly, the condition (1.6) is generalized to the condition g n0 = g n0+1 = . . . = g ∞ .

Main result
The question of current interest is to find stationary distributions for the gap process Z(t). Let us first rigorously define this concept. Take an infinite system X of competing Brownian particles with drift coefficients g 1 , g 2 , . . .; let Z be its gap process.
Definition 1.5. A probability measure π on R ∞ + is called a stationary gap distribution or a quasi-stationary distribution for the system X if there exists in the weak sense a unique in law version of (1.2) with Z(0) ∼ π, and for this version we have: Z(t) ∼ π for every t ≥ 0.
Let Exp(λ) denote the exponential distribution with mean λ −1 , i.e. having density λe −λx dx, x > 0. The following stationary distribution of the gap process of the Atlas model (1.1) was derived by Pal and Pitman [PP08]: Exp(2). (1.7) Samples from this distribution are configurations of particles on R + of roughly uniform density 2, where the value 2 arises from the balancing between the unit drift g 1 = 1 and the push-back from the crowd of particles, as heuristically explained in [Ald03]. It was further shown in [DT15] that, under (1.7), each ranked particle Y k (t) typically deviates O(t 1/4 ) from its starting location Y k (0) for large t.
Here, we provide a one-parameter family of stationary gap distributions π a , with drastically distinct behaviors: the density grows exponentially as x → ∞ and each rank particle Y k travels linearly in time (in expectation). Denote the average of the first n drift coefficients by g n : g n := 1 n (g 1 + . . . + g n ) . (1.8) The following is the main result of this paper.
Theorem 1.6. Consider an infinite system of competing Brownian particles from (1.2) with drift coefficients satisfying (1.5). Take any real number a such that a > −2 inf n≥1 g n .
(1.9) (a) The following measure π a is supported on V, and is a stationary distribution for the gap process: Exp (2(g 1 + . . . + g n ) + na) .
(1.10) (b) If Z(0) ∼ π a : the system is in this stationary distribution, then We now provide some important special cases of the general systems considered in Theorem 1.6. Example 1.7. Infinite Atlas model: g 1 = 1, and g k = 0 for k ≥ 2. Then inf n≥1 g n = 0, so for a > 0, we have the following family of stationary distributions: Exp(2 + na).
(1.11) Example 1.8. Independent Brownian motions: g 1 = g 2 = . . . = 0, so inf n≥1 g n = 0, and for a > 0 we have the following family of stationary distributions: Exp(na). Example 1.9. The "inverted Atlas" model, where the bottom particle has negative drift: . . = 0. Then inf n≥1 g n = −1, and for a > 2 we get: Exp(−2 + na). Remark 1.10. Actually, the condition (1.5) does not play a crucial role in the proof of Theorem 1.6. More precisely, under the weaker condition sup |g n | < ∞, our proof of Theorem 1.6 still applies for constructing a copy of the infinite system with Z(t) ∼ π a for all t ≥ 0. The stronger condition (1.5) is assumed merely to ensure that the solution to (1.2) is unique in law, so that the notion of stationary gap distribution is well-defined. Theorem 1.6 shows that the stationary gap distributions for systems of competing Brownian particles (and in particular for the infinite Atlas model) are not unique. In fact, as we further show in Appendix A, the distributions π a are mutually singular for different values of a. This result in particular resolves the conjecture [PP08, Conjecture 2] of Pal and Pitman in the negative. As mentioned previously, for any a satisfying (1.9), the distribution π a exhibits exponentially growing density as x → ∞. To see why this is true, assuming the condition (1.6) for simplicity, for (ζ k ) ∞ k=1 ∼ π a , we note that For the discrete-time analogue of independent Brownian particles from Example 1.8, quasi-stationary distributions of the type π a already appeared in the study of the Sherrington-Kirkpatrick model of spin glasses [RA05]. Such a distribution arises naturally for independent Brownian particles. However, it is far from obvious that similar quasi-stationary distributions should appear in the context of competing Brownian particles, since rank-based drifts introduce complicated dependence among particles.
Rather, the product-of-exponential distribution π a arises from the study of Reflected Brownian Motion (RBM). We give a heuristic derivation of the distribution π a using RBM in the infinite-dimensional positive orthant R ∞ + in Section 1.5. To justify this heuristic derivation (i.e. to prove Theorem 1.6) requires taking a sequence of finite systems of competing Brownian particles with suitable drift coefficients (g k,N ) N k=1 and showing that the sequence converges to the infinite system. Even for the Atlas model, where g 1 = 1 and g 2 = g 3 = . . . = 0, we need to construct g k,N that varies in a suitable way over k = 2, . . . , N , in order to simulate the pressure caused by the exponentially dense particles at x → ∞; see (2.7). This is in sharp contrast with the derivation of the measure π (1.7), where (g k,N ) N k=1 can be taken to be (1, 0, . . . , 0).
Theorem 1.6 further demonstrates a sharp contrast between finite and infinite systems of competing Brownian particles, regarding the criteria for having stationary gap distributions. For a finite system to have a stationary gap distribution, the stability condition g k > g N , k = 1, . . . , N − 1 (1.12) must hold (see Proposition 2.2), as (1.12) imposes a "crowding" mechanism on the rank particles. On the other hand, for an infinite systems, the stationary gap distribution π a may exist even without any form of crowding mechanisms from the drifts. As we see in Example 1.8, the drifts are not in effect. In Example 1.9, the drifts introduce a "repelling" mechanism-the opposite of a crowding mechanism. The sharp contrast between finite and infinite systems is due to the additional crowding effect, in infinite systems, caused by pressure from exponentially growing density under π a .

Conjectures
Here we state some conjectures related to Theorem 1.6. First we recall that, for more general systems of competing Brownian particles than the Atlas model, [Sar16a] derived the following stationary gap distribution Exp (2(g 1 + . . . + g k )) .
(1.15) By Theorem 1.6(a), we have ζ ∈ V a.s. Combining this with (1.15) yields ζ ∈ V a.s. This stationary gap distribution (1.13) generalizes the distribution (1.7) for the Altas model. Here we use the notation π 0 to unify notation with (1.10). Note that under the conditions (1.5) and (1.14), we necessarily have inf n g n = 0. With this, under the preceding notations, π a is a stationary gap distribution for all a ∈ [0, ∞) = R + , including a = 0. We now conjecture that, the mixtures of these measures, over different values of a ∈ R + , exhaust all stationary gap distributions: Conjecture 1.12. Under the conditions (1.5) and (1.14), any stationary gap distribution of an infinite system of competing Brownian particles is of the following form, for some probability measure ρ on R + : Remark 1.13. For the discrete time analog of the driftless system (i.e. g 1 = g 2 = . . . = 0), [RA05] has already proven the analogous statement as in Conjecture 1.12. Driftless systems differ from the systems considered in Conjecture 1.12 in that the former does not satisfies the condition (1.14). Consequently, driftless systems lack stationary gap distribution of the type π 0 , and the statement in [RA05] involves only the parameter a > 0.
A natural open problem following Theorem 1.6 is the large time behavior of each rank particle Y k (t). In view of Theorem 1.6(b), here we conjecture: Conjecture 1.14. Fix (g n ) n≥1 and the parameter a as in Theorem 1.6. Initiating the system of competing Brownian particles at the configuration X 1 (0) = 0 and (Z k (0)) ∞ k=1 ∼ π a , we have that, for any fixed k ∈ Z >0 , See also [Tsa17, Corollary 1.3] for a related result on the infinite Atlas model in the stationary gap distribution π a for a > 0.

Motivation and literature review
The Atlas model and the more general systems of competing Brownian particles are models of interest in mathematical finance. In particular, finite systems of competing Brownian particles (with rank-based drifts and rank-based diffusion coefficients) were introduced in [BFK05] for the purposes of stock market modeling. Weak existence and uniqueness in law for these systems follows from the earlier work of [BP87]. Specific applications to mathematical finance include the study of: stability of the capital distribution [CP10], market models with splits and mergers [KS16], and portfolio optimization in [JR15]. Furthermore, finite systems of competing Brownian are of interest in their due to their intruding mathematical features. There has been extensive study on various aspects of their properties, including: deriving the unique stationary gap distribution [PP08,BFIKP11]; weak convergence to this stationary distribution [IPS13,Sar15a]; the stochastic monotonicity [Sar15]; small noise limits [JR14]; propagation of chaos [JM08]; refined properties of two dimensional systems [FIKP13]; and the question of triple collision (when three or more particles occupy the same position at the same time) [IK10,IKS13,BS15,Sar15b]. The last question is important because the strong solution of a finite system of competing Brownian particles is only proved to exist until the first triple collision, [IKS13].
In addition to their role in mathematical finance, systems of competing Brownian particles arise as the continuum limit of exclusion processes [KPS12], and also serve as a discrete analogue of a nonlinear diffusion governed by a McKean-Vlasov stochastic differential equation. In fact, a nonlinear diffusion can be approximated by finite systems competing Brownian particles, see [Shk12,JR13,Rey15,DSVZ16].
Infinite systems arise as natural models of large systems. Specifically, infinite systems of competing Brownian particles were first introduced in [PP08] for a special case of the infinite Atlas model, and later in [Shk11,IKS13] for the general case, as well as in [Sar15c] for two-sided systems X = (X n ) n∈Z . Existence and uniqueness were established in [Shk11,IKS13,Sar16a]. As mentioned previously, these infinite models exhibit stationary gap distributions π 0 from (1.13) (in particular, π from (1.7) for the infinite Atlas model) of the desired product-of-exponential form. This is shown in [PP08] for the infinite Atlas model and in [Sar16a] for general systems. In the latter paper [Sar16a], the question of weak convergence of Z(t) as t → ∞ was also studied. As models of large systems, the infinite Atlas model is naturally related to a certain stochastic partial differential equation [DT15]. Furthermore, as mentioned previously, the driftless system already appeared in the description of the infinite volume limit of the Sherrington-Kirkpatrick model. See [RA05,AA09,Shk11] and the references therein.
There are several generalizations of these models: systems of competing Lèvy particles, [Shk11,Sar16]; competing Brownian particles on the positive half-line, [Shk11,IKP13] (in the former paper, these are called regulated systems); competing Brownian particles with elastic collisions, [FIK13,FIKP13]; the case of asymmetric collisions, when particles behave after collision as if they had different mass, [KPS12]; second-order models, where drift and diffusion coefficients depend on both name and rank of the particle, [BFIKP11,FIK13].

A heuristic derivation of π a
Here we give a heuristic derivation of the measure π a , explaining how it arises from the theory of Reflected Brownian Motion (RBM). We shall not give detailed definition of an RBM here, and instead refer the readers to the classical survey [Wil95]. Recall from [Sar16a] that, under conditions of Proposition 1.2, the system Y = (Y k ) k≥1 of ranked particles solves the following infinite system of SDEs: (1.17) , and R is the reflection matrix a tridiagonal matrix given by For finite-dimensional RBM in the orthant, a sufficient condition for having product-ofexponential stationary distributions is the skew-symmetry condition (see, e.g. [Sar16a, Proposition 2.1] or [Wil95]). It is straightforward to verify that finite dimensional truncations of (1.17) (i.e. (2.2) in the following) satisfy the skew-symmetry condition, and have the stationary distribution given by (1.19) which gives rise to the measure π 0 in (1.13). This solution, however, is not unique: solving for the null vector Rη = 0, we have which yields η = (1, 2, 3, . . .). With this, we have the following general solution to (1.18): λ := λ * + aη, i.e. λ k := 2(g 1 + g 2 + . . . + g k ) + ka, with the extra condition (1.9) on a to ensure that each component of λ is positive. The solution (1.20) then suggests that π a should be also be a stationary distribution of Z.

Organization
In Section 2, we introduce finite systems of competing Brownian particles together with the necessary tools, and define the finite systems that will be used to prove Theorem 1.6. In Section 3, we prove Theorem 1.6 by establishing the convergence of the finite systems to the corresponding infinite system. Appendix A is devoted to establishing properties of the measure π a mentioned in Section 1.2.

Acknowledgements
We would like to thank Michael Aizenman, E.

Finite systems of competing Brownian particles
To define a finite system of competing Brownian particles, we fix N ≥ 2 to be the number of particles, and let g 1 , . . . , g N denote the drift coefficients. Here p x ( · ) : {1, . . . , N } → {1, . . . , N } denote the analogous ranking permutation for x ∈ R N , which is unique by resolving ties in the lexicographic order. Note that unlike in infinite dimensions, any x ∈ R N is rankable.
In the sequel we will also need to consider the dynamics for the ranked particles Y k . To this end, we let L (k,k+1) = (L (k,k+1) (t), t ≥ 0) be the local time process at zero of Z k , for k = 1, . . . , N − 1, and call L (k,k+1) the local time of collision between the ranked particles Y k and Y k+1 . For consistency of notation, we let L (0,1) (t) ≡ 0 and L (N,N +1) (t) ≡ 0. It was shown in [BG08,BFIKP11] that the dynamics of ranked particles is given by where the following processes are i.i.d. standard Brownian motions: Our strategy of proving Theorem 1.6 is to approximate the infinite system (1.2) by certain finite systems. To this end, let us recall the following result (proved in [PP08, BFIKP11, Sar16a]) on the necessary and sufficient condition for the existence of stationary gap distributions for finite systems.
Proposition 2.2. Recall the notation g k from (1.8). There exists a stationary distribution for the gap process if and only if the stability condition (1.12). In this case, this stationary distribution is unique and is given by Exp (2k (g k − g N )) .
(2.5) Now, let us define the finite systems that will be used in the proof of Theorem 1.6. For every m ≥ 2, we let X (m) = (X  (2.11) The assumption (1.9) ensures that λ (m) k > 0, for m = 1, . . . , m 2 − 1. This, by (2.9), is equivalent to g  denote the corresponding ranked particles and the gap process for the system X (m) . We initiate X (m) at the stationary gap distribution π (m) a , (2.12). That is, we let
Applying [Sar16a, Lemma 4.5], we complete the proof that the distribution π a is supported on V.
Step 2. For n ≥ n, we let [x] ↓n : (x 1 , . . . , x n ) → (x 1 , . . . , x n ) denote the projection onto the first n coordinates. Fixing arbitrary n and T ∈ R + , our goal is to show that [X (m) ] ↓n converges to a limit process [X] ↓n as m → ∞, such that X solves (2.1) and has a stationary gap distribution given by π a . Toward this end, we will need to truncate the large system (X (m) k ) m 2 k=1 at some fixed dimension. This is done with the help of the following lemma. Hereafter, to simplify notation, we use the letter c for any generic positive constant that depends only on g 1 , g 2 , . . ., a and T . Slightly abusing notation, we use the same letter c even if there are multiple such constants within the same formula. (3.2) Remark 3.2. The following proof actually applies even if the term k −2 in (3.2) is replaced by k − , for arbitrarily large , but doing so makes various constants depend also on .
Here we prove (3.2) only for = 2 as it suffices for our purpose.
Proof. Throughout this proof, for R-valued random variables X, Y , the notation X Y means that X stochastically dominates Y , and likewise for X Y . Define the standard Gaussian density and the tail distribution function: We begin by showing (3.1). Since |g k | ≤ g * < ∞, with b m defined in (2.8), we have that b m → −a/2 as m → ∞. This implies that {b m } m≥1 is bounded, and hence there exists a constant g * * such that for all m ≥ 2, k = 1, . . . , m 2 , |g (m) k | ≤ g * * < ∞.
where W (t) is a standard Brownian motion.
This concludes the desired bound (3.1). We now turn to the proof of (3.2). Similarly to the preceding, here we have defined in (2.10)-(2.11), we clearly have that λ (3.8) We consider the cases k ≤ k * and k > k * separately. For the former, as x → Ψ(x) is decreasing and ξ k > 0, we bound the r.h.s. of (3.8) by 2Ψ( −g * * T −u √ T ). By (3.6), the last expression is bounded by ce cu , so This concludes the desired inequality (3.2) for k ≤ k * .
The case k > k * requires more refined estimates. Fixing k ∈ {k * + 1, . . . , m 2 }, we begin by establishing a bound on the lower tail of ξ k . To this end, we consider the "truncated" variable  Recall that ζ j ∼ Exp( c * j), and E ζ j = ( c * j) −1 . With ξ k defined in (3.9), using (3.7), we calculate this function (3.10) explicitly as defined for all |v| < c * (k * + 1). We further express this as (3.11) To bound the r.h.s. of (3.11), apply Taylor's formula f (y) = f (0)+f (0)y + y 0 (y −z)f (z)dz with f (y) = log(1 + y) − y to obtain Apply this inequality for y = v/( c * j) in (3.11), for j = k * + 1, . . . , k. With ∞ j=1 j −2 < ∞, we obtain f k (v) ≤ e cv 2 for |v| ≤ c * k * . Combine the result with the Chernov bound to obtain P(| ξ k − E( ξ k )| ≥ x) ≤ e −xv+cv 2 , and substitute in v = c * k * . We arrive at (3.12) This yields a tail bound on the variable ξ k . To relate the bound back to a lower tail bound on ξ k , we use ξ k ≥ ξ k , followed by using (3.12), whereby obtaining Further, as E( ξ k ) and E( ξ k ) differ by E( ξ k * ) ≤ c, we conclude (3.13) Going back to proving (3.2), we let F k (x) := P( ξ k ≤ x) and G k (x) := 1 − F k (x) denote the cumulative distribution function and the tail distribution function of ξ k , respectively. Let µ k := E( ξ k ) denote the expected value. By (3.8) we have

Apply integration by parts
) < 0, we drop the last term in (3.14). Further using F k (µ k ) + G k (µ k ) = 1, summing (3.14)-(3.15), we obtain: (3.16) Next, to further bound the r.h.s. of (3.16), we first note that (3.17) Using this and (3.6), we bound the first term on the r.h.s. of (3.16) as (3.18) (Here we put c 2 just to clarify that the second inequality follows by making the constant in the exponential smaller.) As for the integral term in (3.16), we use the tail estimate (3.13) to bound F k (x) by ce − c * k * (µ k −x) , and replace the integral over (−∞, µ k ] by an integral over the entire R, followed by the change of variable µ k −x √ T → x. This yields The last integral is explicitly evaluated to be e ( c * k * √ T ) 2 /2 e − c * k * (µ k −g * * T −u) ≤ ce − c * k * (µ k −u) . Combining this with the estimate of µ k (3.17), followed by using c 2 * k * ≥ 2, we conclude (3.19) Inserting (3.18)-(3.19) into (3.16), we conclude the desired estimate (3.2) for k > k * .
Step 3. We now return (3.21) The limit process X = (X i ) i≥1 is taken to be independent of T and n by a standard diagonal argument.
Step 4. We now proceed to show that X has gap distribution π a . Fix T > 0 and n = 1, 2, . . .
Fix an arbitrarily small ε > 0. With n, T already being fixed, we now choose a sufficiently large u ∈ R + such that P X (3.23) Since ∞ k=1 (e −c(log k−u) 2 + + k −2 e cu ) < ∞, the last expression in (3.23) clearly tends to zero as n → ∞. Fix some n ≥ n such that P X (m) That is, with probability at least 1 − ε/2, none of the name particles X n+1 , X n+2 , . . . ever reaches a level below u within [0, T ]. Let R : R n → R n , R(x) := (x px(i) ) n i=1 , denote the ranking map of an n-tuple. By (3.22) and (3.24), we have that Namely, with probability at least 1 − ε, to obtain the first n ranked particles Y (3.25) a , letting m → ∞ we further obtain that We have thus concluded that the gap process Z(t) of the system X is distributed according to π a for all t ∈ R + .
Step 5. Finally, we still need to show that X solves (1.2). Doing so requires first showing that Y solves the corresponding equation (1.16). Indeed, the ranked process Y (m) solves the following finite system of SDEs: Note that although we take (W k ) ∞ k=1 to be fixed (i.e. independent of m), the Brownian motions B Performing this m → ∞ procedure inductively for k = 2, 3, . . ., we further obtain where L (k,k+1) (t) is defined inductively through the following relation Next, each L (k,k+1) is continuous, nondecreasing, and starts from zero: L (k,k+1) (0) = 0. This is so because each L (m) (k,k+1) has all these properties, and they are preserved in limits under the uniform topology of C[0, T ]. Furthermore, L (k,k+1) increases only when Y k = Y k+1 . To see this, we consider a generic t ∈ [0, T ] such that Y k (t) < Y k+1 (t). By the continuity of Y k (t) and Y k+1 (t), we must also have Y k (s) < Y k+1 (s) for s ∈ [t , t ], for some t < t ∈ [0, T ]. With this, for all large enough m, we have Y Letting m → ∞ yields L (k,k+1) (t ) = L (k,k+1) (t ), which proves that L (k,k+1) increases only when Y k = Y k+1 . With the aforementioned properties of L (k,k+1) , we conclude that L (k,k+1) is the local time of collision between Y k and Y k+1 , and hence that Y solves (1.16).
We now return to proving that X solves (1.2). This is done by taking the m → ∞ limit of the finite system of equations (2.6) similarly to the way we did it for Y (m) . However, unlike (3.26), the diffusion coefficients in (2.6) are generally discontinuous due the exchange of ranks. We resolve this problem following [Sar16a], by first showing: Stationary gap distributions Lemma 3.3. Define the following random set: Then P-a.s. the Lebesgue measure of N is equal to zero.
Proof. As Y solves (1.16) (as proven above), the desired result follows once we show that the infinite system (1.16) can be reduced to a finite system at any given level u ∈ R. More precisely, fixing arbitrary u ∈ R and T ∈ R + , we aim at showing inf 0≤t≤T Y k (t) < u, for only finitely many k, a.s.
With this Lemma 3.3, the rest of the proof follows by the same argument to the end of the proof of [Sar16a, Theorem 3.3], starting from [Sar16a, Lemma 6.5] up to the end of the proof of this theorem. That is, as m → ∞, the solution X (m) k of the finite system (2.1) converges to a solution of the infinite system (1.1).

Proof of Part (b)
Fix t ∈ R + and an integer k. It follows from Proposition 2.2 that E Y     k (t) ∼ Exp(λ k ) for k ≤ m, (3.31) clearly follows once we prove it for k = 1. Proceeding to prove (3.31) for k = 1, we recall that denotes stochastic dominance. Apply comparison techniques from [Sar15, Corollary 3.7] to obtain that, for all m ≥ 1, Y (m) 1 (t) g 1 t + W (t), where W is some standard Brownian motion. From this it follows that sup m≥1 E Y (m) k (t) + 2 ≤ E (g 1 t + W (t)) + 2 < ∞. (3.34) Combining this with (3.2), followed by using e −c(u+log k) 2 ≤ e −cu 2 −c(log k) 2 , we obtain P(Y   (t) has an exponential lower tail which is uniformly in m, so in particular sup m≥1 E(Y

A
Here we provide bounds on the number of particles under the measure π a . Proposition A.1. Fix g 1 , g 2 , . . . satisfying the condition (1.6) and fix a ∈ R satisfying (1.9). Let 0 = ξ 1 < ξ 2 < ξ 3 < . . . ∈ R + be a random configuration of points with the gap distribution (ζ k := ξ k+1 − ξ k ) ∞ k=1 ∼ π a . Let N (x) := # {i ≥ 1 | ξ i ≤ x} denote the random number of ξ k -particles on the interval We next show that the measures π a are all mutually singular for different values of a.
Proposition A.2. Fixing g 1 , g 2 , . . . satisfying the condition (1.5) and a > a > −2 inf nḡn , we have that the measures π a and π a are mutually singular.
Under the conditions (1.5) and a, a > −2 inf n g n , it is straightforward to show that 1 n n k=1 log 2kg k + ka 2kg k + ka → log a a .
Adding this to (A.3) yields 1 n n k=1 log Z k 2kg k + ka → µ + log a a , π a -a.s. This, with a = a , concludes that π a and π a are mutually singular.