Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations
OOrthogonal polynomial duality ofboundary driven particle systems andnon-equilibrium correlations
Simone Floreani , Frank Redig , Federico Sau July 17, 2020
ABSTRACT
We consider symmetric partial exclusion and inclusion processes in a generalgraph in contact with reservoirs, where we allow both for edge disorder and well-chosen site disorder. We extend the classical dualities to this context and thenwe derive new orthogonal polynomial dualities. From the classical dualities, wederive the uniqueness of the non-equilibrium steady state and obtain correlationinequalities. Starting from the orthogonal polynomial dualities, we show univer-sal properties of n -point correlation functions in the non-equilibrium steady statefor systems with at most two different reservoir parameters, such as a chain withreservoirs at left and right ends. K EYWORDS — Interacting Particle Systems, Boundary Driven Systems, Duality, OrthogonalPolynomial Duality, Non-Equilibrium Stationary Measure, Non-Equilibrium Stationary Corre-lations, Symmetric Exclusion Process, Symmetric Inclusion Process.
Exactly solvable models have played an important role in the understanding of fundamentalproperties of non-equilibrium steady states such as the presence of long-range correlations andthe non-locality of large deviation free energies [6, 7, 2, 12]. An important class of particlesystems which is slightly broader than exactly solvable models are the models which satisfyself-duality or, more generally, duality properties. Such systems when coupled to appropriatereservoirs are dual to systems where the reservoirs are replaced by absorbing boundaries, and thecomputation of n -point correlation functions in the original system reduces to the computationof absorption probabilities in a dual system with n particles. Even when these absorptionprobabilities cannot be obtained in closed form, e.g. when Bethe ansatz is not available, still theconnection between the non-equilibrium system coupled to reservoirs and the absorbing dualturns out to be very useful to obtain macroscopic properties such as the hydrodynamic limit,fluctuations, propagation of chaos and local equilibrium (see e.g. [13, 15, 18]).In recent works (self-)duality with orthogonal polynomials has been studied in several particlesystems including generalized symmetric exclusion processes (SEP), symmetric inclusion process Delft Institute of Applied Mathematics, TU Delft, Delft, The Netherlands. s.fl[email protected]. Delft Institute of Applied Mathematics, TU Delft, Delft, The Netherlands. [email protected]. Institute of Science and Technology, IST Austria, Klosterneuburg, Austria. [email protected]. a r X i v : . [ m a t h . P R ] J u l SIP) and associated diffusion processes such as the Brownian momentum process. Orthogonalpolynomials in the occupation number variables are a natural extension of the higher ordercorrelation functions studied in SEP in [7]. Orthogonal polynomial duality is very useful inthe study of fluctuation fields [1], identifies a set of functions with positive time dependentcorrelations and is useful in the study of speed of relaxation to equilibrium [4]. So far, orthogonalpolynomial duality has not been obtained in the context of boundary driven systems.In this paper we start extending the classical dualities from [3] for a generalized class ofboundary driven systems, where we allow both for edge disorder and well-chosen site disorder.We then use a symmetry of the dual absorbing system in order to derive duality with orthogonalpolynomials for these systems.More precisely, we consider three classes of interacting particle systems: partial symmetricexclusion [8] where we allow edge-dependent conductances and a site-varying maximal occu-pancy, symmetric inclusion where we allow edge-dependent conductances and a site-varying“attraction parameter”, and independent walkers. We couple these systems to two reservoirs,with reservoir parameters θ L and θ R . The precise meaning of the reservoir parameters θ L and θ R will be explained in detail later; for the moment one can think of them – roughly – as beingproportional to the densities of left and right reservoirs, respectively. Moreover, the bulk systemcan be defined on any graph. Hence, our setting includes the standard one of a chain coupledto reservoirs at left and right ends, but it is in no way restricted to that setting. The only im-portant geometrical requirement is the presence of precisely two reservoirs. When θ L = θ R = θ the system is in equilibrium, with a unique reversible product measure µ θ . When θ L (cid:54) = θ R thesystem evolves towards a unique non-equilibrium stationary measure µ θ L ,θ R . At stationarity, bymeans of classical dualities with a dual system that has two absorbing sites, corresponding tothe reservoirs in the original system, we obtain correlation inequalities, thereby extending andstrengthening those from [11]. In particular, the dual particle system dynamics does not dependon the reservoir parameters θ L and θ R .Next, for the same pair of boundary driven and purely absorbing systems, we introduceorthogonal polynomial dualities. The orthogonal duality functions are in product form and thefactors associated to the bulk sites are the same orthogonal polynomials as those appearing forthe same particle systems not coupled to reservoirs (see e.g. [9, 19]), while the remaining factorscorresponding to the absorbing sites have a form depending on the reservoir parameters. Theorthogonal polynomials carry themselves a parameter θ which corresponds to the equilibriumreversible product measure µ θ w.r.t. which they are orthogonal.We then give various applications of these orthogonal polynomial dualities to properties ofcorrelation functions in the non-equilibrium stationary measure µ θ L ,θ R . First we prove that thecorrelations of order n of the occupation variables at different locations x , . . . , x n , as well asthe cumulants of order n , are of the form ( θ L − θ R ) n multiplied by a universal function ψ whichdepends only on x , . . . , x n and the dual particle system dynamics, thus, not depending on θ L and θ R . We prove, in fact, a stronger result, namely that whenever the system is started froma local equilibrium product measure, then, at any later time t >
0, the n -point correlations areof the form ( θ L − θ R ) n multiplied by a universal function ψ t which, again, does not depend onthe reservoir parameters θ L and θ R , but only on the dual system dynamics.Finally, we relate the joint moment generating function of the occupation variables to anexpectation in the absorbing dual. Despite the fact that this quantity can in general not beobtained in analytic form, the relation is useful, both from point of view of simulations, as wellas from the point of view of computing macroscopic limits such as density fluctuation fields andlarge deviations of the density profile.The rest of our paper is organized as follows. In Section 2 we introduce the aforementionedboundary driven particle systems and their dual absorbing processes as well as introducing theclassical duality functions. In Section 3 we study properties and correlation inequalities for the quilibrium and non-equilibrium stationary measures. In Section 4 we derive orthogonal dualityfunctions between the boundary driven and the absorbing systems. In Section 5 we obtain auniversal expression for the higher order correlations in the non-equilibrium steady state. In thesame section, the same structure is recovered for more general correlations at finite times whenstarted from a local equilibrium product measure. Section 6 is devoted to a relation betweenweighted exponential generating functions of the occupation variables at stationarity and thecorrelation functions obtained in the previous section. In conclusion, Appendix A contains partof the proof of Theorem 3.3 in Section 3. In this section, we start by introducing the common geometry and the disorder on which theparticle dynamics takes place. Then, we couple this “bulk” system to two reservoirs at possiblydifferent densities.
We consider three particle systems with either an exclusion, inclusion or no interaction. All thesesystems will evolve on a set of sites V = { , . . . , N } ( N ∈ N ) and the rate of particle exchangesbetween two sites x and y ∈ V will be proportional to some given (symmetric) conductance ω { x,y } ∈ [0 , ∞ ). Sites x and y ∈ V for which ω { x,y } (cid:54) = 0 will be considered as connected,indicated by x ∼ y . In what follows, we will assume that ω { x,x } = 0 for all x ∈ V and that theinduced graph ( V, ∼ ) is connected. We will further attach to each site x ∈ V a value α x ∈ N .While the conductances ω = { ω { x,y } : x, y ∈ V } represent the bond environment, the collection α = { α x : x ∈ V } stands for the quenched site disorder.The set V endowed with the environment ( ω , α ) is referred to as bulk of the system. Thisbulk is in contact with a left and a right reservoir through respectively site 1 and site N ∈ V .Particle exchanges between the bulk sites and the reservoirs is tuned by a set of non-negativeparameters ω L , ω R , θ L , θ R , α L and α R as explained in the paragraph below. In this setting, for each choice of the parameter σ ∈ {− , , } , we introduce a boundary drivenparticle system { η t : t ≥ } as a Markov process with X , given by X = (cid:89) x ∈ V { , . . . , α x } if σ = − (cid:89) x ∈ V { , , . . . } = N V otherwise , denoting the configuration space, with η ∈ X standing for a particle configuration and with η ( x ) indicating the number of particles at site x ∈ V for the configuration η ∈ X . The particledynamics is described by the infinitesimal generator L , whose action on bounded functions f : X → R reads as follows: L f ( η ) = L bulk f ( η ) + L L,R f ( η ) . (2.1)In the above expression, the generator L bulk describes the bulk part of the dynamics and isgiven by L bulk f ( η ) = (cid:88) x ∼ y ω { x,y } L { x,y } f ( η ) (2.2) here the summation above runs over the unordered pairs of sites and with the single-bondgenerator L { x,y } given by L { x,y } f ( η ) = η ( x ) ( α y + ση ( y )) ( f ( η x,y ) − f ( η ))+ η ( y ) ( α x + ση ( x )) ( f ( η y,x ) − f ( η )) , where η x,y = η − δ x + δ y ∈ X , i.e. the configuration in which a particle (if any) has beenremoved from x ∈ V and placed at y ∈ V . The boundary part of the dynamics is described bythe generator L L,R in (2.1) as follows: L L,R f ( η ) = ω L L L f ( η ) + ω R L R f ( η ) , (2.3)with L L f ( η ) = η (1) ( α L + σα L θ L ) ( f ( η , − ) − f ( η ))+ α L θ L ( α + ση (1)) ( f ( η , + ) − f ( η )) (2.4)and L R f ( η ) = η ( N ) ( α R + σα R θ R ) (cid:0) f ( η N, − ) − f ( η ) (cid:1) + α R θ R (cid:0) α N + ση ( N )) ( f ( η N, + ) − f ( η ) (cid:1) , (2.5)where η x, − ∈ X , resp. η x, + ∈ X , denotes the configuration obtained from η by removing, resp.adding, a particle from, resp. to, site x ∈ V . In the above dynamics, creation and annihilationof particles occurs at sites x = 1 and x = N due to the interaction with a reservoir.We note that, depending on the choice of the value σ ∈ {− , , } in the definition of thegenerator L in (2.1), we recover either the symmetric partial exclusion process (SEP) for σ = − independent random walkers (IRW) for σ = 0 or the symmetric inclusion process (SIP) for σ = 1 in contact with left and right reservoirs and in presence of disorder. Figure 1: Schematic description of the partial exclusion process (SEP) dynamics in contactwith left and right reservoirs.
The parameters α = { α x : x ∈ V } ⊂ N have the interpretation of maximal occupancies forSEP ( σ = −
1) of the sites of V (see Fig. 1). For IRW ( σ = 0) and SIP ( σ = 1), α x ∈ N stands for he site attraction parameter of the site x ∈ V . We observe that the choice α ⊂ N rather than(0 , ∞ ) is needed only in the context of the exclusion process; however, for the sake of uniformityof notation, we adopt N -valued site parameters α for all three choices of σ ∈ {− , , } .Moreover, while ω L and ω R shall be interpreted as conductances between the boundariesand the associated bulk sites, the parameters α L > α R > α . The parameters θ L and θ R are responsible for the scaling of thereservoirs’ densities ρ L and ρ R , i.e. ρ L = α L θ L and ρ R = α R θ R , (2.6)and, for this reason, we refer to them as scale parameters . In particular, while in general weonly require that θ L , θ R ∈ [0 , ∞ ) , for the case of the exclusion process ( σ = − θ L , θ R ∈ [0 , R EMARK ( notational comparison with [3] ) . If we choose ω x,y = {| x − y | =1 } and α x = j for some j ∈ N if σ = − , if σ = 0 , k for some k > if σ = − , (2.7) for all x, y ∈ V , we recover exactly the same bulk dynamics of the models studied in [3]. Forwhat concerns the reservoir dynamics, the authors of [3] employ the following notation (see e.g.[3, Figs. 1–2]) α := α L θ L γ := α R θ R β := α R + σα R θ R δ := α L + σα L θ L . However, we believe that the parametrization of the bulk-boundary interaction through α L , α R , θ L and θ R yields more transparent formulas as, for instance, for the duality functions in presenceof disorder. R EMARK ( more general reservoirs geometries ) . We emphasize that our results maybe stated for boundary driven particle systems with the same bulk dynamics – as described bythe generator L – and a more general boundary part of the dynamics, in which creation andannihilation of particles due to the reservoir interaction occur at more than two bulk sites. Moreprecisely, the results stated in this section and Sections 3 and 4 below – namely, the dualityrelations and the correlation inequalities – naturally extend if L L,R in (2.1) , (2.3) is replaced by L res f ( η ) = (cid:88) x ∈ V ω res x L res x f ( η ) , with L res x f ( η ) := η ( x ) ( α res x + σα res x θ res x ) (cid:0) f ( η x, − ) − f ( η ) (cid:1) + α res x θ res x ( α x + ση ( x )) (cid:0) f ( η x, + ) − f ( η ) (cid:1) , for some set of non-negative parameters α res = { α res x : x ∈ V } , θ res = { θ res x : x ∈ V } and ω res = { ω res x : x ∈ V } . Also the results in Sections 5 and 6 below extend to this more generalboundary dynamics as long as the scale parameters θ res = { θ res x : x ∈ V } take at most two values,say θ L and θ R . .2 Duality In this section, for each one of the particle systems presented in the section above, we derive twotypes of duality relations with a particle system in contact with purely absorbing boundaries.In particular, by duality relation for the particle system { η t : t ≥ } on X , we mean that thereexist a dual particle system { ξ t : t ≥ } on (cid:99) X and a measurable function D : (cid:99) X × X → R – referred to as duality function – for which the following relation holds: for all configurations η ∈ X , ξ ∈ (cid:99) X and times t ≥
0, we have (cid:98) E ξ [ D ( ξ t , η )] = E η [ D ( ξ, η t )] , (2.8)where (cid:98) E ξ , resp. E η , denotes expectation w.r.t. the law (cid:98) P ξ of { ξ t : t ≥ } with initial condition ξ = ξ , resp. the law P η of { η t : t ≥ } with initial condition η = η . More in general, for agiven probability measure µ on X , E µ denotes the expectation w.r.t. the law P µ of { η t : t ≥ } initially distributed according to µ . Notice that, with a slight abuse of notation, when we write E µ [ D ( ξ, η )] we mean (cid:82) X D ( ξ, η ) d µ ( η ).If (cid:99) L and L denote the infinitesimal generators associated to the processes { ξ t : t ≥ } and { η t : t ≥ } respectively, the duality relation (2.8) is equivalent to the following relation: for allconfigurations η ∈ X and ξ ∈ (cid:99) X , we have (cid:99) L left D ( ξ, η ) = L right D ( ξ, η ) , (2.9)where the subscript “left”, resp. “right”, indicates that the generator acts as an operator on thefunction D ( · , · ), viewed as a function of the left, resp. right, variables. More precisely, (cid:99) L left D ( ξ, η ) = (cid:99) L D ( · , η )( ξ ) and L right D ( ξ, η ) = L right D ( ξ, · )( η ) . In what follows, first we present the dual particle systems and, then, the duality relations.More specifically, we study in Sections 2.2.1 and 4 below, duality relations with two types ofduality functions, which we call, respectively, “classical” and “orthogonal” for reasons that willbe explained below.
For each choice of σ ∈ {− , , } , we define a particle system with purely absorbing reservoirs,which we prove to be dual (see Propositions 2.3 and 4.1 below) to the corresponding system incontact with reservoirs of Section 2.1. For such systems, particles hop on V ∪ { L, R } followingthe same bulk dynamics as the particle systems of Section 2.1 but having { L, R } as absorbingsites. More in detail, { ξ t : t ≥ } denotes such particle systems having (cid:99) X = X × N { L,R } as configuration space and infinitesimal generator (cid:99) L given by (cid:99) L f ( ξ ) = (cid:99) L bulk f ( ξ ) + (cid:99) L L,R f ( ξ ) , (2.10)where, for all bounded functions f : (cid:99) X → R , (cid:99) L bulk f ( ξ ) = (cid:88) x ∼ y ω { x,y } (cid:99) L { x,y } f ( ξ )= (cid:88) x ∼ y ω { x,y } (cid:40) ξ ( x ) ( α y + σξ ( y )) ( f ( ξ x,y ) − f ( ξ ))+ ξ ( y ) ( α x + σξ ( x )) ( f ( ξ y,x ) − f ( ξ )) (cid:41) , nd (cid:99) L L,R f ( ξ ) = ω L (cid:99) L L f ( ξ ) + ω R (cid:99) L R f ( ξ )= ω L α L ξ (1) (cid:0) f ( ξ ,L ) − f ( ξ ) (cid:1) + ω R α R ξ ( N ) (cid:0) f ( ξ N,R ) − f ( ξ ) (cid:1) , with, for all x, y ∈ V ∪ { L, R } , ξ x,y = ξ − δ x + δ y ∈ (cid:99) X .For all configurations ξ ∈ (cid:99) X , let | ξ | denote the total number of particles of the configuration ξ , i.e. | ξ | := ξ ( L ) + ξ ( R ) + (cid:88) x ∈ V ξ ( x ) . (2.11)Once the total number of particles is fixed, due to the conservation of particles under the dy-namics, the assumption of connectedness of the graph ( V, ∼ ) (see Section 2.1) and the positivityof ω L and ω R , the particle system { ξ t : t ≥ } is irreducible on (cid:99) X n := (cid:110) ξ ∈ (cid:99) X : | ξ | = n (cid:111) whenever n = | ξ | and admits a unique stationary measure fully supported on configurations (cid:110) ξ ∈ (cid:99) X n : ξ ( x ) = 0 for all x ∈ V (cid:111) , i.e. all particles will get eventually absorbed in the sites { L, R } . Furthermore, the evolution ofthe particle systems { ξ t : t ≥ } does not depend on θ L and θ R , but only on the following set ofparameters: ω = { ω { x,y } : x, y ∈ V } , (2.12) α = { α x : x ∈ V } and { ω L , ω R , α L , α R } . (2.13)For this reason, in the sequel we will refer to V ∪ { L, R } endowed with the above parameters asthe underlying geometry of our particle systems. In this section, we generalize to the quenched environment setting the duality relations alreadyappearing in e.g. [3]. In particular, these duality functions are in factorized – jointly in theoriginal and dual configuration variables – form over all sites, i.e., for all η ∈ X and ξ ∈ (cid:99) X , D ( ξ, η ) = d L ( ξ ( L )) × (cid:32) (cid:89) x ∈ V d x ( ξ ( x ) , η ( x )) (cid:33) × d R ( ξ ( R )) , (2.14)with the factors { d x ( · , · ) : x ∈ V } ∪ { d L ( · ) , d R ( · ) } named single-site duality functions . More-over, we refer to them as “classical” because the duality functions consist in weighted factorialmoments of the occupation variables of the configuration η generalizing to IRW and SIP therenown duality relations for the symmetric simple exclusion process, see e.g. [17, Theorem 1.1,p. 363].The precise form of these classical duality functions is contained in the following proposition.The proof of this duality relation boils down to directly check identity (2.9) and we omit it beingit a straightforward rewriting of the proof of [3, Theorem 4.1]. We remark that in (2.16) belowand in the rest of the paper, we adopt the conventions 0 := 1, Γ( v + (cid:96) )Γ( v ) := v ( v + 1) · · · ( v + (cid:96) − v ≥ (cid:96) ∈ N , whileΓ( v + (cid:96) )Γ( v ) := (cid:96) = 0 v ( v + 1) · · · ( v + (cid:96) −
1) if (cid:96) ∈ { , ..., | v |} v ∈ Z ∩ ( −∞ ,
0) and (cid:96) ∈ N . ROPOSITION ( classical duality functions ) . For each choice of σ ∈ {− , , } , let L and (cid:99) L be the infinitesimal generators given in (2.1) and (2.10) , respectively, associated to theparticle systems { η t : t ≥ } and { ξ t : t ≥ } . Then the duality relations in (2.8) and (2.9) holdwith the duality function D c(cid:96) : (cid:99) X × X → R defined as follows: for all configurations η ∈ X and ξ ∈ (cid:99) X , D c(cid:96) ( ξ, η ) = d c(cid:96)L ( ξ ( L )) × (cid:32) (cid:89) x ∈ V d c(cid:96)x ( ξ ( x ) , η ( x )) (cid:33) × d c(cid:96)R ( ξ ( R )) , where, for all x ∈ V and k, n ∈ N , d c(cid:96)x ( k, n ) = n !( n − k )! 1 w x ( k ) { k ≤ n } (2.15) and d c(cid:96)L ( k ) = ( θ L ) k and d c(cid:96)R ( k ) = ( θ R ) k , (2.16) where w x ( k ) = α x !( α x − k )! { k ≤ α x } if σ = − α kx if σ = 0Γ( α x + k )Γ( α x ) if σ = 1 . (2.17) The long run behavior of the boundary driven particle systems of Section 2.1, encoded in theirstationary measures, is explicitly known when the particle systems are not in contact with thereservoirs. Indeed, if ω L = ω R = 0, the particle systems { η t : t ≥ } admit a one-parameterfamily of stationary – actually reversible – product measures { µ θ = ⊗ x ∈ V ν x,θ : θ ∈ Θ } (3.1)with Θ = [0 ,
1] if σ = − Θ = [0 , ∞ ) if σ = 0 (IRW) and σ = 1 (SIP) and marginalsgiven, for all x ∈ V , by ν x,θ ∼ Binomial( α x , θ ) if σ = − α x θ ) if σ = 0Negative-Binomial( α x , θ θ ) if σ = 1 . (3.2)More concretely, for all n ∈ N , ν x,θ ( n ) = w x ( n ) z x,θ (cid:16) θ σθ (cid:17) n n ! , (3.3)with the functions { w x : x ∈ V } as given in (2.17) and z x,θ = (1 − θ ) − α x if σ = − e α x θ if σ = 0(1 + θ ) α x if σ = 1 , (3.4) here, for σ = −
1, we set ν x, ( n ) := { n = α x } . Reversibility of these product measures for thedynamics induced by L in (2.1) follows by a standard detailed balance computation (see e.g. [3]for an analogous statement with site-independent parameters α ). We note that, in analogy with(2.6), the parameterization of these product measures and corresponding marginals is chosen insuch a way that the density of particles ρ x := E µ θ [ η ( x )] (3.5)at site x ∈ V w.r.t. µ θ is given by the product of α x and θ , i.e. ρ x = α x θ , x ∈ V . (3.6)
In presence of interaction with only one of the two reservoirs, e.g. ω L > ω R = 0 andwith scale parameters given by θ L and θ R , respectively, the same detailed balance computationshows that the systems have µ θ (see (3.1)) with θ = θ L as reversible product measures. Thestationary measures remain the same as long as the systems are in contact with both reservoirs,i.e. ω L , ω R >
0, and the two reservoirs are given equal scale parameters θ L = θ R ∈ Θ . We referto such stationary measures as equilibrium stationary measures . As for non-equilibrium stationary measures , i.e. the stationary measures of the particle systemswhen ω L , ω R > θ L (cid:54) = θ R , none of the measures { µ θ = ⊗ x ∈ V ν x,θ : θ ∈ Θ } above isstationary. However, for each of the particle systems, the non-equilibrium stationary measureexists, is unique and we denote it by µ θ L ,θ R . Moreover, while for the case of independent randomwalkers µ θ L ,θ R is in product form, for the case of exclusion and inclusion particle systems in non-equilibrium µ θ L ,θ R is non-product and has non-zero two-point correlations. This is the contentof Theorem 3.3 below. In particular, the result on two-point correlations (item (b)) will becomplemented with the study of the signs of such correlations in Theorem 3.4 and Lemma 3.5below. We recall that, for the special case of the exclusion process with α = { α x : x ∈ V } satisfying α x = 1 for all x ∈ V and with nearest-neighbor unitary conductances, i.e. ω { x,y } = {| x − y | =1 } , x, y ∈ V , the unique non-product non-equilibrium stationary measure µ θ L ,θ R has been characterized interms of a matrix formulation (see e.g. [6] and [16, Part III. Section 3]). Goal of Section 5below is to provide a partial characterization of the non-equilibrium stationary measure of thesesystems by expressing suitably centered factorial moments – related to the orthogonal dualityfunctions of Section 4 below – in terms of the product of a suitable power of ( θ L − θ R ) and acoefficient which does not depend on neither θ L nor θ R .In what follows, for all x ∈ V , we introduce the non-equilibrium stationary profile of theclassical duality functions:¯ θ x := E µ θL,θR (cid:20) η ( x ) α x (cid:21) = E µ θL,θR (cid:104) D c(cid:96) ( δ x , η ) (cid:105) . (3.7)We recall that (cid:98) P ξ denotes the law of the dual particle system started from the deterministicconfiguration ξ ∈ (cid:99) X . Then, by stationarity and duality (Proposition 2.3), we obtain, for all x ∈ V , ¯ θ x = lim t →∞ E µ θL,θR (cid:20) η t ( x ) α x (cid:21) = (cid:98) p ∞ ( δ x , δ L ) θ L + (cid:98) p ∞ ( δ x , δ R ) θ R θ R + (cid:98) p ∞ ( δ x , δ L ) ( θ L − θ R ) , (3.8)where, for all ξ, ξ (cid:48) ∈ (cid:99) X , (cid:98) p ∞ ( ξ, ξ (cid:48) ) := lim t →∞ (cid:98) p t ( ξ, ξ (cid:48) ), with (cid:98) p t ( ξ, ξ (cid:48) ) := (cid:98) P ξ ( ξ t = ξ (cid:48) ) . Equivalently, stationarity and duality imply that { ¯ θ x : x ∈ V } solves the following differenceequations: for all x ∈ V , (cid:88) y ∈ V ω { x,y } α y (¯ θ y − ¯ θ x )+ { x =1 } ω L α L ( θ L − ¯ θ ) + { x = N } ω R α R ( θ R − ¯ θ N ) = 0 . (3.9) R EMARK ( non-equilibrium stationary profile for a chain ) . In the particular in-stance of a chain, i.e. ω { x,y } > if and only if | x − y | = 1 , the solution to the system (3.9) is given by: ¯ θ x = θ R + (cid:98) p ∞ ( δ x , δ L ) ( θ L − θ R )= θ R + ω R α R α N + (cid:80) N − y = x ω { y,y +1 } α y α y +1 ω L α L α + (cid:16)(cid:80) N − y =1 1 ω { y,y +1 } α y α y +1 (cid:17) + ω R α R α N ( θ L − θ R ) . If, additionally, the conductances and site parameters ω and α are constant, α L = α R = α x and ω L = ω R = ω { x,x +1 } , the profile x (cid:55)→ ¯ θ x is linear (cf. [3, Eq. (4.24)]): ¯ θ x = θ R + (cid:18) − xN + 1 (cid:19) ( θ L − θ R ) . (3.10)Before stating the main result of this section, we introduce the following definition. D EFINITION ( local equilibrium product measure ) . Given ¯ θ := { ¯ θ x : x ∈ V } thestationary profile introduced in (3.8) , we define the following product measure µ ¯ θ := ⊗ x ∈ V ν x, ¯ θ x , (3.11) and refer to it as the local equilibrium product measure . T HEOREM
For each choice of σ ∈ {− , , } and provided that ω L ∨ ω R > , for all θ L , θ R ∈ Θ there exists a unique stationary measure µ θ L ,θ R for the particle system { η t : t ≥ } .Moreover, (a) If σ = 0 (IRW) , the stationary measure µ θ L ,θ R is in product form and is given by µ θ L ,θ R = µ ¯ θ . (3.12)(b) If either σ = − or σ = 1 (SIP) and, additionally, ω L , ω R > and θ L (cid:54) = θ R , thereexists x, y ∈ V with x (cid:54) = y for which E µ θL,θR (cid:20)(cid:18) η ( x ) α x − ¯ θ x (cid:19) (cid:18) η ( y ) α y − ¯ θ y (cid:19)(cid:21) (cid:54) = 0 . As a consequence, the unique non-equilibrium stationary measure µ θ L ,θ R is not in productform. ROOF . The proof of existence and uniqueness of the stationary measure µ θ L ,θ R is trivial forthe exclusion process, which is a finite state irreducible Markov chain. We postpone the prooffor the case of independent random walkers and inclusion process to Appendix A. Although thisresult is standard, it does not appear, to the best of our knowledge, in the literature.For what concerns item (a) in which σ = 0, let us compute, for all ξ ∈ (cid:99) X , (cid:90) X L right D c(cid:96) ( ξ, η ) µ ¯ θ (d η ) . By duality, the following relation (see e.g. [19]) (cid:88) n ∈ N d c(cid:96)x ( k, n ) ν x, ¯ θ x ( n ) = (¯ θ x ) k , k ∈ N , (3.13)and (3.9), we obtain, for all ξ ∈ (cid:99) X , (cid:90) X L right D c(cid:96) ( ξ, η ) µ ¯ θ (d η ) = (cid:90) X (cid:99) L left D c(cid:96) ( ξ, η ) µ ¯ θ (d η )= (cid:88) x ∈ V (cid:18)(cid:90) X D c(cid:96) ( ξ − δ x , η ) µ ¯ θ (d η ) (cid:19) ξ ( x ) (cid:80) y ∈ V ω { x,y } α y (¯ θ y − ¯ θ x )+ { x =1 } ω L α L ( θ L − ¯ θ )+ { x = N } ω R α R ( θ R − ¯ θ N ) = 0 , Because the products of Poisson distributions are completely characterized by their factorialmoments { D c(cid:96) ( ξ, · ) : ξ ∈ (cid:99) X } , we get (3.12).For item (b) in which σ (cid:54) = 0, let us suppose by contradiction that all two-point correlationsare zero, i.e. for all x, y ∈ V with x (cid:54) = y , E µ θL,θR (cid:20) η ( x ) α x η ( y ) α y (cid:21) = E µ θL,θR (cid:104) D c(cid:96) ( δ x + δ y , η ) (cid:105) = ¯ θ x ¯ θ y . (3.14)If we use the following shortcut ¯ θ (cid:48)(cid:48) x := E µ θL,θR (cid:104) D c(cid:96) (2 δ x , η ) (cid:105) , by stationarity, duality and (3.14), we obtain, for all x ∈ V , (cid:90) X L right D c(cid:96) (2 δ x , η ) µ θ L ,θ R (d η ) = (cid:90) X (cid:99) L left D c(cid:96) (2 δ x , η ) µ θ L ,θ R (d η )= 2 (cid:88) y ∈ V ω { x,y } α y (¯ θ x ¯ θ y − ¯ θ (cid:48)(cid:48) x )+ 2 (cid:8) { x =1 } ω L α L ( θ L ¯ θ − ¯ θ (cid:48)(cid:48) ) + { x = N } ω R α R ( θ R ¯ θ N − ¯ θ (cid:48)(cid:48) N ) (cid:9) = 0 . By adding and subtracting2 (cid:88) y ∈ V ω { x,y } α y (¯ θ x ) + { x =1 } ω L α L (¯ θ ) + { x = N } ω R α R (¯ θ N ) o the identity above and by relation (3.9), we get (cid:0) (¯ θ x ) − ¯ θ (cid:48)(cid:48) x (cid:1) (cid:88) y ∈ V ω { x,y } α y + { x =1 } ω L α L + { x = N } ω R α R = 0 . Because the above identity holds for all x ∈ V and by the positivity of the expression in curlybrackets due to the connectedness of ( V, ∼ ), we get¯ θ (cid:48)(cid:48) x = (¯ θ x ) , for all x ∈ V . (3.15)In view of (3.14), (3.15), stationarity of µ θ L ,θ R and duality, we have (cid:90) X L right D c(cid:96) ( δ x + δ y , η ) µ θ L ,θ R (d η ) = (cid:90) X (cid:99) L left D c(cid:96) ( δ x + δ y , η ) µ θ L ,θ R (d η )= ¯ θ y (cid:80) z ∈ V ω { x,z } α z (¯ θ z − ¯ θ x )+ { x =1 } ω L α L ( θ L − ¯ θ )+ { x = N } ω R α R ( θ R − ¯ θ N ) + ¯ θ x (cid:80) z ∈ V ω { y,z } α z (¯ θ z − ¯ θ y )+ { y =1 } ω L α L ( θ L − ¯ θ )+ { y = N } ω R α R ( θ R − ¯ θ N ) + σ ω { x,y } (¯ θ x − ¯ θ y ) = σ ω { x,y } (¯ θ x − ¯ θ y ) . (3.16)Therefore, because σ ∈ {− , } , as a consequence of the connectedness of ( V, ∼ ), we have (cid:88) x ∼ y (cid:18)(cid:90) X L right D c(cid:96) ( δ x + δ y , η ) µ θ L ,θ R (d η ) (cid:19) = σ (cid:88) x ∼ y ω { x,y } (¯ θ x − ¯ θ y ) = 0 (3.17)if and only if ¯ θ x = ¯ θ y , for all x, y ∈ V . (3.18)However, because θ L (cid:54) = θ R , the latter condition (3.18) contradicts (3.9). In the following theorem we prove that as soon as the system has interaction, i.e. σ ∈ {− , } ,the local equilibrium product measure expectations of classical duality functions decrease (resp.increase) for exclusion (resp. inclusion) in the course of time. This implies, in particular, negative(resp. positive) two-point correlations for exclusion (resp. inclusion) particle systems. Thisstrengthens previous results on correlation inequalities in [11], indeed here we obtain strictinequalities. The proof of this theorem is based on Lemma 3.5 below, which is of interest initself because it provides an explicit expression of the l.h.s. in (3.19). T HEOREM ( sign of two-point correlations ) . If ω L , ω R > and ξ ∈ (cid:99) X is such that (cid:80) x ∈ V ξ ( x ) ≥ , then, for all θ L , θ R ∈ Θ with θ L (cid:54) = θ R and t > , ddt E µ ¯ θ (cid:104) D c(cid:96) ( ξ, η t ) (cid:105) (cid:40) < if σ = − > if σ = 1 . (3.19) As a consequence, for all x, y ∈ V with x (cid:54) = y , E µ θL,θR (cid:20)(cid:18) η ( x ) α x − ¯ θ x (cid:19) (cid:18) η ( y ) α y − ¯ θ y (cid:19)(cid:21) (cid:40) < if σ = − > if σ = 1 . ROOF . The local equilibrium product measures µ ¯ θ satisfy the hypothesis of Lemma 3.5 below(cf. (3.13)). Then, (3.19) is recovered as a consequence of the first equality of (3.21) from thesame lemma. L EMMA
For all n ∈ N , let µ be a probability measure on X such that E µ (cid:104) D c(cid:96) ( ξ, η ) (cid:105) = H ( ξ, ¯ θ ) (3.20) holds for all ξ ∈ (cid:99) X with | ξ | ≤ n , where ¯ θ = { ¯ θ x : x ∈ V } and, for all θ = { θ x : x ∈ V } ⊂ Θ , H ( ξ, θ ) := ( θ L ) ξ ( L ) (cid:32) (cid:89) x ∈ V ( θ x ) ξ ( x ) (cid:33) ( θ R ) ξ ( R ) . Then ddt E µ (cid:104) D c(cid:96) ( ξ, η t ) (cid:105) = σ (cid:88) x ∼ y ω { x,y } (cid:0) ¯ θ y − ¯ θ x (cid:1) (cid:98) E ξ (cid:20) ξ t ( x )¯ θ x ξ t ( y )¯ θ y E µ (cid:104) D c(cid:96) ( ξ t , η ) (cid:105)(cid:21) = σ (cid:88) x ∼ y ω { x,y } (cid:98) E ξ (cid:104)(cid:0) ¯ θ y − ¯ θ x (cid:1) ∂ θ x θ y H ( ξ t , ¯ θ ) (cid:105) (3.21) holds for all ξ ∈ X with | ξ | ≤ n and t ≥ . P ROOF . By duality, we obtain, for all ξ ∈ (cid:99) X , ddt E µ (cid:104) D c(cid:96) ( ξ, η t ) (cid:105) = (cid:90) X L right E η (cid:104) D c(cid:96) ( ξ, η t ) (cid:105) µ (d η )= (cid:90) X L right (cid:98) E ξ (cid:104) D c(cid:96) ( ξ t , η ) (cid:105) µ (d η ) = (cid:90) X (cid:98) E ξ (cid:104) (cid:99) L left D c(cid:96) ( ξ t , η ) (cid:105) µ (d η )= (cid:88) x ∈ V (cid:98) E ξ ξ t ( x ) (cid:80) y ∈ V ω { x,y } α y (cid:0) E µ (cid:2) D c(cid:96) ( ξ x,yt , η ) (cid:3) − E µ (cid:2) D c(cid:96) ( ξ t , η ) (cid:3)(cid:1) + { x =1 } ω L α L ( E µ (cid:104) D c(cid:96) ( ξ ,Lt , η ) (cid:105) − E µ (cid:2) D c(cid:96) ( ξ t , η ) (cid:3) )+ { x = N } ω R α R ( E µ (cid:104) D c(cid:96) ( ξ N,Rt , η ) (cid:105) − E µ (cid:2) D c(cid:96) ( ξ t , η ) (cid:3) ) + σ (cid:88) x ∈ V (cid:98) E ξ (cid:88) y ∈ V ω { x,y } ξ t ( x ) ξ t ( y ) (cid:16) E µ (cid:104) D c(cid:96) ( ξ x,yt , η ) (cid:105) − E µ (cid:104) D c(cid:96) ( ξ t , η ) (cid:105)(cid:17) . By (3.20), for all x, y ∈ V and ξ ∈ (cid:99) X with | ξ | ≤ n , we have E µ (cid:104) D c(cid:96) ( ξ x,y , η ) (cid:105) − E µ (cid:104) D c(cid:96) ( ξ, η ) (cid:105) = E µ (cid:2) D c(cid:96) ( ξ, η ) (cid:3) ¯ θ x (cid:0) ¯ θ y − ¯ θ x (cid:1) , and, similarly, E µ (cid:104) D c(cid:96) ( ξ ,L , η ) (cid:105) − E µ (cid:104) D c(cid:96) ( ξ, η ) (cid:105) = E µ (cid:2) D c(cid:96) ( ξ, η ) (cid:3) ¯ θ (cid:0) θ L − ¯ θ (cid:1) E µ (cid:104) D c(cid:96) ( ξ N,R , η ) (cid:105) − E µ (cid:104) D c(cid:96) ( ξ, η ) (cid:105) = E µ (cid:2) D c(cid:96) ( ξ, η ) (cid:3) ¯ θ N (cid:0) θ R − ¯ θ N (cid:1) . s a consequence, we further obtain ddt E µ (cid:104) D c(cid:96) ( ξ, η t ) (cid:105) = (cid:90) X L right E η (cid:104) D c(cid:96) ( ξ, η t ) (cid:105) µ (d η )= (cid:88) x ∈ V (cid:98) E ξ ξ t ( x )¯ θ x E µ (cid:104) D c(cid:96) ( ξ t , η ) (cid:105) (cid:80) y ∈ V ω { x,y } α y (cid:0) ¯ θ y − ¯ θ x (cid:1) + { x =1 } ω L α L (cid:0) θ L − ¯ θ (cid:1) + { x = N } ω R α R (cid:0) θ R − ¯ θ N (cid:1) + σ (cid:88) x ∼ y ω { x,y } (cid:0) ¯ θ y − ¯ θ x (cid:1) (cid:98) E ξ (cid:20) ξ t ( x )¯ θ x ξ t ( y )¯ θ y E µ (cid:104) D c(cid:96) ( ξ t , η ) (cid:105)(cid:21) . The observation that each of the expressions between curly brackets above equals zero becauseof the choice of the scale parameters { ¯ θ x : x ∈ V } (cf. (3.7)) concludes the proof. R EMARK (a)
For all ξ ∈ (cid:99) X with (cid:80) z ∈ V ξ ( z ) ≥ , for all times t > and for all sites x, y ∈ V , the geometric assumption on the connectedness of ( V, ∼ ) implies that (cid:98) P ξ ( ξ t ( x ) ξ t ( y ) > > . As a consequence, the sign of the time derivative in (3.21) for ξ ∈ (cid:99) X with (cid:80) z ∈ V ξ ( z ) ≥ and for t > is determined by σ ∈ {− , , } . In particular, if the probability measure µ and the configuration ξ ∈ (cid:99) X are given as in Theorem 3.4, the convergence E µ (cid:104) D c(cid:96) ( ξ, η t ) (cid:105) −→ t →∞ E µ θL,θR (cid:104) D c(cid:96) ( ξ, η ) (cid:105) is strictly monotone in time: decreasing for σ = − and increasing for σ = 1 . (b) In the particular situation in which ξ = δ x + δ y for some x, y ∈ V and the probabilitymeasure µ satisfies the hypothesis of Theorem 3.4 for n ≥ , the expression in (3.21) further simplifies yielding, for all t > , E µ (cid:104) D c(cid:96) ( δ x + δ y , η t ) (cid:105) − ¯ θ x ¯ θ y = E µ (cid:104) D c(cid:96) ( δ x + δ y , η t ) (cid:105) − E µ (cid:104) D c(cid:96) ( δ x + δ y , η ) (cid:105) = σ (cid:90) t (cid:88) z ∼ w ω { z,w } (cid:0) ¯ θ w − ¯ θ z (cid:1) (cid:98) E ξ = δ x + δ y [ ξ s ( z ) ξ s ( w )] ds = σ (cid:90) t (cid:88) z ∼ w ω { z,w } (cid:0) ¯ θ w − ¯ θ z (cid:1) (cid:98) P ξ = δ x + δ y ( ξ s ( z ) = 1 and ξ s ( w ) = 1) ds . (3.22) If, additionally, we impose α x = α L = α R and ω L = ω R = 1 ω { x,y } = {| x − y | =1 } , for all x, y ∈ V , we further get (cf. (3.10) ) E µ (cid:104) D c(cid:96) ( δ x + δ y , η t ) (cid:105) − ¯ θ x ¯ θ y = σ ( θ L − θ R ) ( N + 1) (cid:90) t (cid:98) P ξ = δ x + δ y (cid:32) N − (cid:88) z =1 ξ s ( z ) ξ s ( z + 1) = 1 (cid:33) ds . (3.23) Orthogonal dualities
By orthogonal dualities we refer to a specific subclass of duality functions D ( ξ, η ) in the form(2.14). This subclass consists of jointly factorized functions whose each “bulk” single-site dualityfunction ( k, n ) ∈ N × N (cid:55)→ d x ( k, n ) ∈ R is a family of polynomials in the n -variables and orthogonal w.r.t. a suitable probability measure ν x on N , i.e. for all k, (cid:96) ∈ N , ∞ (cid:88) n =0 d x ( k, n ) d x ( (cid:96), n ) ν x ( n ) = { k = (cid:96) } (cid:107) d x ( k, · ) (cid:107) L ( ν x ) . Orthogonal duality functions for exclusion, inclusion and independent particle systems with nointeraction with reservoirs have been first introduced in [9] by direct computations and thencharacterized in [19] through generating function techniques. There the dual particle systemshave the same law of the original particle system, therefore orthogonal dualities are actuallyself-dualities. Moreover, for each σ ∈ {− , , } , these jointly factorized orthogonal dualitiesconsist of products of hypergeometric functions of the following two types: either F (cid:20) − k − n − ; − u (cid:21) := k (cid:88) (cid:96) =0 (cid:18) k(cid:96) (cid:19) (cid:18) n !( n − (cid:96) )! { (cid:96) ≤ n } (cid:19) u (cid:96) (4.1)or F (cid:20) − k − nv ; u (cid:21) := k (cid:88) (cid:96) =0 (cid:18) k(cid:96) (cid:19) (cid:18) Γ( v )Γ( v + (cid:96) ) n !( n − (cid:96) )! { (cid:96) ≤ n } (cid:19) u (cid:96) , (4.2)with k, n ∈ N and u, v ∈ R . More specifically, these orthogonal single-site self-duality functionsare Kravchuk polynomials for SEP ( σ = − σ = 0) and Meixnerpolynomials for SIP ( σ = 1) (see e.g. [14]). It turns out that such single-site self-duality functionsare orthogonal families w.r.t. the single-site marginals of the stationary (actually reversible)product measures of the corresponding particle system; in particular, Kravchuk polynomialsare orthogonal w.r.t. Binomial distributions, Charlier polynomials w.r.t. Poisson distributionsand Meixner polynomials w.r.t. Negative Binomial distributions. More precisely, because in thissetting there exists a one-parameter family of stationary product measures for each of the threeparticle systems (see also Section 3 above), this corresponds to the existence of a one-parameterfamily of orthogonal duality functions.This correspondence between orthogonal duality functions and stationary measures maysuggest that, knowing a stationary measure of a particle system, an orthogonal family of observ-ables of this system would correspond, in general, to duality functions. This program, however,besides not being generally verifiable, does not apply to the case of particle systems in contactwith reservoirs, for which the non-equilibrium stationary measures are, generally speaking, notin product form and not explicitly known (see also Section 3.2).Nevertheless, from an algebraic point of view (see e.g. [10]), new duality relations may begenerated from the knowledge of a duality relation and a symmetry of one of the two generatorsinvolved in the duality relation. In brief, given the following duality relation (cid:99) L left D ( ξ, η ) = L right D ( ξ, η )for all ξ ∈ (cid:99) X , η ∈ X , and a symmetry (cid:99) K for the generator (cid:99) L , i.e., for all f : (cid:99) X → R and ξ ∈ (cid:99) X , (cid:99) K (cid:99) L f ( ξ ) = (cid:99) L (cid:99) K f ( ξ ) , (4.3) hen, if F ( (cid:99) K ) with F : R → R is a well-defined operator, the function ( F ( (cid:99) K )) left D ( ξ, η ) is aduality function between L and (cid:99) L . Indeed, for all η ∈ X and ξ ∈ (cid:99) X , we have (cid:99) L left ( F ( (cid:99) K )) left D ( ξ, η ) = ( F ( (cid:99) K )) left (cid:99) L left D ( ξ, η )= ( F ( (cid:99) K )) left L right D ( ξ, η )= L right ( F ( (cid:99) K )) left D ( ξ, η ) . This latter approach is the one we follow here (Theorem 4.1 below) to recover a one-parameterfamily of orthogonal duality functions for boundary driven particle systems. Its proof combinestwo ingredients: first, as already proved in [5], we observe that the so-called annihilation operatoron V ∪ { L, R } given, for all f : (cid:99) X → R , by (cid:99) K f ( ξ ) = (cid:99) K bulk f ( ξ ) + (cid:99) K L,R f ( ξ ) , (4.4)where (cid:99) K bulk f ( ξ ) = (cid:88) x ∈ V (cid:99) K x f ( ξ ) = (cid:88) x ∈ V ξ ( x ) f ( ξ − δ x )and (cid:99) K L,R f ( ξ ) = (cid:99) K L f ( ξ ) + (cid:99) K R f ( ξ ) = ξ ( L ) f ( ξ − δ L ) + ξ ( R ) f ( ξ − δ R ) , is a symmetry for the generator (cid:99) L associated to the particle systems with purely absorbingreservoirs and defined in (2.10). Then, we obtain the candidate orthogonal dualities by act-ing with suitable exponential functions of this symmetry (cid:99) K on the classical duality functionsappearing in Proposition 2.3. We recall that in (4.6) below, the convention 0 := 1 holds. T HEOREM ( orthogonal duality functions ) . For each choice of σ ∈ {− , , } , let L and (cid:99) L be the infinitesimal generators given in (2.1) and (2.10) , respectively, associated to theparticle systems { η t : t ≥ } and { ξ t : t ≥ } . Then the duality relations in (2.8) and (2.9) hold with the duality functions D orθ : (cid:99) X × X → R defined, for all θ ∈ Θ , as follows: for allconfigurations η ∈ X and ξ ∈ (cid:99) X , d orx,θ ( k, n ) = ( − θ ) k × F (cid:20) − k − n − α x ; 1 θ (cid:21) σ = − F (cid:20) − k − n − ; − θα x (cid:21) σ = 0 F (cid:20) − k − nα x ; − θ (cid:21) σ = 1 , (4.5) and d orL,θ ( k ) = ( θ L − θ ) k and d orR,θ ( k ) = ( θ R − θ ) k . (4.6) P ROOF . We start with the observation that, for each σ ∈ {− , , } , the commutation relation(4.3) between the annihilation operator (cid:99) K in (4.4) and the generator (cid:99) L (2.10) holds (for adetailed proof, we refer to e.g. [5, Section 5]).As a consequence, for all θ ∈ Θ , the following function( e − θ (cid:99) K ) left D c(cid:96) ( ξ, η ) (4.7) s a duality function between L and (cid:99) L . In particular, recalling the definitions of single-siteclassical duality functions in (2.15)–(2.16) and hypergeometric functions in (4.1)–(4.2), due tothe factorized form of both symmetry e − θ (cid:99) K and classical duality function, the combination of( e − θ (cid:99) K L ) d c(cid:96)L ( k ) = k (cid:88) (cid:96) =0 (cid:18) k(cid:96) (cid:19) d c(cid:96)L ( (cid:96) ) ( − θ ) k − (cid:96) = ( θ L − θ ) k ( e − θ (cid:99) K R ) d c(cid:96)R ( k ) = k (cid:88) (cid:96) =0 (cid:18) k(cid:96) (cid:19) d c(cid:96)R ( (cid:96) ) ( − θ ) k − (cid:96) = ( θ R − θ ) k and ( e − θ (cid:99) K x ) left d c(cid:96)x ( k, n ) = k (cid:88) (cid:96) =0 (cid:18) k(cid:96) (cid:19) d c(cid:96)x ( (cid:96), n ) ( − θ ) k − (cid:96) (4.8)= ( − θ ) k × F (cid:20) − k − n − α x ; 1 θ (cid:21) σ = − F (cid:20) − k − n − ; − θα x (cid:21) σ = 0 F (cid:20) − k − nα x ; − θ (cid:21) σ = 1 , for all x ∈ V , concludes the proof. R EMARK
The proof of Theorem 4.1 reveals that the classical duality functions may be seenas a particular instance of the orthogonal duality functions if the scale parameter θ ∈ Θ is setequal to zero, i.e. D c(cid:96) ( ξ, η ) = D orθ =0 ( ξ, η ) . (4.9) Moreover, we remark that the following formula connecting orthogonal and classical dualities isreminiscent of the Newton binomial formula: d orx,θ ( k, n ) = k (cid:88) (cid:96) =0 (cid:18) k(cid:96) (cid:19) d c(cid:96)x ( (cid:96), n ) ( − θ ) k − (cid:96) . (4.10) R EMARK ( orthogonality relations ) . In general, the orthogonal duality functions ofTheorem 4.1 are not orthogonal w.r.t. the stationary measure of the particle dynamics in non-equilibrium. In fact, for each choice of σ ∈ {− , , } and θ ∈ Θ , the orthogonal duality function D orθ ( ξ, η ) gives rise to an orthogonal basis { e ξ : ξ ∈ (cid:99) Y } of L ( X , µ θ ) , where µ θ is given in (3.1) , e ξ := D orθ ( ξ, · ) and (cid:99) Y := { ξ ∈ (cid:99) X : ξ ( L ) = ξ ( R ) = 0 } . While in equilibrium, i.e. θ L = θ R = θ ∈ Θ , we have seen (see Section 3) that the measure µ θ is stationary for the particle system { η t : t ≥ } , in non-equilibrium, i.e. θ L (cid:54) = θ R , µ θ fails to bestationary. Nevertheless, the aforementioned orthogonality relations still hold in both contexts,regardless of the stationarity of µ θ . As an immediate consequence of Theorem 4.1, we can compute the following expectationsof the orthogonal duality functions. ROPOSITION
Let β ∈ R such that θ := θ R + β ( θ L − θ R ) ∈ Θ . (4.11)
Then, for all t ≥ and for all configurations ξ ∈ (cid:99) X , we have E µ θ [ D orθ ( ξ, η t )] = ( θ L − θ R ) | ξ | φ t,β ( ξ ) , (4.12) where µ θ is the product measure (cf. (3.1) ) with scale parameter θ = θ R + β ( θ L − θ R ) and φ t,β ( ξ ) := ( − β ) | ξ | (cid:98) E ξ (cid:34)(cid:18) β − β (cid:19) ξ t ( L ) { ξ t ( L )+ ξ t ( R )= | ξ |} (cid:35) . Moreover, for all configurations ξ ∈ (cid:99) X , we have E µ θL,θR [ D orθ ( ξ, η )] = ( θ L − θ R ) | ξ | φ β ( ξ ) , (4.13) where φ β ( ξ ) := ( − β ) | ξ | (cid:98) E ξ (cid:34)(cid:18) β − β (cid:19) ξ ∞ ( L ) (cid:35) . In particular, φ β,t and φ β do not depend on neither θ L nor θ R , but only on β , σ ∈ {− , , } and the underlying geometry of the system. P ROOF . As a consequence of duality (Theorem 4.1), orthogonality of the single-site dualityfunctions (cid:110) d orx,θ ( k, · ) : k ∈ N (cid:111) w.r.t. the marginal ν x,θ (see also Remark 4.3) and the observationthat d orx,θ (0 , · ) ≡ , x ∈ V , we have E µ θ [ D orθ ( ξ, η t )] = (cid:98) E ξ [ E µ θ [ D orθ ( ξ t , η )]]= (cid:98) E ξ (cid:104) ( θ L − θ ) ξ t ( L ) ( θ R − θ ) ξ t ( R ) { ξ t ( L )+ ξ t ( R )= | ξ |} (cid:105) = (cid:98) E ξ (cid:104) ( θ L − θ ) ξ t ( L ) ( θ R − θ ) | ξ |− ξ t ( L ) { ξ t ( L )+ ξ t ( R )= | ξ |} (cid:105) = ( θ R − θ ) | ξ | (cid:98) E ξ (cid:104) ( θ L − θ ) ξ t ( L ) ( θ R − θ ) − ξ t ( L ) { ξ t ( L )+ ξ t ( R )= | ξ |} (cid:105) . By the definition of θ as a function of β in (4.11), we get (4.12). By sending t → ∞ , theuniqueness of the stationary measure yields (4.13). R EMARK
For the choice β = and, thus, θ = θ L + θ R , (4.12) and (4.13) further simplifyas E µ θ [ D orθ ( ξ, η t )] = (cid:18) θ L − θ R (cid:19) | ξ | (cid:98) E ξ (cid:104) ( − | ξ |− ξ t ( L ) { ξ t ( L )+ ξ t ( R )= | ξ |} (cid:105) (4.14) and E µ θL,θR [ D orθ ( ξ, η )] = (cid:18) θ L − θ R (cid:19) | ξ | (cid:98) E ξ (cid:104) ( − | ξ |− ξ ∞ ( L ) (cid:105) . (4.15) Higher order correlations in non-equilibrium
In this section, we study higher order space correlations for the non-equilibrium stationarymeasures presented in Section 3. In particular, we show in Theorem 5.1 below, by using the or-thogonal duality functions of Section 4, that the n -point correlation functions in non-equilibriummay be factorized into a first term, namely ( θ L − θ R ) n , and a second term, which we call ψ andwhich is independent of the values θ L and θ R . This result may be seen as a higher order gener-alization of the decomposition obtained for the simple symmetric exclusion process in [7, Eqs.(2.3)–(2.8)]. There the authors exploit the matrix formulation of the non-equilibrium station-ary measure to recover the explicit expression for the first, second and third order correlationfunctions.While the coefficients ψ in (5.3) for the case of independent random walkers are identicallyzero (see item (b) after Theorem 5.1 below), for the interacting case ( σ ∈ {− , } ) they areexpressed in terms of absorption probabilities of both interacting and independent dual particles.These absorption probabilities – apart from some special instances, see e.g. [7] and [3, Section 6.1]– are not explicitly known. Nonetheless, Theorem 5.1 – and the related Theorem 5.6 – highlightthe common structure of the higher order correlations for all three particle systems consideredin this paper. In particular, this common structure arises for all values of the parameters θ L and θ R ∈ Θ and with all quenched random environments ( ω , α ) and parameters { ω L , ω R , α L , α R } asin (2.12)–(2.13). Moreover, along the same lines, we show that all higher order space correlationsat any finite time t > For each choice of σ ∈ {− , , } , we recall that µ θ L ,θ R denotes the non-equilibrium stationarymeasure of the particle system { η t : t ≥ } with generator L given in (2.1). Moreover, letus recall the definition of { ¯ θ x : x ∈ V } in (3.7) and introduce the following ordering of dualconfigurations: for all ξ ∈ (cid:99) X , ζ ≤ ξ if and only if ζ ∈ (cid:99) X and ζ ( L ) ≤ ξ ( L ) , ζ ( R ) ≤ ξ ( R ) ζ ( x ) ≤ ξ ( x ) , for all x ∈ V . (5.1)Analogously, we say that ζ < ξ if ζ ≤ ξ and at least one of the inequalities in (5.1) is strict.Finally, given ξ, ζ ∈ (cid:99) X , let ξ ± ζ denote the configuration with ξ ( x ) ± ζ ( x ) particles at site x ,for all x ∈ V ∪ { L, R } , as long as ξ ± ζ ∈ (cid:99) X .In what follows, for all choices of σ ∈ {− , , } , (cid:98) P and (cid:98) E denote the law and expectation,respectively, of the dual process with either exclusion ( σ = − σ = 1) or no interac-tion ( σ = 0), while we adopt (cid:98) P IRW and (cid:98) E IRW to refer to the law and corresponding expectation,respectively, of the dual process consisting of non-interacting random walks ( σ = 0). T HEOREM ( stationary correlation functions ) . For all n ∈ N with n ≤ N and forall x , . . . , x n ∈ V with x i (cid:54) = x j if i (cid:54) = j , by setting ξ = δ x + · · · + δ x n , we have E µ θL,θR (cid:34) n (cid:89) i =1 (cid:18) η ( x i ) α x i − ¯ θ x i (cid:19)(cid:35) = ( θ L − θ R ) n ψ ( ξ ) ( θ L − θ R ) n ψ ( δ x + · · · + δ x n ) , (5.2) where ψ ( ξ ) = (cid:88) ζ ≤ ξ (cid:98) P IRW ξ − ζ (( ξ − ζ ) ∞ ( L ) = | ξ − ζ | ) (cid:98) P ζ ( ζ ∞ ( L ) = | ζ | ) . (5.3) In particular, ψ ( ξ ) ∈ R and it does not depend on neither θ L nor θ R , but only on σ ∈ {− , , } and the underlying geometry (see Eqs. (2.12) – (2.13) ) of the system. As an immediate consequence we have the following corollary on the stationary non-equilibriumjoint cumulants. C OROLLARY ( joint cumulants ) . For all n ∈ N and x , . . . , x n ∈ V with x i (cid:54) = x j if i (cid:54) = j ,let κ ( δ x + . . . + δ x n ) denote the joint cumulant of the random variables (cid:26) η ( x i ) α x i − ¯ θ x i : x , . . . , x n ∈ V (cid:27) . Then, we have κ ( δ x + · · · + δ x n ) = ( θ L − θ R ) n ϕ ( δ x + · · · + δ x n ) , where ϕ ( δ x + · · · + δ x n ) ∈ R does not depend on neither θ L nor θ R , but only on σ ∈ {− , , } and the underlying geometry of the system. P ROOF . After recalling that κ ( δ x + · · · + δ x n ) = (cid:88) γ ∈ T ( | γ | − − | γ |− (cid:89) U ∈ γ E µ θL,θR (cid:89) y ∈ U (cid:18) η ( y ) α y − ¯ θ y (cid:19) , where T = T ( { x , . . . , x n } ) denotes the set of partitions of { x , . . . , x n } ⊂ V , the result followsby (5.2) with ϕ ( { x , . . . , x n } ) given by ϕ ( δ x + · · · + δ x n ) = (cid:88) γ ∈ T ( | γ | − − | γ |− (cid:89) U ∈ γ ψ ( U ) , where ψ ( U ) := ψ ( (cid:80) x ∈ U δ x ). ψ We collect below some further properties of the coefficients ψ in (5.2):(a) For all σ ∈ {− , , } , if | ξ | = 0, i.e. the dual configuration is empty, then ψ ( ξ ) = 1.(b) For σ = 0, ψ ( ξ ) = 0 for all ξ ∈ (cid:99) X such that | ξ | ≥ σ ∈ {− , , } and for all x ∈ V , ψ ( δ x ) = 0.(d) If σ ∈ {− , } and θ L (cid:54) = θ R , as a consequence of Theorem 3.4 and ( θ L − θ R ) > ψ ( δ x + δ y ) is negative for σ = − σ = 1 for all x, y ∈ V .(e) Because ψ ( δ x + · · · + δ x n ) depends only on the underlying geometry of the system andnot on θ L , θ R , exchanging the role of θ L and θ R does not affect the value of the stationary n -point correlation functions if n ∈ N is even, while it involves only a change of sign if n ∈ N is odd. More precisely, for all n ∈ N and x , . . . , x n ∈ V , E µ θL,θR (cid:34) n (cid:89) i =1 (cid:18) η ( x i ) α x i − ¯ θ x i (cid:19)(cid:35) = ( − n E µ θR,θL (cid:34) n (cid:89) i =1 (cid:18) η ( x i ) α x i − ¯ θ x i (cid:19)(cid:35) . f) As we will see in the course of the next section 5.2, ψ ( ξ ) in (5.2)–(5.3) can be defined forany ξ ∈ (cid:99) X and equivalently expressed in terms of a parameter β ∈ R . More precisely,given ξ ∈ (cid:99) X and β ∈ R , we have ψ ( ξ ) = (cid:88) ζ ≤ ξ ( − | ξ |−| ζ | (cid:32) (cid:89) x ∈ V (cid:18) ξ ( x ) ζ ( x ) (cid:19) ( (cid:98) p ∞ ( δ x , δ L ) − β ) ξ ( x ) − ζ ( x ) (cid:33) (cid:98) E ζ (cid:104) (1 − β ) ζ ∞ ( L ) ( − β ) ζ ∞ ( R ) (cid:105) . (5.4)Notice that, by setting ξ = δ x + · · · + δ x n with x i (cid:54) = x j if i (cid:54) = j , all the binomial coefficientsin (5.4) are equal to one. The choice β = 0 corresponds then to the expression on thel.h.s. of (5.2), while choosing β = 1 leads to ψ ( ξ ) = (cid:88) ζ ≤ ξ ( − ζ (cid:98) P IRW ξ − ζ (( ξ − ζ ) ∞ ( R ) = | ξ − ζ | ) (cid:98) P ζ ( ζ ∞ ( R ) = | ζ | ) . (5.5)In particular, since ψ ( ξ ) does not depend on β , we have thatd ψ ( ξ )d β = 0 , (5.6)which is an equation giving information on the absorption probabilities. If we consider,for instance, the case ξ = δ x + δ y with x (cid:54) = y , (5.4) and (5.6) yield2 (cid:98) P ξ = δ x + δ y ( ξ ∞ ( L ) = 2) + (cid:98) P ξ = δ x + δ y ( ξ ∞ ( L ) = 1) = (cid:98) p ∞ ( δ x , δ L ) + (cid:98) p ∞ ( δ y , δ L ) , (5.7)which corresponds to the recursive relation found in [5, Proposition 5.1]. More generally,by matching the two expressions of ψ ( ξ ) for ξ = δ x + · · · + δ x n with x i (cid:54) = x j if i (cid:54) = j , in(5.3) and (5.5), the relation that we find is (cid:98) P ξ ( ξ ∞ ( L ) = | ξ | ) − ( − | ξ | (cid:98) P ξ ( ξ ∞ ( R ) = | ξ | )= (cid:88) ζ<ξ (cid:40) (cid:98) P IRW ξ − ζ (( ξ − ζ ) ∞ ( R ) = | ξ − ζ | ) (cid:98) P ζ ( ζ ∞ ( R ) = | ζ | ) − ( − | ξ | (cid:98) P IRW ξ − ζ (( ξ − ζ ) ∞ ( L ) = | ξ − ζ | ) (cid:98) P ζ ( ζ ∞ ( L ) = | ζ | ) (cid:41) . In other words, the above equation relates the probabilities of having all | ξ | dual particlesabsorbed at the same end with a linear combination of analogous probabilities for systemswith a strictly smaller number of particles. Theorem 5.1 follows from a more general result. This is the content of Theorem 5.6 below.There, we show that a decomposition reminiscent of that in (5.2) holds also for expectations atsome fixed positive time of generalizations of the n -point correlation functions of Theorem 5.1when the particle system starts from a suitable product measure. The aforementioned gener-alizations of the correlation functions are constructed by suitably recombining the orthogonalduality functions of Section 4 so to obtain a family of functions orthogonal w.r.t. what we call“interpolating product measures” given in the following definition. D EFINITION ( interpolating product measures ) . We call interpolating product mea-sure with interpolating parameters β = { β x : x ∈ V } (5.8) he measure given by µ θ L ,θ R , β := ⊗ x ∈ V ν x,θ x , (5.9) with θ x := θ R + β x ( θ L − θ R ) , (5.10) where the marginals { ν x,θ : x ∈ V } appearing in (5.9) are those given in (3.2) and β in (5.8) – (5.10) is chosen such that, for each choice of σ ∈ {− , , } , the product measure µ θ L ,θ R , β is aprobability measure, i.e., for all x ∈ V , the following conditions hold: β x ∈ R and θ x = θ R + β x ( θ L − θ R ) ∈ Θ . (5.11)In particular, if we choose β x = (cid:98) p ∞ ( δ x , δ L ) =: ¯ β x , x ∈ V , as corresponding interpolating product measure we recover the local equilibrium product mea-sure µ ¯ θ (Definition 3.2): µ θ L ,θ R , ¯ β = µ ¯ θ . (5.12)Let us now introduce what we call the “interpolating orthogonal functions”. D EFINITION ( interpolating orthogonal functions ) . Recalling the definition of or-thogonal polynomial dualities in (4.5) – (4.6) and the definition of interpolating parameters β in (5.8) , we define the interpolating orthogonal function with interpolating parameters β as follows: D orθ L ,θ R , β ( ξ, η ) := d orL,θ L ( ξ ( L )) × (cid:32) (cid:89) x ∈ V d orx,θ x ( ξ ( x ) , η ( x )) (cid:33) × d orR,θ R ( ξ ( R )) , (5.13) where the parameters { θ x : x ∈ V } are defined in terms of θ L , θ R and β as in (5.10) . In analogy with (5.12), we define D or ¯ θ ( ξ, η ) := D orθ L ,θ R , ¯ β ( ξ, η )= d orL,θ L ( ξ ( L )) × (cid:32) (cid:89) x ∈ V d orx, ¯ θ x ( ξ ( x ) , η ( x )) (cid:33) × d orR,θ R ( ξ ( R )) . (5.14) R EMARK
We note that, despite the analogy in notation, in general these functions are not duality functions for the particle system { η t : t ≥ } , unless we assume the system to be atequilibrium, i.e. θ L = θ R = θ ∈ Θ . Only in the latter case, D orθ L ,θ R , β ( ξ, η ) = D orθ ( ξ, η ) for allchoices of β . With the definition (5.13), we have (cf. Remark 4.3) that D orθ L ,θ R , β ( ξ, · ) = 0 , if ξ ∈ (cid:99) X \ (cid:99) Y , (5.15)and that the family of functions (cid:110) D orθ L ,θ R , β ( ξ, · ) : ξ ∈ (cid:99) Y (cid:111) is an orthogonal basis in L ( X , µ θ L ,θ R , β ). Now we are ready to state the main result of thissection, whose Theorem 5.1 is a particular instance. HEOREM
Let us consider two set of interpolating parameters β = { β x : x ∈ V } and β (cid:48) = (cid:8) β (cid:48) x : x ∈ V (cid:9) both satisfying (5.10) . Then, for all ξ ∈ (cid:99) Y ⊂ (cid:99) X and t ≥ , we have E µ θL,θR, β (cid:104) D orθ L ,θ R , β (cid:48) ( ξ, η t ) (cid:105) = ( θ L − θ R ) | ξ | ψ t, β , β (cid:48) ( ξ ) , (5.16) where ψ t, β , β (cid:48) ( ξ ) := (cid:88) ζ ≤ ξ ( − | ξ |−| ζ | (cid:32) (cid:89) x ∈ V (cid:18) ξ ( x ) ζ ( x ) (cid:19) ( β (cid:48) x ) ξ ( x ) − ζ ( x ) (cid:98) E ζ (cid:34) { ζ t ( R )=0 } (cid:32) (cid:89) x ∈ V ( β x ) ζ t ( x ) (cid:33)(cid:35)(cid:33) , (5.17) and ψ t, β , β (cid:48) ( ξ ) does not depend on neither θ L nor θ R , but only on β , β (cid:48) , σ ∈ {− , , } and theunderlying geometry of the system. Moreover, by sending t to infinity in (5.16) we obtain, forall ξ ∈ (cid:99) Y ⊂ (cid:99) X , E µ θL,θR (cid:104) D orθ L ,θ R , β (cid:48) ( ξ, η ) (cid:105) = ( θ L − θ R ) | ξ | ψ β (cid:48) ( ξ ) , (5.18) where ψ β (cid:48) ( ξ ) := (cid:88) ζ ≤ ξ ( − | ξ |−| ζ | (cid:32) (cid:89) x ∈ V (cid:18) ξ ( x ) ζ ( x ) (cid:19) ( β (cid:48) x ) ξ ( x ) − ζ ( x ) (cid:33) (cid:98) P ζ [ ζ ∞ ( L ) = | ζ | ] . (5.19) Again, ψ β (cid:48) ( ξ ) is independent of θ L and θ R . R EMARK
From the proof of Theorem 5.6, the results of the theorem extend to configurations ξ ∈ (cid:99) X \ (cid:99) Y and, by (5.15) , ψ t, β , β (cid:48) ( ξ ) = 0 , if ξ ∈ (cid:99) X \ (cid:99) Y . (5.20)Before moving to the next section, Section 5.3, in which we provide the proof of Theorem5.6, we show how this latter result implies Theorem 5.1. P ROOF OF T HEOREM
Recall that, by the definitions of hypergeometric functions (4.1)–(4.2) and of single-site orthogonal duality functions in (4.5), we have, for all n ∈ N and η ∈ X , D or ¯ θ ( δ x + · · · + δ x n , η ) = n (cid:89) i =1 (cid:18) η ( x i ) α x i − ¯ θ i (cid:19) (5.21)anytime x , . . . , x n ∈ V with x i (cid:54) = x j if i (cid:54) = j . By choosing for any x ∈ V , β (cid:48) x = (cid:98) p ∞ ( δ x , δ L ), theresult follows immediately from Theorem 5.6. ψ Theorem 5.1 may be seen as a particular instance of Theorem 5.6 with the choice t = ∞ , ξ ∈ (cid:99) X consisting of finitely many particles all sitting at different sites in the bulk and β (cid:48) x = (cid:98) p ∞ ( δ x , δ L )for every x ∈ V . In fact, Theorem 5.6 extends the relation (5.2) to all ξ ∈ (cid:99) X , i.e. E µ θL,θR (cid:2) D or ¯ θ ( ξ, η ) (cid:3) = ( θ L − θ R ) | ξ | ψ ( ξ ) , (5.22) ith, ψ ( ξ ) := (cid:88) ζ ≤ ξ (cid:18) ξζ (cid:19) ( − ξ − ζ (cid:98) P IRW ξ − ζ (( ξ − ζ ) ∞ ( L ) = | ξ − ζ | ) (cid:98) P ζ ( ζ ∞ ( L ) = | ζ | ) , (5.23)where (cid:0) ξζ (cid:1) := (cid:81) x ∈ V (cid:0) ξ ( x ) ζ ( x ) (cid:1) and (cid:98) P IRW refers to the law of the dual process for σ = 0, consisting ofnon-interacting random walks.In order to obtain a more probabilistic interpretation of (5.23), we define(a) the probability measure γ ξ on (cid:99) X given by γ ξ ( ζ ) = (cid:0) ξζ (cid:1) | ξ | { ζ ≤ ξ } , (5.24)i.e. the distribution of uniformly chosen sub-configuration of ξ (i.e. ζ ≤ ξ );(b) the function Ψ ξ : (cid:99) X → R given byΨ ξ ( ζ ) := { ζ ≤ ξ } (cid:98) P IRW ξ − ζ (( ξ − ζ ) ∞ ( L ) = | ξ − ζ | ) (cid:98) P ζ ( ζ ∞ ( L ) = | ζ | ) , i.e., the function that assigns to any ζ ≤ ξ the probability that, in a system composed bythe superposition of the configuration ζ of interacting dual particles and the configuration ξ − ζ of independent dual random walks , independent between each other, all the particlesare eventually absorbed at L .The function ψ ( ξ ) in (5.23) can, then, be rewritten as follows: ψ ( ξ ) = 2 | ξ | (cid:90) (cid:99) X ( − | ξ − ζ | Ψ ξ ( ζ )d γ ξ ( ζ ) . Similarly, for all t ≥ ξ ∈ (cid:99) X and for the special choice β = β (cid:48) and β x = β (cid:48) x = (cid:98) p ∞ ( δ x , δ L ) , the identity in (5.16) yields, as a particular case, E µ ¯ θ (cid:2) D or ¯ θ ( ξ, η t ) (cid:3) = ( θ L − θ R ) | ξ | ψ t ( ξ ) , (5.25)where ψ t ( ξ ) := (cid:88) ζ ≤ ξ ( − ξ − ζ (cid:98) P IRW ξ − ζ (( ξ − ζ ) ∞ ( L ) = | ξ − ζ | ) (cid:98) E ζ (cid:104)(cid:98) P IRW ζ t ( ζ ∞ ( L ) = | ζ | ) (cid:105) (5.26)= 2 | ξ | (cid:90) (cid:99) X ( − | ξ |−| ζ | Ψ t,ξ ( ζ )d γ ξ ( ζ ) , where the integral in the last identity is w.r.t. the probability measure γ ξ defined in (5.24) andΨ t,ξ ( ζ ) := { ζ ≤ ξ } (cid:98) P IRW ξ − ζ (( ξ − ζ ) ∞ ( L ) = | ξ | − | ζ | ) (cid:98) E ζ (cid:104)(cid:98) P IRW ζ t ( ζ ∞ ( L ) = | ζ | ) (cid:105) . .3 Proof or Theorem 5.6 We prove Theorem 5.6 in two steps.First we obtain a formula to relate the functions D orθ L ,θ R , β (cid:48) ( ξ, η ) in (5.13) appearing in thestatement of Proposition 5.6 to the orthogonal duality functions D orθ ( ξ, η ) in Section 4, for some θ ∈ Θ . L EMMA
For each choice of σ ∈ {− , , } and β ∈ R , we define θ := θ R + β ( θ L − θ R ) . (5.27) Then, for all configurations η ∈ X and ξ ∈ (cid:99) X , D orθ L ,θ R , β (cid:48) ( ξ, η ) = (cid:88) ζ ≤ ξ ( θ L − θ R ) | ξ |−| ζ | ( − | ξ |−| ζ | E β (cid:48) ,β ( ζ, ξ ) D orθ ( ζ, η ) , where E β (cid:48) ,β ( ζ, ξ ) is defined as E β (cid:48) ,β ( ζ, ξ ) := E L,β ( ζ ( L ) , ξ ( L )) × (cid:32) (cid:89) x ∈ V E x,β (cid:48) x ,β ( ζ ( x ) , ξ ( x )) (cid:33) × E R,β ( ζ ( R ) , ξ ( R )) , (5.28) where, for all x ∈ V , E x,β (cid:48) x ,β ( (cid:96), k ) := (cid:18) k(cid:96) (cid:19) ( β (cid:48) x − β ) k − (cid:96) { (cid:96) ≤ k } , and E L,β ( (cid:96), k ) := (cid:18) k(cid:96) (cid:19) (1 − β ) k − (cid:96) { (cid:96) ≤ k } E R,β ( (cid:96), k ) := (cid:18) k(cid:96) (cid:19) ( − β ) k − (cid:96) { (cid:96) ≤ k } . P ROOF . By definition of the orthogonal duality functions in Theorem 4.1 (see also (4.7)) andof the functions D orθ L ,θ R , β (cid:48) in (5.13), we have D orθ = (cid:16) e − θ (cid:99) K (cid:17) left D c(cid:96) and D orθ L ,θ R , β (cid:48) = (cid:18) e − θ L (cid:99) K L − (cid:16)(cid:80) x ∈ V θ (cid:48) x (cid:99) K x (cid:17) − θ R (cid:99) K R (cid:19) left D c(cid:96) , where θ (cid:48) x := θ R + β (cid:48) x ( θ L − θ R ) , x ∈ V .
Next, we get D orθ L ,θ R , β (cid:48) = (cid:18) e − θ L (cid:99) K L − (cid:16)(cid:80) x ∈ V θ (cid:48) x (cid:99) K x (cid:17) − θ R (cid:99) K R + θ (cid:99) K (cid:19) left (cid:16) e − θ (cid:99) K (cid:17) left D c(cid:96) = (cid:18) e − ( θ L − θ ) (cid:99) K L − (cid:16)(cid:80) x ∈ V ( θ (cid:48) x − θ ) (cid:99) K x (cid:17) − ( θ R − θ ) (cid:99) K R (cid:19) left D orθ , where the latter identity is a consequence of the fact that all the operators { (cid:99) K x : x ∈ V } ∪{ (cid:99) K L , (cid:99) K R } commute. The expressions in terms of ( θ L − θ R ) of the parameters { θ (cid:48) x : x ∈ V } in(5.11) and θ in (5.27) yield the final result. hen, we derive an analogue of Theorem 5.6 for the orthogonal duality functions. L EMMA
For each choice of σ ∈ {− , , } and β ∈ R and θ ∈ R as in (5.27) and such that θ ∈ Θ , we have, for all configurations ζ ∈ (cid:99) X , E µ θL,θR, β [ D orθ ( ζ, η t )] = ( θ L − θ R ) | ζ | φ t, β ,β ( ζ ) , (5.29) where φ t, β ,β ( ζ ) ∈ R is defined as φ t, β ,β ( ζ ) := (cid:98) E ζ (cid:34) (1 − β ) ζ t ( L ) × (cid:32) (cid:89) x ∈ V ( β x − β ) ζ t ( x ) (cid:33) × ( − β ) ζ t ( R ) (cid:35) (5.30) and, in particular, it does not depend on neither θ L nor θ R , but only on β , β , σ ∈ {− , , } and the underlying geometry of the system. P ROOF . Recall the definition of µ θ L ,θ R , β in (5.9) and of the scale parameters { θ x : x ∈ V } in(5.10). By duality (Theorem 4.1), we have E µ θL,θR, β [ D orθ ( ζ, η t )]= (cid:88) ζ (cid:48) ∈ (cid:99) X (cid:98) p t ( ζ, ζ (cid:48) ) E µ θL,θR, β (cid:2) D orθ ( ζ (cid:48) , η ) (cid:3) = (cid:88) ζ (cid:48) ∈ (cid:99) X (cid:98) p t ( ζ, ζ (cid:48) ) (cid:40) ( θ L − θ ) ζ (cid:48) ( L ) × (cid:32) (cid:89) x ∈ V ( θ x − θ ) ζ (cid:48) ( x ) (cid:33) × ( θ R − θ ) ζ (cid:48) ( R ) (cid:41) , where this last identity is a consequence of (cid:88) n ∈ N d orx,θ ( k, n ) ν x,θ x ( n ) = ( θ x − θ ) k for all x ∈ V and k ∈ { , . . . , α x } if σ = − k ∈ N if σ ∈ { , } (see e.g. [19]). We obtain(5.29) with the function φ t, β ,β as in (5.30) by rewriting in terms of the parameters β and β theexpression above between curly brackets.A combination of Lemma 5.8 and Lemma 5.9 concludes the proof of Theorem 5.6. Indeed, E µ θL,θR, β (cid:104) D orθ L ,θ R , β (cid:48) ( ξ, η t ) (cid:105) = (cid:88) ζ ≤ ξ ( θ L − θ R ) | ξ |−| ζ | ( − | ξ |−| ζ | E β (cid:48) ,β ( ζ, ξ ) E µ θL,θR, β [ D orθ ( ζ, η t )]= ( θ L − θ R ) | ξ | (cid:88) ζ ≤ ξ ( − | ξ |−| ζ | E β (cid:48) ,β ( ζ, ξ ) φ t, β ,β ( ζ ) , which yields (5.16) with ψ t, β , β (cid:48) ( ξ ) given by ψ t, β , β (cid:48) ( ξ ) = (cid:88) ζ ∈ (cid:99) X ( − | ξ |−| ζ | E β (cid:48) ,β ( ζ, ξ ) φ t, β ,β ( ζ ) . (5.31)We note that, because the l.h.s. in (5.16) and ( θ L − θ R ) | ξ | do not depend on the parameter β ∈ R ,the whole expression in (5.31) is independent of β , and in particular, we obtain (5.16) for thechoice β = 0. By passing to the limit as t goes to infinity on both sides in (5.16), by uniquenessof the stationary measure µ θ L ,θ R , we obtain (5.18)–(5.19). Exponential moments and generating functions
In this section we use the fact that the orthogonal dualities have explicit and simple generatingfunctions in order to produce a formula for the joint moment generating function of the occu-pation variables in the non-equilibrium stationary state, in terms of the absorbing dual startedfrom a random configuration ξ of which the distribution is related to the reservoir parameters. T HEOREM
Let λ = { λ x : x ∈ V } ∈ R N be such that, for all x ∈ V , Λ x := 1 + λ x σλ x (1 + ¯ θ x ) ≥ , (6.1) and κ x := λ x ( θ L − θ R )1 + σλ x (1 − ( θ L − θ R )) ∈ Θ . (6.2)
Then, we have E µ θL,θR (cid:34) (cid:89) x ∈ V (Λ x ) η ( x ) (cid:35) = (cid:32) (cid:89) x ∈ V J θ L ,θ R ,λ x (cid:33) E µ k [ ψ ] , (6.3) and, for all t ≥ , E µ ¯ θ (cid:34) (cid:89) x ∈ V (Λ x ) η t ( x ) (cid:35) = (cid:32) (cid:89) x ∈ V J θ L ,θ R ,λ x (cid:33) E µ k [ ψ t ] , (6.4) where ψ and ψ t are given in (5.2) and (5.26) , respectively, µ k = ⊗ x ∈ V ν x,κ x is the probabilitymeasure defined in (3.1) with parameters κ = { κ x : x ∈ V } , viewed as a probability measure on (cid:99) X concentrated on (cid:99) Y , and J θ L ,θ R ,λ x := e α x λ x (¯ θ x +( θ L − θ R )) if σ = 0 (cid:18) σλ x (1 + ¯ θ x )1 + σλ x (1 − ( θ L − θ R )) (cid:19) σα x if σ ∈ {− , } . (6.5) R EMARK ( conditions (6.1) & (6.2)) . Condition (6.1) is obtained for λ x ⊂ (cid:18) −∞ ,
11 + θ x (cid:21) ∪ (cid:20) θ x , ∞ (cid:19) if σ = − − , ∞ ) if σ = 0 (cid:18) −∞ , −
11 + θ x (cid:21) ∪ (cid:20) −
12 + θ x , ∞ (cid:19) if σ = 1 , (6.6) while condition (6.2) for (i) Case θ L − θ R ≥ : λ x ⊂ [1 , ∞ ) if σ = − , ∞ ) if σ = 0[ θ L − θ R − , ∞ ) if σ = 1 , (6.7)(ii) Case θ L − θ R ≤ : λ x ⊂ ( −∞ , − θ L + θ R ] if σ = − −∞ , − if σ = 0( −∞ , θ L − θ R − if σ = 1 . (6.8) e devote the remaining of this section to the proof of Theorem 6.1. To this purpose, let usrecall the definition of { w x : x ∈ V } and { z x, · : x ∈ V } in (2.17) and (3.4), respectively. D EFINITION ( single-site generating functions ) . For each choice of σ ∈ {− , , } ,for all x ∈ V and for all functions f : N → R , we define ( Υ x f ) ( λ ) := ∞ (cid:88) k =0 w x ( k ) k ! (cid:16) λ σλ (cid:17) k z x,λ f ( k ) , (6.9)( Υ L f )( λ ) := ∞ (cid:88) k =0 ( α L λ ) k k ! f ( k ) e − α L λ and ( Υ R f )( λ ) := ∞ (cid:88) k =0 ( α R λ ) k k ! f ( k ) e − α R λ for all λ ∈ R such that the above series absolutely converge. Moreover, we define Υ := Υ L ⊗ ( ⊗ x ∈ V Υ x ) ⊗ Υ R , (6.10) acting on functions f : N N +20 → R . R EMARK If λ ∈ Θ then, for all x ∈ V and f : N → R , ( Υ x f ) ( λ ) = E ν x,λ [ f ] , where ν x,λ x is given in (3.3) . As a first step, we investigate the action of the operators { Υ x : x ∈ V } on the dualityfunctions. L EMMA ( duality and generating functions ) . For each choice of σ ∈ {− , , } , forall θ ∈ Θ and for all x ∈ V , ( Υ x ) left d c(cid:96)x ( · , n )( λ ) = (cid:16) λ σλ (cid:17) n z x,λ (6.11) and ( Υ x ) left d orx,θ ( · , n )( λ ) = (cid:16) λ σλ (1+ θ ) (cid:17) n z x,λ (1+ θ ) . (6.12) Moreover Υ left D orθ ( · , η )( λ ) = e − α L λ L (1+ θ − θ L ) (cid:89) x ∈ V (cid:16) λ x σλ x (1+ θ ) (cid:17) η ( x ) z x,λ x (1+ θ ) e − α R λ R (1+ θ − θ R ) , and, analogously, Υ left D or ¯ θ ( · , η )( λ ) = e − α L λ L (cid:89) x ∈ V (cid:16) λ x σλ x (1+¯ θ x ) (cid:17) η ( x ) z x,λ x (1+¯ θ x ) e − α R λ R . (6.13) EMARK
In order to guarantee the absolute convergence of the series in the definition ofthe operators Υ in Definition 6.3, for the case σ = 1 we have to choose λ and θ such that (cid:12)(cid:12)(cid:12)(cid:12) θλ λ (cid:12)(cid:12)(cid:12)(cid:12) < . P ROOF . By (4.9), we prove (6.12) from which, by setting θ = 0, (6.11) follows. By definitionof Υ x in (6.9), relation (4.10) and the form of the functions { w x : x ∈ V } (see (2.17)), we obtain( Υ x ) left d orx,θ ( · , n )( λ ) = ∞ (cid:88) k =0 w x ( k ) k ! (cid:16) λ σλ (cid:17) k z x,λ d orx,θ ( k, n )= n (cid:88) (cid:96) =0 (cid:18) n(cid:96) (cid:19) (cid:16) λ σλ (cid:17) (cid:96) z x,λ ∞ (cid:88) k = (cid:96) w x ( k ) w x ( (cid:96) )( k − (cid:96) )! (cid:18) − θλ σλ (cid:19) k − (cid:96) = n (cid:88) (cid:96) =0 (cid:18) n(cid:96) (cid:19) (cid:16) λ σλ (cid:17) (cid:96) z x,λ F x ( θ, λ, (cid:96) ) , where, as long as (cid:12)(cid:12)(cid:12) θλ λ (cid:12)(cid:12)(cid:12) < σ = 1 and for all λ ∈ R otherwise, F x ( θ, λ, (cid:96) ) = (cid:18) σλ (1 + θ )1 + σλ (cid:19) − ( σα x + (cid:96) ) if σ ∈ {− , } e − α x θλ if σ = 0 . P ROOF OF T HEOREM
We start by proving (6.4). By (6.13), the exchange between Υ left and the expectation w.r.t. the variables η , (5.26), (5.20) and the definition of µ κ , we rewrite thel.h.s. in (6.3) as E µ ¯ θ (cid:2) Υ left D or ¯ θ ( · , η t )( λ ) (cid:3) e α L λ L + α R λ R (cid:32) (cid:89) x ∈ V z x,λ x (1+¯ θ x ) (cid:33) = Υ (cid:16) ( θ L − θ R ) |·| ψ t, β ( · ) (cid:17) ( λ ) e α L λ L + α R λ R (cid:32) (cid:89) x ∈ V z x,λ x (1+¯ θ x ) (cid:33) = (cid:88) ξ ∈ (cid:99) X (cid:89) x ∈ V w x ( ξ ( x ))( ξ ( x ))! (cid:16) λ x ( θ L − θ R )1+ σλ x (cid:17) ξ ( x ) z x,λ x ψ t ( ξ ) (cid:32) (cid:89) x ∈ V z x,λ x (1+¯ θ x ) (cid:33) = (cid:32) (cid:89) x ∈ V z x,λ x (1+¯ θ x ) z x,κ x z x,λ x (cid:33) (cid:88) ξ ∈ (cid:99) X µ κ ( ξ ) ψ t ( ξ ) . The explicit form of { z x, · : x ∈ V } given in (3.4) yields (6.3). Sending t → ∞ in (6.4), by theuniqueness of the stationary measure, we obtain (6.3). Existence and uniqueness of the equilibrium and non-equilibrium station-ary measure
In this appendix, we treat with full details the issue of existence and uniqueness of the stationarymeasure for IRW and SIP in equilibrium and non-equilibrium. In what follows we take either σ = 0 or σ = 1.We recall that a probability measure µ on the countable space X (endowed with the discretetopology) is the unique stationary measure for the particle system { η t : t ≥ } if, for all boundedfunctions f : X → R and for all probability measures µ (cid:48) on X , the following holds:lim t →∞ E µ (cid:48) [ f ( η t )] = E µ [ f ( η )] . (A.1)Out of all probability measures µ on X , we say that µ is tempered if it is characterized by theintegrals E µ (cid:104) D c(cid:96) ( ξ, η ) (cid:105) , for all ξ ∈ (cid:99) X . To the purpose of determining whether a probability measure µ is tempered or not, we adoptthe following strategy. First, we recall that the functions { D c(cid:96) ( ξ, · ) : ξ ∈ (cid:99) X } are weightedproducts of factorial moments of the variables { η ( x ) : x ∈ V } (see Proposition 2.3). Then, weexpress these weighted factorial moments in terms of moments. We conclude by means of amultidimensional Carleman’s condition.By following the aforementioned ideas, we provide in the following lemma a sufficient con-dition for a measure to be tempered. L EMMA
A.1.
Let µ be a probability measure on X . If there exists θ ∈ Θ = [0 , ∞ ) such that E µ (cid:104) D c(cid:96) ( ξ, η ) (cid:105) ≤ θ | ξ | (A.2) for all ξ ∈ (cid:99) X , then µ is tempered. P ROOF . Let us start by expressing the moments of η ( x ) in terms of single-site classical dualityfunctions in (2.15): for all x ∈ V and for all k, n ∈ N , n k = k (cid:88) (cid:96) =0 (cid:26) k(cid:96) (cid:27) d c(cid:96)x ( (cid:96), n ) w x ( (cid:96) ) , where (cid:8) k(cid:96) (cid:9) denotes the Stirling number of the second kind given by (cid:26) k(cid:96) (cid:27) = 1 (cid:96) ! (cid:96) (cid:88) j =0 ( − (cid:96) − j (cid:18) (cid:96)j (cid:19) j k . (A.3)In view of (A.2), we obtain E µ (cid:104) ( η ( x )) k (cid:105) = k (cid:88) (cid:96) =0 (cid:26) k(cid:96) (cid:27) E µ (cid:104) D c(cid:96) ( (cid:96)δ x , η ) (cid:105) w x ( (cid:96) ) ≤ k (cid:88) (cid:96) =0 w x ( (cid:96) ) (cid:96) ! E µ (cid:104) D c(cid:96) ( (cid:96)δ x , η ) (cid:105) (cid:96) (cid:88) j =0 (cid:18) (cid:96)j (cid:19) j k ≤ k k k (cid:88) (cid:96) =0 (2 θ ) (cid:96) (cid:96) ! w x ( (cid:96) ) . y recalling the definition of w x ( (cid:96) ) in (2.17), in both cases with σ = 0 and σ = 1, we get E µ (cid:104) ( η ( x )) k (cid:105) ≤ ( a x k ) k , (A.4)for all k ∈ N , with a x = (1 + 2 θα x ) for σ = 0 and a x = (cid:98) α x (cid:99) !(1 + 2 θ ) (cid:98) α x (cid:99) +1 for σ = 1. Therefore,if m x ( k ) := E µ (cid:2) ( η ( x )) k (cid:3) , (A.4) yields ∞ (cid:88) k =1 m x (2 k ) − k ≥ a x ∞ (cid:88) k =1 k = ∞ . Because the above condition holds for all x ∈ V , the multidimensional Carleman condition(see e.g. [20, Theorem 14.19]) applies. Hence, µ is completely characterized by the moments { m x ( k ) : x ∈ V, k ∈ N } and, in turn, is tempered.Now, by means of duality, we observe that, for all η ∈ X and ξ ∈ (cid:99) X with | ξ | = k ,lim t →∞ E η (cid:104) D c(cid:96) ( ξ, η t ) (cid:105) = lim t →∞ (cid:98) E ξ (cid:104) D c(cid:96) ( ξ t , η ) (cid:105) = k (cid:88) (cid:96) =0 θ (cid:96)L θ k − (cid:96)R (cid:98) P ξ ( ξ ∞ = (cid:96)δ L + ( k − (cid:96) ) δ R ) . (A.5)We note that the expression above does not depend on η ∈ X and, moreover,lim t →∞ E η (cid:104) D c(cid:96) ( ξ, η t ) (cid:105) ≤ ( θ L ∨ θ R ) | ξ | for all ξ ∈ (cid:99) X . Therefore, by Lemma A.1, there exists a unique probability measure µ (cid:63) on X such that E µ (cid:63) (cid:104) D c(cid:96) ( ξ, η ) (cid:105) = | ξ | (cid:88) (cid:96) =0 θ (cid:96)L θ | ξ |− (cid:96)R (cid:98) P ξ ( ξ ∞ = (cid:96)δ L + ( | ξ | − (cid:96) ) δ R ) . Furthermore, because the convergence in (A.5) for all ξ ∈ (cid:99) X implies convergence of all marginalmoments and because the limiting measure is uniquely characterized by these limiting moments,then, for all f : X → R bounded and for all η ∈ X , we havelim t →∞ E η [ f ( η t )] = E µ (cid:63) [ f ( η )] . (A.6)By dominated convergence, (A.6) yields, for all probability measures µ on X and f : X → R ,lim t →∞ E µ [ f ( η t )] = E µ (cid:104) lim t →∞ E η [ f ( η t )] (cid:105) = E µ (cid:63) [ f ( η )] , i.e. µ (cid:63) is the unique stationary measure of the process { η t : t ≥ } . A CKNOWLEDGMENTS.
The authors would like to thank Gioia Carinci and Cristian Giardin`afor useful discussions. F.R. and S.F. thank Jean-Ren´e Chazottes for a stay at CPHT (InstitutPolytechnique de Paris), in the realm of Chaire d’Alembert (Paris-Saclay University), wherepart of this work was performed. S.F. acknowledges Simona Villa for her support in creating thepicture. S.F. acknowledges financial support from NWO via the TOP1 grant 613.001.753. F.S.acknowledges financial support from the European Union’s Horizon 2020 research and innovationprogramme under the Marie-Sk(cid:32)lodowska-Curie grant agreement No.754411. EFERENCES [1]
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