Multiplicative functionals on ensembles of non-intersecting paths

www.imstat.org/aihp 2015, Vol. 51, No. 1, 28–58

DOI:10.1214/13-AIHP579

© Association des Publications de l’Institut Henri Poincaré, 2015

Multiplicative functionals on ensembles of non-intersecting paths

Alexei Borodin

, Ivan Corwin

and Daniel Remenik

aDepartment of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA, and Institute for Information Transmission Problems, Moscow, Russia. E-mail:[email protected]

bDepartment of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA, and Clay Mathematics Institute, 10 Memorial Blvd. Suite 902, Providence, RI 02903, USA, and Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue,

Cambridge, MA 02139-4307, USA. E-mail:[email protected]

cDepartamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile. E-mail:[email protected]

Received 7 February 2013; revised 30 August 2013; accepted 8 September 2013

Abstract. The purpose of this article is to develop a theory behind the occurrence of “path-integral” kernels in the study of extended determinantal point processes and non-intersecting line ensembles. Our first result shows how determinants involving such kernels arise naturally in studying ratios of partition functions and expectations of multiplicative functionals for ensembles of non-intersecting paths on weighted graphs. Our second result shows how Fredholm determinants with extended kernels (as arise in the study of extended determinantal point processes such as the Airy₂process) are equal to Fredholm determinants with path-integral kernels. We also show how the second result applies to a number of examples including the stationary (GUE) Dyson Brownian motion, the Airy₂process, the Pearcey process, the Airy₁and Airy_2→1processes, and Markov processes on partitions related to thez-measures.

Résumé. Le but de cet article est de développer une théorie autour des noyaux de la forme « intégrale de chemin » qui apparaissent dans l’étude des processus déterminantaux et des familles de chemins sans intersection. Notre premier résultat montre comment des déterminants avec de tels noyaux apparaissent naturellement dans l’étude du quotient de fonctions de partition et d’espérances de fonctionnelles pour des familles de chemins sans intersection sur des graphes avec des pondérations. Notre second résultat montre comment les déterminants de Fredholm avec des noyaux étendus (comme ceux que l’on trouve dans le cas du processus déterminantal Airy₂) sont égaux à des déterminants de Fredholm avec des noyaux de la forme « intégrale de chemin ». Nous montrons aussi comment ce second résultat s’applique à une grande variété d’exemples dont le mouvement Brownien stationnaire de Dyson, le processus Airy₂, le processus de Pearcey, les processus Airy₁et Airy_2→1ainsi que les processus de Markov sur les partitions reliées auxz-mesures.

MSC:60B20; 60G55

Keywords:Non-intersecting paths; Determinantal point process

1. Introduction

The Airy₂ process is a universal scaling limit of a wide variety of probabilistic systems including random ma- trix theory, random growth processes, interacting particle systems and directed polymers in random media (see [28,48] and references therein). Denoted Airy₂(·), it is defined via its consistent finite dimensional distributions:

fort₁< t₂<· · ·< t_n, P

_n

i=1

Airy₂(t_i)≤s_i

=det

I−χ K₂^ext

L²({t1,...,tn}×R,μ). (1)

Hereχis an operator which acts on functionsf:{t1, . . . , t_n} ×R→Ras χf (t_i, x):=1x>sif (t_i, x).

The operatorK₂^extacts as K₂^extf (t_i, x):=

n j=1

RdyK₂^ext(t_i, x;t_j, y)f (t_j, y), whereK₂^ext(s, x;t, y)is the “extended” Airy₂kernel given by

K₂^ext(s, x;t, y):= ^0∞dλe^{−λ(s−t )}Ai(x+λ)Ai(y+λ) ifs≥t,

−₀

−∞dλe^{−λ(s−t )}Ai(x+λ)Ai(y+λ) ifs < t,

with Ai(x)the classical Airy function. The right-hand side of (1) is the Fredholm determinant of the identity minus a trace class operator (see Section3.1for definition and details) and the measureμappearing there is the product of counting measure on{t1, . . . , t_n}and Lebesgue measure onR.

The formula given in (1) for the finite dimensional distributions of the Airy₂process becomes increasingly cum- bersome asnincreases. This is due to then-dependence in theL²space on which the operators act. When taking a limit of a sequence of operators, or their determinants, it is convenient to have the operators all act on the sameL² space, rather than a sequence of different spaces.

In Prähofer and Spohn’s initial work on the Airy₂process (see Section 5 of [43] forn=2 or [26,44,46] forn≥2) the extended kernel formula is shown to be equivalent to the following “path-integral” kernel formula:

P _n

i=1

Airy₂(t_i)≤s_i

=det

I−K₂+ ¯P_s1e^(t1−t2)HP¯_s2· · ·e^{(tn−1−tn)H}P¯_sne^(tn−t1)HK₂

L²(R). (2)

Here K₂(x, y)=K₂^ext(0, x;0, y) is called the Airy₂ kernel, P¯_sg(x)=1x≤sg(x)is a projection operator and H=

−Δ+xis called the Airy Hamiltonian becauseHAi(· −s)=sAi(· −s)(Δis the Laplacian onR). The dependence onnhas been absorbed into the operator rather than theL²space, and it is now plausible to take a largenlimit.

The reason we call this a path-integral kernel is because a portion of it can be written in terms of the expectation of a certain path-integral. By the Feynman–Kac formula (cf. [35]),

P¯_s1e^(t1−t2)HP¯_s2· · ·e^{(tn−1−tn)H}P¯_snf (x)=Eb(t₁)=x

f

b(t_n) e⁻

_tn

t1 b(s)dsn i=1

1b(t_i)≤s_i

, (3)

whereb:[t1, t_n] →Ris the trajectory of a Brownian motion with diffusion coefficient 2 starting atb(t1)=x. Lett1<· · ·< tnfill out the interval[, r]and letsi =h(ti)for some functionh: [, r] →R. Then asngoes to infinity, the above operator has a limit in trace norm (see [26] or Section4.2below for details) given by

Γ_,r^h f (x)=Eb()=x

f b(r)

e⁻

r b(s)ds

1b≤h

,

where{b≤h}denotes the event{b(s)≤h(s),∀s∈ [, r]}. Thus, it is shown in [26], Theorem 2, or Eq. (41) below that

P

Airy₂(s)≤h(s),∀s∈ [, r]

=det

I−K₂+Γ_,r^h e^(r−)HK₂

L²(R). (4)

The above formula proved useful in [26] in providing a direct proof that the value of the maximum of the Airy₂ process minus a parabola is distributed according to the (GOE) Tracy–Widom distribution; and in [39] (see also [9, 22,50,51]) in computing the joint distribution for the value and location (int) of the maximum. The two-time path- integral kernel formula in [43] was utilized to compute asymptotics of the two-time covariance of the Airy₂process, since the extended kernel does not easily yield this. Note that the left-hand side in the last formula presupposes the existence of a continuous version of the Airy₂process. This was first shown to exist in [32].

What is remarkable about formula (4) is that the right-hand side is simple (despite the cumbersome finite dimen- sional distributions given above) and the event in question in the left-hand side has a clear translation into the operator Γ_,r^h . As a further application of the Feynman–Kac formula as well as the Cameron–Martin–Girsanov formula (see [26]), the integral kernel ofΓ_,r^h can be expressed as

Γ_,r^h (x, y)=P_b()=x−²_,b(r)=y−r²

b(s)≤h(s)−s²for alls∈ [, r] ,

whereb is now a Brownian bridge run from x−² at time toy −r² at time r (this means that Γ_,r^h f (x)= dyΓ_,r^h (x, y)f (y)). In other words, the probability that the Airy₂process hits a functionhcan be expressed as the Fredholm determinant of an operator which is partly expressed by the probability that a Brownian bridge hits the same function (minus a parabola).

We will now see how these formulas for the Airy₂process are part of a more general result.

1.1. Extended kernels and the path-integral kernel in a general setting

There are many other examples of extended determinantal point processes (some given in Section4) and our aim is to find path-integral kernel formulas for these other processes. This may have further applications, although we do not address them here. For example, besides the previous work of [44], in [46] a path-integral kernel formula was discovered for the Airy₁process and used to prove existence of a continuous version of the process and its Hölder regularity.

When the Airy₂process was introduced, it arose as the top layer of the multi-layer Airy₂process, which will be denoted{Airy2(i;t ): i∈Z_≥1, t∈R}and is such that Airy₂(i;t ) >Airy₂(j;t )fori < j. Ast varies,_∞

i=1δ_{Airy2(i;t )} forms anR-indexed collection of point processes which has the structure of an (extended) determinantal point process (see [10] and references therein) with correlation kernelK₂^ext(s, x;t, y).

This has the following consequence. Fort₁<· · ·< t_nfixq_ti:R→Rand letq¯_ti=1−q_ti. Forg:{t₁, . . . , t_n} →R defineq(g):=n

i=1qt_i(g(ti))and likewise defineq(g). Then (given some conditions on¯ q to ensure convergence – see Section4.2below)

E _∞

i=1

¯ q

Airy₂(i; ·)

=det

I−QK₂^ext

L²({t₁,...,tn}×R,μ), (5)

whereQf (ti, x):=qt_i(x)f (ti, x). The left-hand side above is referred to here as the expectation of a “multiplicative functional” of the multi-layer process. Whenqt_i(x)=1x>s_i (and henceq¯t_i(x)=1x≤s_i), the above formula reduces to (1) withQ=χ.

Our first result shows that the type of identity between extended and path-integral kernel Fredholm determinants which one gets by equating the right-hand sides of (1) and (2) is quite general and dependent on a few structural properties of the kernels.

In stating our results presently, we leave out a number of technical assumptions (see Section3for these details) and assume that we have the following collection of operators on functionsf:{t₁, . . . , t_n} →R:

• For each 1≤i, j≤n,Wti,tj (with the conventionWti,ti=I).

• For each 1≤i≤n,K_ti.

• A diagonal operatorQsuch thatQf (t_i,·):=Q_tif (t_i,·)where for 1≤i≤nandg:R→R,Q_tig(x):=q_ti(x)g(x).

Theorem 1.1 (Theorem3.3, with technical assumptions suppressed). Lett1<· · ·< tn and assume that for all 1≤i≤j≤k≤nthe following holds:

• Right-invertibility:Wti,tjWtj,tiK_ti=K_ti.

• Semigroup property:Wti,tjWtj,tk=Wti,tk.

• Reversibility relation:Wt_i,t_jK_tj=K_tiWt_i,t_j. Then

det

I−QK^ext

L²({t₁,...,t_n}×R)=det(I−K_t1+Q_t1Wt1,t2Q_t2· · ·Wtn−1,tnQ_tnWtn,t1K_t1)_L2(R), where

K^ext(t_i, x;t_j, y)=

Wti,tjK_tj(x, y) ifi≥j,

−Wti,tj(I−K_tj)(x, y) ifi < j,

andQ_ti=I−Q_ti.

This is proved in Section3.3essentially via linear algebra.

Letting Wti,tj =e^{−(tj−ti)H}, K_ti =K₂ andQ_si =P_si we recover the equality (2) for the Airy₂ process. More generally, the result implies that the expectation of a multiplicative functional of the multi-layer Airy₂process (5) can be expressed in a similar way, by replacing eachPs_i byQs_ion the right-hand side of (2).

In Section 4we apply this theorem to a variety of examples of extended determinantal point processes such as the stationary (GUE) Dyson Brownian motion, the Airy₂ process, the Pearcey process, and Markov processes on partitions related to thez-measures. We also show how the identity applies to signed extended determinantal point processes such as the Airy₁and Airy_2→1processes. In the case of the stationary (GUE) Dyson Brownian motion and the Airy₂process, we also obtain the continuum limits of the corresponding path-integral kernel formulas (which is likely doable in other cases as well).

1.2. Ensembles of non-intersecting paths and the path-integral kernel

The multi-layer Airy₂process arises as the scaling limit of a variety of ensembles of non-intersecting paths (see for instance [31]). The occurrence of extended kernel determinants in such ensembles is a consequence of the Eynard–

Mehta theorem, which implies the existence of an extended determinantal point process structure [20,27,32,40,54].

The equivalence of the extended kernel determinant formula with the path-integral kernel determinant formula which is given in Theorem1.1(see also Theorem3.3) is via linear algebra, but does not indicate why such a path-integral kernel formula exists. Theorem1.3below provides a direct link between ensembles of non-intersecting paths and path- integral kernel determinant formulas, and its proof boils down to the Lindström–Gessel–Viennot Lemma (recorded below as Lemma2.1). By first proving the path-integral kernel determinant formula and then relating it to an extended kernel determinant formula, this provides another proof of the extended determinant point process structure for these ensembles (i.e., the Eynard–Mehta theorem), see Section1.2.1.

Let us now introduce, for givenT ∈Z_≥0andN∈Z_≥1, the ensembles ofNnon-intersecting paths of lengthT. Let G=(V , E)be a finite directed acyclic planar graph with vertex setV =V₀V₁ · · · V_T (hererepresents the disjoint union of sets) and directed edge setE=E_0→1E_1→2 · · · E_T−1→T whereE_n→n+1only contains edges fromx→y withx∈Vnandy∈Vn+1. Herex→ydenotes an edge directed fromxtoy.

As an example, letVn be vertices ofZ² of the form(n, i)forn−i≡0 mod 2 (and |i| ≤M for someM) and letE_n contain all directed edges from(x, n)to(x±1, n+1). Paths in this directed graph are trajectories of simple symmetric random walks (constrained to stay within distanceMfrom the origin), cf. Fig.1.

We define apathπas a sequence of edges(e₀=x₀→x₁, e₁=x₁→x₂, . . . , e_T−1=x_T−1→x_T)wherex_i∈V_i for 0≤i≤T. For such a path, letπ(n)=x_ndenote thenth vertex in the path.

To edgese∈Ewe associate weightswe∈R, and to a pathπ we associate a weightw(π )given by the product of the weightsw(e_n)along the edgese_nofπ. Forx∈V0andy∈V_T we define a transition matrix

W(x, y):=

π:x→y

w(π ),

Fig. 1. An example of a graphGwith source verticesX on the left-hand side and sink verticesY on the right-hand side. Here there areN=3 non-intersecting paths which are shown in grey, with starting pointsπ_i(b)and ending pointsπ_i(d)fori=1,2,3.

where the summation is over all paths π from x toy. Instead of considering just a single path, we may consider ensembles ofN non-intersecting paths from elements ofV0to elements ofVT (by which we mean paths which use disjoint collections of vertices). We define the collection of all such paths as

N.I.(N ):=

Π= {π₁, . . . , π_N}: ∀i, π_igoes fromV₀toV_T and no two paths intersect .

We will describe a measure on such an ensemble. This requires the introduction of two additional families of functions. ForN fixed, consider functionsψi:V0→R, 1≤i≤N, andϕj:VT →R, 1≤j≤N. Define the weight ofΠ∈N.I.(N )as

Wt(Π ):=det ψi

πj(0)N i,j=1

_N

i=1

w(πi)

det ϕi

πj(T )N

i,j=1, (6)

and the partition function as Z=

Π∈N.I.(N )

Wt(Π ).

Ifψ_i andϕ_j areδ-functions then the ends of the paths are fixed.

Assuming thatZ=0 we may define a measure (not necessarily positive but with total integral 1) onΠ∈N.I.(N ) as

ν(Π )=Wt(Π ) Z .

When each of the three factors on the right-hand side of (6) are positive (and thus, in particular,ν is a probability measure), one may think ofνas follows. The determinants det[ψi(πj(0))]^N_i,j=1and det[φi(πj(0))]^N_i,j=1define mea- sures on the collections ofN initial points inV₀andN final points inV_T. The weightsw(π_i)in the middle factor in (6) describe the transition probabilities forN independent paths(π₁, . . . , π_N)connecting these points. The measure νis restricted to non-intersecting paths, and the division by the normalizing constantZmeans thatνcorresponds to a probability measure conditioned on theNpaths not intersecting.

Defineϕ_j⁽⁰⁾:V₀→Rby ϕ_j⁽⁰⁾(x):=

y∈V_TW(x, y)ϕ_j(y). Note that this implies thatW has a right-inverse on span{ϕ_i⁽⁰⁾}^N_i=1which is given byW⁻¹ϕ_j⁽⁰⁾=ϕ_j.

We will make the followingbiorthogonality assumptionon the{ψ_i}^N_i=1and{ϕ_j⁽⁰⁾}^N_j=1:

x∈V₀

ψ_i(x)ϕ_j⁽⁰⁾(x)=1i=j.

Remark 1.2. Assuming Z=0, one can show that it is always possible to perform a linear transformation in span{ψi}^N_i=1in such a way that the biorthogonality assumption is satisfied andνremains unchanged.

The final concept we introduce is that of apath-integral functional, which is any function f :E0→1×E1→2× · · · ×ET−1→T →R

such that

f (e0, e1, . . . , eT−1)=

T−1 n=0

fn(en)

for functionsf_n:E_n→n+1→R. This definition extends to a directed pathπfromV₀toV_T by settingf (π )equal tof applied to the ordered sequence of edges inπ. From the functionf define a second set of edge weightsw˜_e:=f_n(e)w_e werenis such thate∈En→n+1. With respect to these weights{ ˜we}e∈E define a transition matrix

W(x, y)=

π:x→y

˜ w(π ).

The following theorem is a consequence of Theorem2.2, which is a similar result for a more general graph setting.

The theorem shows how the path-integral kernel determinant naturally arises from ensembles of non-intersecting paths. A proof of the below result appears in Section2.

Theorem 1.3. For any path-integral functionalf as above

Π=(π₁,...,π_N)∈N.I.(N )

N i=1

f (π_i)ν(Π )=det

I−K+WW ⁻¹K

L²(V₀),

whereK:L²(V0)→L²(V0)is given by its kernel

K(x1, x2)=

N i=1

ϕ⁽⁰⁾_i (x1)ψi(x2).

As we will explain in the proof of Corollary 1.4below, the above result can also be seen as a consequence of Theorem 1.1and the known determinantal structure for ensembles of non-intersecting paths. Instead, we provide a direct and simple linear algebraic proof of Theorem1.3using only the Lindström–Gessel–Viennot Lemma. This provides an explanation of the appearance of path-integral kernel formulas.

1.2.1. Recovering the determinantal structure

As an application of Theorems 1.1and1.3, let us see how to recover the determinantal structure of the ensemble of non-intersecting paths associated to the measure ν. We would like to show that for any collection of vertices {x1, . . . , xk} ∈V, theν-measure of the set

Π∈N.I.(N ): all of thex_iare visited by paths inΠ

can be written as det[K(xi, xj)]^k_i,j=1for some fixed matrixKwith rows and columns indexed by the set of verticesV. This property can be seen as a consequence of Corollary1.4below which we show.

Consider any collection of functionsq_n:V_n→R, 0≤n≤T −1. Consider the space of matrices with rows and columns indexed byV, and for notational convenience denotex∈Vn as(n, x)so that matrix elements of a matrix M are written asM(n, x;m, y). Define a matrix Qso thatQf (n, x)=q_n(x)f (n, x). For m≤nandx∈V_m,y∈ Vn defineWm,n(x, y):=

π:x→yw(π ) (for m=nlet this be the identity matrix). For x∈Vn defineϕ_j⁽ⁿ⁾(x):=

y∈V_T Wn,T(x, y)ϕj(y). Forx1, x2∈VndefineKn(x1, x2):=N

i=1ϕ_i⁽ⁿ⁾(x1)ψi(x2). Note that form≤n,Wm,nhas

a right-inverse on span{ϕ_i^(m)}^N_i=1which is given byW_m,n⁻¹ϕ_j^(m)=ϕ_j⁽ⁿ⁾. We will write this inverse asWn,m. On account of this we may define the following (extended kernel) matrix

K^ext(m, x;n, y)=

Wm,nK_n(x, y) ifm≥n,

−Wm,n(I−K_n)(x, y) ifm < n.

Corollary 1.4. For any collection of functionsq_n:V_n→R, 0≤n≤T −1

Π=(π₁,...,π_N)∈N.I.(N )

N i=1

T−1 n=0

¯ q_n

π_i(n)

ν(Π )=det

I−QK^ext

L²(V ), (7)

where we recall thatπ(n)denotes the vertex inV_nthrough whichπpasses and thatq(x)¯ =1−q(x).

This result is essentially a version of the Eynard–Mehta Theorem.

Proof of Corollary1.4. We use Theorem3.3(the more general version of Theorem1.1). The technical assumptions are immediately satisfied since we are dealing with a finite vector space. The right-invertibility, semigroup property and reversibility relation are all readily checked from the definitions of theWm,nandKn. As a consequence of that theorem we find that we may rewrite the right-hand side of (7) as

RHS(7)=det(I−K0+Q₀W0,1Q₁W1,2· · ·Q_T−1WT−1,TWT ,0K0)_L2(V₀), (8) where forx∈Vn,Q_nf (n, x):= ¯qn(x)f (n, x).

Define a path-integral functionalf so that for an edgee∈E_n→n+1fromx∈V_ntoy∈V_n+1,f_n(e)= ¯q_n(x). As a consequence, for any pathπ fromV0toVT,f (π )=T−1

n=0q¯n(π(n)). This observation and Theorem1.3then imply that we may rewrite the left-hand side of (7) as

LHS(7)=det(I−K₀+W0,TWT ,0K₀)_L2(V₀). (9)

Note that we have introduced the subscripts on the right-hand side to be consistent with the notation introduced before the statement of the corollary. Due to the specific type of path-integral functionalf, it is now straight-forward to see that

W0,T =Q₀W0,1Q₁W1,2· · ·Q_T−1WT−1,T.

This implies that the right-hand sides of Eqs. (8) and (9) match and therefore completes the proof of the corollary.

1.3. Outline

In Section2 we prove a result about ensembles of non-intersecting paths on directed graphs (which implies The- orem1.3above). In Section 3we prove a general result (which implies Theorem1.1above) showing the equality between certain extended kernel and path-integral Fredholm determinants. In Section4we apply this equality be- tween Fredholm determinants to a variety of extended kernels from the literature, and in theAppendixwe check the technical assumptions necessary in order to do this.

2. Non-intersecting directed paths on weighted graphs 2.1. A general combinatorial result

Let G=(V , E)be a finite directed acyclic planar graph with vertices V and edges E. Fix source vertices X = {x₁, . . . , x_N}and sink verticesY = {y₁, . . . , y_N}. Fix edge weightsw_efor each directed edgee∈E.

A directed pathπis a sequence of vertices connecting a vertex inXto a vertex inY via directed edges inE. Denote the source vertex ofπ byπ(b)and the sink vertex ofπbyπ(d).¹We say that two pathsintersectif their vertex sets

1We usebto denote the source, or base vertex; anddto denote the sink, or destination vertex.

have non-empty intersection. To a directed pathπ we associate a weightw(π )which is given by the product ofw_e over edgeseofπ. Define

W(x, y)=

π:x→y

w(π ), (10)

where the sum is over directed pathsπfrom source vertexxto sink vertexy.

Define the ensemble ofN directed non-intersecting paths from the elements ofXto the elements ofY as N.I.(N;X→Y )=

{π1, . . . , πN}: ∀i, πi(b)∈X, πi(d)∈Y and no two paths intersect .

WriteΠ= {π₁, . . . , π_N}for an element ofN.I.(N;X→Y ). Note that the non-intersection condition and the fact thatXandY haveNelements ensures that{π₁(b), . . . , π_N(b)} =Xand{π₁(d), . . . , π_N(d)} =Y.

Lemma 2.1 ([30,36,38,53]). FixN ≥1.For any finite directed acyclic planar graph Gwith source verticesX= {x₁, . . . , x_N},sink verticesY = {y₁, . . . , y_N}and edge weightsw_efor each directed edgee∈E,

det

W(x_i, y_j)N i,j=1=

Π∈N.I.(N;X→Y )

N i=1

w(π_i).

Consider now finite sets of source verticesX⊂V and sink verticesY⊂V (with at leastN vertices in each ofX andY). For suchX andY we can likewise define the ensemble ofN directed non-intersecting paths from elements ofXto elements ofY. Denote this ensembleN.I.(N;X→Y).

Fix functionsψi:X→Rfor 1≤i≤N and functionsϕj:Y→Rfor 1≤j ≤N. For Π= {π1, . . . , πN}with source vertices{π₁(b), . . . , π_N(b)} ⊂X and sink vertices{π₁(d), . . . , π_N(d)} ⊂Ywe define the weight ofΠ as

Wt(Π ):=det ψi

πj(b)N i,j=1

_N

i=1

w(πi)

det ϕi

πj(d)N i,j=1.

Define a partition function for directed non-intersecting ensembles ofN paths fromX toY with respect to weights {w_e}e∈Eand functions{ψ_i}^N_i=1,{ϕ_j}^N_j=1as

Z=Z

X,Y,{w_e}e∈E,{ψ_i}^N_i=1,{ϕ_j}^N_j=1 :=

Π∈N.I.(N;X→Y)

Wt(Π ).

For 1≤j≤N defineϕ^(b)_j :X→Rby ϕ^(b)_j (x):=

y∈Y

W(x, y)ϕj(y) (11)

and further define the operatorK:L²(X)→L²(X)by its kernel (which is just anX-indexed matrix)

K(x1, x2)=

N i=1

ϕ^(b)_i (x1)ψ_i(x2). (12)

Observe thatWhas a right-inverse on span{ϕ_i^(b)}^N_i=1, which we will denote byW⁻¹, given by W⁻¹ϕ_j^(b)=ϕj.

In particular, since the range ofKis contained in span{ϕ_i^(b)}^N_i=1,W⁻¹Kis well defined as an operator mappingL²(X) toL²(Y):

W⁻¹Kf =

N i=1

ψ_i, f_L²_(X)ϕ_i, (13)

where·,·_L²_(X)is the inner product inL²(X).

We say that thebiorthogonality assumptionis satisfied if ψi, ϕ_j^(b)

L²(X)=1i=j for all 1≤i, j≤N.

Theorem 2.2. LetG=(V , E)be a finite directed acyclic planar graph.Fix sets of source verticesX⊂V and sink verticesY⊂V.Fix edge weights w_e and a second set of weightsw˜_e for each directed edgee∈E.Fix functions ψ_i:X→Rfor1≤i≤Nand functionsϕ_j:Y→Rfor1≤j≤Nwhich satisfy the biorthogonality assumption with ϕ_j^(b)defined via thew_eweights.Write

Z=Z

X,Y,{we}e∈E,{ψi}^N_i=1,{ϕj}^N_j=1

and Z=Z

X,Y,{ ˜we}e∈E,{ψi}^N_i=1,{ϕj}^N_j=1

.

Then Z Z =det

I−K+WW ⁻¹K

L²(X),

whereW:L²(Y)→L²(X)is given by(10)withwreplaced byw,˜ andW⁻¹K:L²(X)→L²(Y)is defined in(13).

Remark 2.3. As will be clear from the proof,the biorthogonality assumption implies thatZ=0 (and,in fact,that Z=1).Conversely,one can show that ifZ=0 then there exists a linear change of basis in the space spanned by {ψ_i}^N_i=1 (or equally well the space spanned by{ϕ_j}^N_j=1)which leads back to the biorthogonality assumption being satisfied and does not change the ratioZ/Z.

Before turning to the proof of the theorem, let us check that it implies Theorem1.3.

Proof of Theorem 1.3. Recall that for the path-integral functional f, we defined a set of weightsw˜e =fn(e)we

wheree∈E_n→n+1. LetZ=Z(V₀, V_T,{w_e}e∈E,{ψ_i}^N_i=1,{ϕ_j}^N_j=1)andZ=Z(V₀, V_T,{ ˜w_e}e∈E,{ψ_i}^N_i=1,{ϕ_j}^N_j=1).

We claim that

Π∈N.I.(N;V₀→V_T)

dν(Π ) N i=1

f (π_i)=Z Z =det

I−K+WW ⁻¹K

L²(V0).

The second equality is an immediate corollary of Theorem2.2. To see the first equality above observe that Z

Z =

Π∈N.I.(N;V0→VT)

N i=1

T−1 n=0

f_n

π_i(n)→π_i(n+1)Wt(Π ) Z

=

Π∈N.I.(N;V₀→V_T)

dν(Π ) N i=1

T−1 n=0

fn

πi(n)→πi(n+1)

=

Π∈N.I.(N;V₀→V_T)

dν(Π ) N i=1

f (πi)

as desired.

Proof of Theorem 2.2. The proof is linear algebra. We may rewrite Zby first summing over the subsets of X andY which host the source and sink vertices, and then considering all non-intersecting paths between these sets.

Thus Z=

X={x₁,x₂,...,x_N}⊂X Y={y₁,y₂,...,y_N}⊂Y

det

ψ_i(x_j)N i,j=1

Π∈N.I.(N;X→Y )

N i=1

˜ w(π_i)

det

ϕ_i(y_j)N i,j=1

=

{x₁,x₂,...,x_N}⊂X {y₁,y₂,...,y_N}⊂Y

det

ψ_i(x_j)N

i,j=1det W(x_i, y_j)N i,j=1det

ϕ_i(y_j)N i,j=1.

The second line follows by an application of Lemma2.1.

We may now apply the Cauchy–Binet identity twice. The first application is with respect to the summation in the x’s, and it yields

{x₁,x₂,...,x_N}⊂X

det

ψ_i(x_j)N

i,j=1det W(x_i, y_j)N

i,j=1=det

x∈X

ψ_i(x)W(x, y_j) N

i,j=1

.

The second application likewise is applied to the summation iny’s and yields that Z=det

x∈X y∈Y

ψ_i(x)W(x, y)ϕ_j(y) N

i,j=1. (14)

Observe that by the same argument we can obtain an analogous expression forZwithWreplaced byW. However, by the definition ofϕ_j^(b)we find that

Z=det

x∈X

ψ_i(x)ϕ_j^(b)(x) N

i,j=1

.

By the biorthogonality assumption,

x∈X

ψ_i(x)ϕ_j^(b)(y)=

ψ_i, ϕ_j^(b)

L²(X)=1i=j,

and henceZ=1. Now observe that we can rewrite the kernel in appearing in (14) as

x∈X y∈Y

ψ_i(x)W(x, y)ϕ_j(y)= Wϕ_i, ψ_jL²(X)

by using the definition of the operatorWand the inner product onL²(X).

Therefore, it remains to prove that det

I−K+WW ⁻¹K

L²(X)=det

Wϕ_i, ψ_jL²(X)

N

i,j=1. (15)

To prove the above statement we will write down the matrix for the operator I −K+WW ⁻¹K in the ba- sis (ϕ₁^(b), . . . , ϕ^(b)_N , (span{ψi}^N_i=1)^⊥), where (span{ψi}^N_i=1)^⊥ represents any basis of the orthogonal complement of span{ψi}^N_i=1inL²(X). Let us consider the action of the operatorKon the basis elements. Onϕ_j^(b) one sees thatK