Algebraic area enumeration for lattice paths St´ ephane Ouvry * Alexios P. Polychronakos † October 19, 2021 Abstract We give a summary of recent progress on the algebraic area enumeration of closed paths on planar lattices. Several connections are made with quantum mechanics and statistical mechanics. Explicit combinatorial formulae are proposed which rely on sums labelled by the multicompositions of the length of the paths. * LPTMS, CNRS, Universit´ e Paris-Saclay, 91405 Orsay Cedex, France [email protected]† Physics Department, the City College of New York, NY 10031, USA and The Graduate Center of CUNY, New York, NY 10016, USA [email protected]1 arXiv:2110.09394v1 [math-ph] 18 Oct 2021
Transcript
Algebraic area enumeration for lattice paths
Stephane Ouvry∗ Alexios P. Polychronakos†
October 19, 2021
Abstract
We give a summary of recent progress on the algebraic area enumeration of closedpaths on planar lattices. Several connections are made with quantum mechanics andstatistical mechanics. Explicit combinatorial formulae are proposed which rely onsums labelled by the multicompositions of the length of the paths.
† Physics Department, the City College of New York, NY 10031, USA andThe Graduate Center of CUNY, New York, NY 10016, [email protected]
1
arX
iv:2
110.
0939
4v1
[m
ath-
ph]
18
Oct
202
1
The seminal problem of the algebraic area enumeration of paths on planar latticesof various kinds has been around for a long time. It is well known that this purelycombinatorial problem can be equivalently reformulated in the realm of Hofstadter-likequantum mechanics models. Recently [1] it has been given a boost in the form of anexplicit enumeration formula which in turn could be reinterpreted [2] in terms of statisticalmechanics models with exclusion statistics, again a purely quantum concept. It is astriking fact that an enumeration quest regarding classical random paths should be in theend so intimately connected to quantum physics, this in so many ways.
In this note we give a summary of this recent progress starting with the original al-gebraic area enumeration problem for closed paths on a square lattice and then enlargingthe perspective to other kind of lattices and paths via the statistical mechanics reinter-
pretation. So the first question we address is : among the(
nn/2
)2closed n-steps paths that
one can draw on a square lattice starting from and returning to a given point –note thatn is then necessarily even n = 2n–, how many of them enclose a given algebraic area A?
The algebraic area enclosed by a path is weighted by its winding numbers: if the pathmoves around a region in counterclockwise (positive) direction, its area counts as positive,otherwise negative; if the path winds around more than once, the area is counted withmultiplicity. These regions inside the path are called winding sectors. In Figure 1 wegive an explicit example of what is meant by algebraic area for a closed path of lengthn = 36. We see inside the path various winding sectors with winding numbers +2,+1,0,-1,-1 and various numbers of lattice cells per winding sectors: respectively 2,14,1,1,1. The0-winding number inside the path arises from a superposition of a +1 and a −1 winding,+1 − 1 = 0. It does not contribute to the algebraic area. Taking into account the non0-winding sectors we end up with an algebraic area A = 2× 2 + 1× 14 + (−1)× 2 = 16.Quite generally, calling Sm the arithmetic area of the m-winding sectors inside a path(i.e. the total number of lattice cells it encloses with winding number m, where m can bepositive or negative) the algebraic area is
A =∞∑
m=−∞
mSm
to be distinguished from the arithmetic area∑∞
m=−∞ Sm.
Winding sectors for continuous Brownian curves as well as for discrete lattice pathshave been the subject of studies for a long time. In this respect we note in the last fewyears some advances in [3] where an explicit formula for the expected area 〈Sm〉 of the m-winding sectors inside square lattice paths is proposed, to the exception of the 0-windingsector, for the simple reason that the latter is difficult to distinguish from the outside–i.e., 0-winding again– sector, which is of infinite size. Taking the continuous limit allowsto recover the results previously obtained in [4] for Brownian curves. One notes that forBrownian curves the expected area 〈S0〉 of the 0-winding sectors is also known by othermeans thanks to the SLE machinery [5]. However it remains an open problem for discretelattice paths.
2
Figure 1: A closed path of length n = 36 starting from and returning to the same bulletred point with its various winding sectors m = +2,+1, 0,−1,−1. Note the double arrowon the horizontal link which indicates that the path has moved twice on this link, here inthe same left direction.
3
Counting on the square lattice the number of closed paths of length n enclosing analgebraic area A amounts in a most straightforward way to introducing two lattice hoppingoperators u and v respectively in the right and up directions and declaring that they donot commute
v u = Q u v
Clearly selecting the u and v independent part in(u+ u−1 + v + v−1
)n=∑A
Cn(A) QA + . . . (1)
provides the number Cn(A) which counts the paths enclosing an algebraic area A. For
example it is not difficult to check that(u + u−1 + v + v−1
)4= 28 + 4Q + 4Q−1 + . . .
indicating that among the(
42
)2= 36 closed paths making 4 steps C4(0) = 28 enclose an
algebraic area A = 0 and C4(1) = C4(−1) = 4 enclose an algebraic area A = ±1.
Now it is immediate to see that, provided Q is rewritten as Q = ei2πΦ/Φo where Φ is theflux of an external magnetic field through the unit lattice cell and Φo the flux quantum,the expression
H = u+ u−1 + v + v−1
can be interpreted as a Hamiltonian modelling a quantum particle hopping on a squarelattice and coupled to a perpendicular magnetic field. This famous model is known underthe name Hofstadter model [6].
Going a step further, a simplification arises when the flux is rational Q = ei2πp/q withp, q two coprime integers: in this case determining the Hofstadter spectrum narrows downto computing the eigenvalues E1, E2, . . . , Eq of the finite q × q Hamiltonian matrix
where one has summed over the q eigenvalues and integrated over the two continuousquantum numbers kx and ky while enforcing the proper normalization Tr Iq = 1, whereIq is the q × q identity matrix.
Selecting as in (1) the u, v independent part of(u + u−1 + v + v−1
)ntranslates in
the quantum world to computing the trace of Hnq so that in view of the algebraic area
4
enumeration one should write
Tr Hnq =
∑A
Cn(A) QA (3)
As a consequence the integer q has to be understood as a priori undetermined, i.e., as afree parameter allowing for the QA basis –where A = 0,±1,±2, . . . stands for the possiblevalues of the algebraic area– on which the enumeration (3) can take place. On the otherhand keeping a particular value of q would amount to counting the algebraic area moduloq.
All the machinery of quantum mechanics is now at our disposal. It is known that thedeterminant of the secular matrix Iq − zHq simplifies to
det(Iq − zHq) =
bq/2c∑n=0
(−1)nZ(n)z2n − 2(
cos(qkx) + cos(qky))zq
where the Z(n)’s are independent of kx and ky and by convention Z(0) = 1. Even more,Kreft [7] was able to rewrite them in a closed form as trigonometric multiple nested sums
Z(n) =
q−2n+2∑k1=1
k1∑k2=1
· · ·kn−1∑kn=1
sk1+2n−2sk2+2n−4 · · · skn−1+2skn (4)
wheresk = 4 sin2(πkp/q) (5)
From the knowledge of Z(n) in (4) the algebraic area enumeration can proceed. Wegive here a summary of the procedure, more details can be found in [1, 2]. First introducethe b(n)’s via
log
bq/2c∑n=0
Z(n)zn
=∞∑n=1
b(n)zn (6)
It is not difficult to see that b(n) selects the part of Z(n) which is obtained by rewriting itas a linear combination of trigonometric single sums plus other terms which are productsof such single sums, which are then ignored. In other words the b(n)’s end up beingproportional to q –i.e., the right scaling for the trace of a q×q matrix– whereas the Z(n)’salso contain terms proportional to q2, . . . , qn. This rewriting is encoded in the coefficientsc(l1, l2, . . . , lj) labeled by the compositions l1, l2, . . . , lj of n (meaning the ordered partitionsof n: there are 2n−1 compositions of n, for example 3 = 3, 2 + 1, 1 + 2, 1 + 1 + 1) so thatb(n) is expressed as
b(n) = (−1)n+1∑
l1,l2,...,ljcomposition of n
c(l1, l2, . . . , lj)
q−j+1∑k=1
sljk+j−1 · · · sl2k+1s
l1k (7)
5
with
c(l1, l2, . . . , lj) =
(l1+l2l1
)l1 + l2
l2
(l2+l3l2
)l2 + l3
· · · lj−1
(lj−1+ljlj−1
)lj−1 + lj
As announced, solely trigonometric single sums appear in (7).
Now one can use the identity log det(I− zM) = Tr log(I− zM) valid for any matrixM where Tr here means the usual matrix trace. After some manipulations one reaches,not surprisingly, that the quantum trace (2) is proportional to b(n)
Tr Hn=2nq = 2n(−1)n+1 1
qb(n)
It follows that, using (7), it can be rewritten as composition-dependent trigonometric sin-
gle sums∑q−j+1
k=1 sljk+j−1 · · · sl2k+1s
l1k weighted by the combinatorial coefficient c(l1, l2, . . . , lj)
and summed over all compositions l1, l2, . . . , lj of the integer n = n/2
Tr Hn=2nq = 2n
∑l1,l2,...,lj
composition of n
c(l1, l2, . . . , lj)1
q
q−j+1∑k=1
sljk+j−1 · · · sl2k+1s
l1k
The trigonometric single sums remain to be computed, which can also be done, stillkeeping in mind that, as said earlier, q is an a priori undetermined free parameter. Finallyone extracts from (3) the desired number of closed paths of length n enclosing a givenalgebraic area A as
Cn(A) =2n∑
l1,l2,...,ljcomposition of n
(l1+l2l1
)l1 + l2
l2
(l2+l3l2
)l2 + l3
. . . lj−1
(lj−1+ljlj−1
)lj−1 + lj
2l3∑k3=0
2l4∑k4=0
. . .
2lj∑kj=0
j∏i=3
(2liki
)(2l1
l1 + A+∑j
i=3(i− 2)(ki − li)
)(2l2
l2 − A−∑j
i=3(i− 1)(ki − li)
)(8)
This formula grows quickly in complexity since the number of compositions on which onehas to sum increases like 2n with the number of steps of the paths. Still it has the benefitof being explicit.We leave as an open problem to the interested reader to prove that inthe continuous limit where the elementary lattice size a→ 0, the number of steps n→∞with the scaling na2 = 2t,
n Cn(A/a2)(n
n/2
)2 → π1
cosh2(πA/t)
i.e., one recovers Levy’s law for the distribution of the algebraic area enclosed by Browniancurves after a time t (the convergence has been checked numerically to improve withincreasing n up to 138).
6
Why in (4) and (6) the particular choice of the notations Zn and bn? In statisticalmechanics Z(n) usually refers to an n-body partition function and b(n) to its associated n-th cluster coefficient. Let us interpret sk in (5) as sk = e−βεk (β is the inverse temperature),i.e., as a spectral function for a quantum 1-body spectrum εk labeled by an integer k. Thestructure of Z(n) in (4) with the +2 shifts in the spectral function arguments then preciselycorresponds to an n-body partition function for a gas of particles with exclusion statisticsg = 2 (no two particles can occupy two adjacent quantum states) and 1-body spectrum εk.Exclusion statistics is a purely quantum (again) concept which describes the statisticalmechanical properties of identical particles. Usual particles are either Bosons (g = 0) orFermions (g = 1). Here for square lattice paths one goes beyond Fermi exclusion withg = 2. In general for g-exclusion the n-body partition function (4) would become
Z(n) =
q−gn+g∑k1=1
k1∑k2=1
· · ·kn−1∑kn=1
sk1+gn−gsk2+gn−2g · · · skn−1+gskn
with a shift in the arguments of the spectral function which is g instead of 2. In line with(6, 7) the associated n-th cluster coefficient would end up rewriting as
b(n) = (−1)n+1∑
l1,l2,...,ljg−composition of n
cg(l1, l2, . . . , lj)
q−j+1∑k=1
sljk+j−1 · · · sl2k+1s
l1k (9)
where
cg(l1, l2, . . . , lj) =(l1 + · · ·+ lg−1 − 1)!
l1! · · · lg−1!
j−g+1∏i=1
(li + · · ·+ li+g−1 − 1
li+g−1
)In (9) one sums over all g-compositions of the integer n obtained by inserting at will insidethe usual compositions (i.e., the 2-compositions) no more than g− 2 zeroes in succession.For example for n = 3 and g = 3 one has 9 such 3-compositions n = 3 = 2 + 1 = 1 + 2 =1 + 1 + 1 = 2 + 0 + 1 = 1 + 0 + 2 = 1 + 0 + 1 + 1 = 1 + 1 + 0 + 1 = 1 + 0 + 1 + 0 + 1.For general g there are gn−1 such g-compositions of the integer n (see [8] for an analysisof these extended compositions, also called multicompositions).
One has reached the conclusion that the square lattice paths algebraic area enumer-ation is described by a quantum gas of particles with statistical exclusion g = 2. To seethis more explicitly on the Hofstadter Hamiltonian itself let us perform on the hoppinglattice operators u and v the modular transformation
u→ −u v , v → v
to get the new Hamiltonian
H = −u v − v−1 u−1 + v + v−1 (10)
still describing the same paths but on the deformed lattice of Figure 2.
7
Figure 2: The deformed square lattice after the modular transformation.
8
The secular matrix corresponding to the Hamiltonian (10) is
Iq − zHq =
1 −(1−Q)z 0 · · · 0 −(1− 1Qq )z
−(1− 1Q
)z 1 −(1−Q2)z · · · 0 0
0 −(1− 1Q2 )z 1 · · · 0 0
......
.... . .
......
0 0 0 · · · 1 −(1−Qq−1)z−(1−Qq)z 0 0 · · · −(1− 1
Qq−1 )z 1
(11)
where one has set kx = ky = 0 for simplicity. The Hofstadter spectral function (5) isrecovered as
sk = (1−Qk)(1− 1
Qk)
(11) is a particular case of the more general class of secular matrices
with below the unity main diagonal an empty sub-diagonal made only of 0’s which is themanifestation of the stronger g = 3 exclusion. The spectral function then follows as
sk = g(k)f(k)f(k + 1)
9
For g-exclusion the generalization of (13) amounts to a Hamiltonian of the form
and a secular matrix with now g − 2 empty sub-diagonals below the main diagonal (hereq is always understood to be larger than g). Clearly the Hofstadter Hamiltonian (10),which rewrites as H = (1 − u)v + v1−2(1 − u−1), is a particular case of (14) with g = 2and F (u) = 1− u, G(u) = 1− u−1.
Let us illustrate this mechanism in the case of g = 3 exclusion with the specificexample of chiral paths on a triangular lattice (Kreweras type paths). The three chiralhopping operators U, V and W = QU−1V −1 described in Figure 3 are such that
V U = Q2UV
The triangular lattice Hamiltonian is, in a self-explanatory form,
H = U + V +W
To bring it to the exclusion form (14) one chooses the representation U = −i u v andV = i u−1 v in which case H rewrites as
H = i(−u+ u−1)v + v−2
It is indeed an Hamiltonian of the type (14) for g = 3 exclusion, F (u) = i(−u + u−1),G(u) = 1 and with spectral parameters
which is indeed of the type (13) with a vanishing bottom-left entry. Note that the nonHermiticity of the triangular Hamiltonian and thus of the secular matrix is a consequenceof the fact that the chiral paths carry an orientation on the lattice.
The triangular algebraic area enumeration follows with an expression similar to (8)provided that the trigonometric single sums appearing in (9) pertaining to the triangularspectral function (15) can be computed [2] and that the sum is made on all 3-compositionsof the length of the triangular paths considered.
In conclusion we have shown how various tools available in quantum and statisticalphysics allowed for an explicit algebraic area enumeration of closed paths on planar lat-tices. The enumeration formulae rely on an explicit sum over compositions whose numbergrows quickly with the length of the path. It would be certainly rewarding to rewrite thissum with a smaller number of terms. Trivially by symmetry one can restrict to mirror-freecompositions weighted twice except for the palindromic ones. We leave this issue as wellas other questions of interest to the lattice path combinatorics community.
References
[1] S. Ouvry and S. Wu, ”The algebraic area of closed lattice random walks”, J. Phys. A:Math. Theor. 52 (2019) 255201.
[2] S. Ouvry and A. P. Polychronakos, ”Exclusion statistics and lattice random walks”,NPB 948 (2019) 114731; ”Lattice walk area combinatorics, some remarkable trigono-metric sums and Apery-like numbers”, NPB Volume 960 (2020) 115174; see also L.Gan, S. Ouvry and A. P. Polychronakos, ”Algebraic area enumeration of random walkson the honeycomb lattice”, arXiv:2107.10851, Phys. Rev. E (to be published).
[3] T. Budd, Winding of simple walks on the square lattice, preprint, arXiv:1709.04042and talk given at the Lattice Paths, Combinatorics and Interactions CIRM Luminyconference (2021).
[4] A. Comtet, J. Desbois and S. Ouvry, ”Winding of planar Brownian curves”, Journalof Physics A : Math. Gen. 23 (1990) 3563.
[5] C. Garban and J. A. Trujillo Ferreras, ”The expected area of the filled planar Brownianloop is π/5”, Commun. Math. Phys. 264, 3 (2006), 797–810.
[6] D. Hofstadter, ”Energy levels and wave functions of Bloch electrons in rational andirrational magnetic fields”, Phys. Rev. B 14 (1976) 2239.
[7] C. Kreft, https://www-sfb288.math.tu-berlin.de/Publications/preprint-list/51/100,SFB 288 Preprint No. 89, TU-Berlin (1993).
[8] B. Hopkins and S. Ouvry,”Combinatorics of multicompositions, Proceedings of theCombinatorial and Additive Number Theory conference, CUNY Graduate Center NYNY, Springer (2020).
