Return probability for the loop-erased random walk and mean height in sandpile : a proof
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Return probability for the loop-erased random walk andmean height in sandpile : a proof
V.S. Poghosyan , V.B. Priezzhev and P. Ruelle Institut de Recherche en Math´ematique et Physique,Universit´e catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, 141980 Dubna, Russia
Single site height probabilities in the Abelian sandpile model, and the corresponding meanheight h h i , are directly related to the probability P ret that a loop erased random walk passesthrough a nearest neighbour of the starting site (return probability). The exact values ofthese quantities on the square lattice have been conjectured, in particular h h i = 25 / P ret = 5 /
16. We provide a rigourous proof of this conjecture by using a local monomer-dimerformulation of these questions.
I. INTRODUCTION
As different as they may appear at first sight, the dimer model, the loop-erased random walk(LERW) and the Abelian sandpile model (ASM) are very closely connected, as they all have analternative formulation in terms of a unifying concept: spanning trees.Among the three models just mentioned, the dimer model is the oldest one as it was formulatedin 1937 by Fowler and Rushbrooke [1], although the first exact results were obtained somewhatlater, in the sixties, by Kasteleyn, Fisher, Temperley and Stephenson [2–6]. The dimer (or domino)model, also known as the perfect matching problem, has been the subject of an increasing numberof works over the last decade, see for instance [7] and the references therein. The correspondencebetween packed dimer configurations (no vacancy or monomer) on the square lattice and spanningtrees on a sublattice was established by Temperley [8], and generalized to general planar graphsby Burton and Pemantle [9]. This correspondence holds in the presence of monomers, and leadsto spanning webs rather than spanning trees [10, 11]. It will be recalled in Section 3.The loop-erased random walk was defined by Lawler [12] as paths generated from a simplesymmetric random walk by removing the loops as they appear. The connection between theLERW and the spanning trees has been discussed in several works [13–17], with the result thatthe probability measure on LERW sample paths coincides with the uniform measure on chemicalpaths of spanning trees.Finally the Abelian sandpile model is an open stochastic dynamical system, which works like anon-linear diffusion process [18, 19]. The ASM is defined in terms of height variables, attached tothe sites of a two-dimensional square lattice. The height variables h i take the four integer values1, 2, 3 and 4, but only a fraction of all height configurations, called recurrent, keep reoccuringwhen the dynamics is run over long periods. The long time behaviour of the model is controlledby the stationary measure on the height configurations. This measure is uniform on the recurrentconfigurations and vanishes on the non-recurrent ones. The connection to spanning trees stemsfrom the burning algorithm [20], which establishes a direct and one-to-one correspondence betweenthe recurrent configurations of the sandpile on a lattice and spanning trees on the same lattice.All three models have non-local features. This is manifest for the dimer model and the LERW,whereas the non-locality in the ASM comes from the recurrence criterion [19], which requires toscan the whole of a height configuration before one can declare it recurrent. These non-localfeatures usually make explicit calculations particularly hard. Let us also mention that the questionof conformal invariance has been addressed in these three models, directly in terms of a conformalfield theory for the dimer model [21–23], spanning webs [24] and the ASM [25–28], and as stochasticLoewner processes (SLE) in the case of LERW and spanning trees [29, 30].The purpose of this work is to compute the value of the return probability for the LERW P ret = , announced in [31]. This value was based on the observation that the return probabilityand the ASM mean height are related by P ret = h h i − , and on an earlier conjecture made in[27] for the mean height h h i = , in the limit of the infinite square lattice. It turns out thatthese two numbers are themselves related to three others, recently introduced by Levine and Peres[32], namely the looping constant ξ , the ratio τ between the number of spanning unicycles and thenumber of spanning trees, and the mean length λ of the cycle in a spanning unicycle, all defined aslimits over increasing finite square grids. In addition these five numbers have all a d -dimensionalanalogue, and remain rationally related in any dimension [32]. In this sense, our result provides aproof for the values of these five numbers in two dimensions, P ret = , h h i = , ξ = , τ = and λ = 8.In Section 2, we recall the expressions of the LERW return probability and of the ASM meanheight in terms of spanning trees with some specific properties. Section 3 contains the proofitself and reduces the counting of the required spanning trees to certain local configurations ofmonomers and dimers. We should emphasize that the proof does not rely on the exact evaluationof the multiple integral on which the conjecture made in [27] is based. On the contrary the proof isessentially combinatorial, and shows that the counting of spanning graphs with certain non-localproperties can be reduced to the counting of local monomer-dimer arrangements, which can thenbe easily carried out. It thus avoids the full complexity of the graph theoretical computations,inherent in [27, 33]. Our proof shows and explains why the LERW return probability and the otherfour related quantities are such simple numbers. II. THE LERW RETURN PROBABILITY AND THE ASM MEAN HEIGHT
The Abelian sandpile model [18] in finite volume is defined by height variables, located at thesites of a finite, two-dimensional square grid and taking the values 1 , , , ∼ . L for a L × L grid). Dhar also gave a criterion to select the recurrent configurations. Its explicit formwill not be important for what follows.As sandpile configurations are made of random variables valued in { , , , } and distributedaccording to the uniform measure on the recurrent subset, it is natural to ask about the statisticalproperties of these variables and their spatial correlations. In particular the first question concernsthe distribution of the height at a single site. As a first step, and in order to avoid boundary effects,we consider a site deep in the middle of the grid, and take the infinite volume limit. We denote by P i , for i = 1 , , ,
4, the resulting probabilities, namely P i is the probability that the height at anyfixed site be equal to i , in the limit of an infinite grid.The first of these four probabilities has been obtained by Majumdar and Dhar [34], P = 2 π − π ≃ . . (2.1)The other three probabilities P , P and P turned out to be more complicated, and are moreconveniently expressed in terms of spanning trees. As mentioned in the Introduction, the burningalgorithm [20], which is the algorithmic translation of the recurrence criterion found by Dhar [19],establishes a one-to-one mapping between the set of recurrent configurations and the set of rooted(oriented) spanning trees on the same lattice. The characterization of those spanning trees whichcorrespond, under this mapping, to recurrent configurations with certain height values at certainpositions, has been worked out in [33]. As a result, the fractions P i of recurrent configurationswhich have a height equal to i at a reference site, are given by the following fractions amongspanning trees, P = X N , P = P + X N , P = P + X N , P = P + X N , (2.2)where X k , for k = 0 , , ,
3, is the number of spanning trees such that the reference site has exactly k predecessors among its four nearest neighbours [33] (a site x is called a predecessor of y if thepath along the tree from x to the root passes through y ). The quantity N is the total number ofspanning trees on the grid. The identities (2.2) are valid for a finite grid. Their infinite volumelimits exist, and define the sought probabilities.Unlike X , the quantities X k for k ≥ X satisfy local constraints).These numbers have been first computed in [33], the results taking the following form, P = 12 − π − π + 12 π + I , (2.3) P = 14 + 32 π + 1 π − π − I − I , (2.4) P = 14 − π + 4 π + I I , (2.5)where I and I are two complicated, multiple integrals. Their numerical evaluation gave thefollowing values for the probabilities, P ≃ . P ≃ . P ≃ . P i on the plane (and the upper half-plane) was reconsideredin [27], which led to an exact relation between P and P ,( π − P + 2( π − P = π − − π + 12 π − π , (2.6)and a similar relation between the two integrals I and I . A conjecture on the value of theremaining integral, based on its numerical evaluation to twelve decimal places (later pushed to 25places), then yielded the following conjectural values for the probabilities [27], P = 14 − π − π + 12 π , P = 38 + 1 π − π , P = 38 − π + 1 π + 4 π , (2.7)and for the stationary mean height (particle density), h h i = P + 2 P + 3 P + 4 P = 258 . (2.8)Conjectured by Grassberger [35] almost 20 years ago, the value 25 / P ret be the probability thata path passes through a fixed nearest neighbour of the origin, say the right neighbour. In termsof uniformly distributed spanning trees, P ret is the probability that the origin is a predecessor ofits right neighbouring site. By listing explicitly the situations where this is the case among thosewhere the right neighbour has k predecessors among its nearest neighbours, the following identityfollows [31] P ret = X N + X N + 3 X N . (2.9)By using the equations (2.2) to trade the X k for the P i (and the identity P + P + P + P = 1),the previous combination turns out to be directly related to the mean height of the sandpile model, P ret = 14 ( − P − P + P + 3 P ) = 12 ( P + 2 P + 3 P + 4 P ) −
54 = h h i − . (2.10)In order to actually compute the return probability, we express directly the fact that the originis a predecessor of its right neighbour, without paying attention to the other nearest neighbours.This is pictured in Fig. 1, where all possible configurations are shown. = + + ++ + + + P ret º (cid:144) FIG. 1: Decomposition of th return probability in terms of chemical paths between two nearest neighbours. If A, B, C denote the corresponding fractions of spanning trees with the chemical paths asshown, we have that P ret = 14 + 2 ( A + B + C ) . (2.11)It remains to compute A, B and C . This will be done in the next section by using particular localarrangements of monomers and dimers. III. MONOMER-DIMER COMPUTATION OF LERW RETURN PROBABILITY
Consider a (2 n − × (2 n −
1) square lattice L with the rightmost lower site removed. Thesites of the lattice can be subdivided into three subsets: (1) black sites forming the sublattice B of sites with odd-odd coordinates; (2) white sites forming the sublattice W of sites with even-evencoordinates; (3) the other remaining sites colored grey. The corner site removed is black anddenoted below by r (for root).Consider a dense packed dimer configuration on L . Each dimer covers two sites, of colors blackand grey, or white and grey. Temperley’s correspondence associates two sets of arrows to the dimerconfiguration in the following way. We replace each dimer covering a black site by an arrow directedfrom the black site to the grey one, and similarly we replace each dimer covering a white site byan arrow directed from the white site to the grey one. The sets of black and white arrows are twoacyclic configurations of arrows, which form spanning trees on the two sublattices B and W , seeFig. 2. The two spanning trees are dual, and from either one, the original dimer configuration canbe entirely reconstructed. H a L H b L H c L r FIG. 2: (a) Dimer tiling; (b) spanning tree on the odd-odd sublattice or (c) even-even sublattice.
A directed path from a site x ∈ B ( x ∈ W ) to a site y ∈ B ( y ∈ W ) along the branches of theblack (white) spanning tree is the chemical path from x to y . As shown in [13–17], the statisticalproperties of the chemical paths coincide with those of the loop erased random walk. Our maininterest here is the return probability of the LERW: the probability that a chemical path startingat a given point on the spanning tree visits one of neighbouring sites of this point, say the rightone, before going off to the root located at r .We consider now packed dimer tilings on the lattice L ′ obtained from the (2 n − × (2 n − r , a grey site i and a black site j . The grey site i hastwo nearest neighbors i , i ∈ B , see Fig.3. Kenyon made the observation (lemma 17 in [36]) thatif j was on the boundary of L ′ , then the sites i and i would have to be in different components ofthe two-component black spanning tree, one component being rooted at r , the other at j . Indeed, i and i cannot be in the same component for otherwise the chemical path from i to i togetherwith the bond of B linking i and i would form a loop enclosing an odd number of lattice sites of H a L H b L H c L ij i i r FIG. 3: (a) Dimer tiling with two monomers inside the lattice; (b) and (c) spanning trees corresponding tocase (I). L ′ which cannot be fully covered by dimers. When j is not on the boundary, this result does nothold because one of the paths from i or from i may form a loop around j (so that one of i , i isdirected to r , the other to a loop winding around j ), or else both i and i are directed to r , thechemical path between the two going around j .For our purposes, we will choose the site j as the left nearest neighbour of i on the sublattice B . Then a closed loop around j is impossible for the lack of space between j and i or i on thesublattice B . Three possibilities remain.(I) The first possibility is depicted in Fig. 3b: the path from i goes to j and the path from i goes to r . By reversing the orientation of the first (red) path and inserting an arrow from i to i ,we obtain a long path from j to r passing through the bond ( i , i ). Depending on the bonds usedby this path around i , we have three possible configurations shown in Fig. 4. = + + M º B C 14 Π FIG. 4: The tree possible path configurations for case (I).
The first two are exactly those we had denoted by B and C in the previous section (up to amirror transformation). The third one corresponds to all spanning trees on B which use the twobonds ( j, i ) and ( i , i ). In terms of dimers on the original lattice L , it is the set of all dimercoverings with two forced dimers on j and i . So the fraction of these is a local dimer-dimercorrelation, computed by Fisher and Stephenson in [6], and equal to 1 / π in the limit of largelattices.(II) The path from i goes to r and the path from i goes to j , see Fig. 5b. Like in case (I), wereverse the orientation the second path and insert an arrow form i to i , to obtain a long pathfrom j to r passing through the bond ( i , i ). It leads to the two possible local configurations A and B shown in Fig. 6. H a L H b L H c L ij i i r FIG. 5: The tiling and spanning trees for the case (II). = + M º A B
FIG. 6: The two possible path configurations for the case (II). (III) The sites i and i belong to the same component, rooted at r , and the chemical pathbetween the two loops around j , almost completely enclosing the component rooted at j , see Fig. 7band Fig. 8b. It implies that the dual graph on W contains one loop around the site j (and onlyone for the lack of space between i and j ) . This loop can have the two orientations, as picturedin red in Fig. 7c and Fig. 8c. Rotating clockwise by π/ i and j , we obtain a set of paths of type B or C , but now on the sublattice W (Fig. 9).The number of paths must be doubled since there are two equivalent orientations of the arrowsalong the loop.Collecting the various contributions from cases (I) to (III), we find how the dimer configurationson L ′ is related to the specific classes A, B, C of spanning trees on B . Dividing by the total numberof dimer configurations on L , we obtain that the correlation P mm of two monomers at i and j isexpressed in terms of the relative fractions of spanning trees of types A, B, C , in the limit of large H a L H b L H c L ij i i r FIG. 7: The tiling and spanning graphs for case (III). H a L H b L H c L ij i i r FIG. 8: The same as previous figure but with the opposite orientation of the loop. = + M º B C
FIG. 9: Configurations of path for case (III). lattices, P mm = M + M + M = 14 π + A + 4 B + 3 C. (3.1)Following [6], the calculation of the correlation is easily carried out. One finds P mm = π , yieldinga first relation for the three unknowns.In order to write a second relation, we repeat the previous calculation in which, in addition to thetwo monomers at i and j , we force a dimer in between them, like shown in Fig. 10b. Thus instead ofa monomer-monomer correlation, we now consider a monomer-dimer-monomer correlation P mdm .0 H a L H b L H c L FIG. 10: (a) two monomers; (b) two monomers and one dimer; (c) two dimers equivalent to case (b).
All the steps above remain, except that case (I) becomes forbidden (the red loop in Fig. 3b wouldenclose an odd number of sites), and only one orientation of the loop in case (III) is allowed.Therefore we can write P mdm = M + 12 M = A + 2 B + C. (3.2)The correlation P mdm is equivalent to have two fixed dimers, as shown in Fig. 10c. From [6], wefind P mdm = − π .Finally we notice the identity A = B as follows from the steps shown on Fig. 11, where wereverse the orientation of the loop, move the vertical up arrow to the horizontal right arrow andeventually apply a (diagonal) mirror transformation. A º = = º B FIG. 11: Three steps proving the equality of A and B . The two equations (3.1) and (3.2) can then be solved, with the result A = B = 332 − π , (3.3) C = 12 π − . (3.4)Plugging these values back in the LERW return probability (2.11) yields P ret = 5 /
16, and in turnthe ASM mean height h h i = 25 / IV. PERSPECTIVE
The present work raises (at least) two natural questions. We have shown how to computethe ASM mean height, a quantity so far thought to have a non-local interpretation in terms ofspanning trees, in a purely local way, though specific local arrangements of dimers and monomers.1It would be interesting to see whether this technique may help computing 2-site height correlationsfor heights larger or equal to 2 (since the others are known [28, 34]).The second question is related to the LERW. Beyond the return probability, or passage probabil-ity to a nearest neighbour, one could ask for the passage probability to a second nearest neighbour(distance √ /
9. Applying the present techniques to this case, and perhaps to the next few cases,would bring valuable results.
Acknowledgments
This work was supported by a Russian RFBR grant No 09-01-00271-a and by the BelgianInteruniversity Attraction Poles Program P6/02, through the network NOSY (Nonlinear systems,stochastic processes and statistical mechanics). P.R. is Senior Research Associate of the BelgianNational Fund for Scientific Research (FNRS). [1] R.H. Fowler and G.S. Rushbrooke, Statistical theory of perfect solutions,
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