We consider the equiprobable distribution of spanning trees on the square
lattice. All bonds of each tree can be oriented uniquely with respect to an
arbitrary chosen site called the root. The problem of predecessors is finding
the probability that a path along the oriented bonds passes sequentially fixed
sites $i$ and $j$. The conformal field theory for the Potts model predicts the
fractal dimension of the path to be 5/4. Using this result, we show that the
probability in the predecessors problem for two sites separated by large
distance $r$ decreases as $P(r) \sim r^{-3/4}$. If sites $i$ and $j$ are
nearest neighbors on the square lattice, the probability $P(1)=5/16$ can be
found from the analytical theory developed for the sandpile model. The known
equivalence between the loop erased random walk (LERW) and the directed path on
the spanning tree says that $P(1)$ is the probability for the LERW started at
$i$ to reach the neighboring site $j$. By analogy with the self-avoiding walk,
$P(1)$ can be called the return probability. Extensive Monte-Carlo simulations
confirm the theoretical predictions.Comment: 7 pages, 2 figure

Poghosyan, V. S.

Priezzhev, V. B.

English

arXiv

We consider the equiprobable distribution of spanning trees on the square lattice. All bonds of each tree can be oriented uniquely with respect to an arbitrary chosen site called the root. The problem of predecessors is to find the probability that a path along the oriented bonds passes sequentially fixed sites i and j. The conformal field theory for the Potts model predicts the fractal dimension of the path to be 5/4. Using this result, we show that the probability in the predecessors problem for two sites separated by large distance r decreases as P(r) ∼ r −3/4. If sites i and j are nearest neighbors on the square lattice, the probability P(1) = 5/16 can be found from the analytical theory developed for the sandpile model. The known equivalence between the loop erased random walk (LERW) and the directed path on the spanning tree states that P(1) is the probability for the LERW started at i to reach the neighboring site j. By analogy with the self-avoiding walk, P(1) can be called the return probability. Extensive Monte-Carlo simulations confirm the theoretical predictions

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Acta Polytechnica Vol. 51 No. 1/2011The Problem of Predecessors on Spanning TreesV. S. Poghosyan, V. B. PriezzhevAbstractWe consider the equiprobable distribution of spanning trees on the square lattice. All bonds of each tree can be orienteduniquely with respect to an arbitrary chosen site called the root. The problem of predecessors is to ﬁnd the probabilitythat a path along the oriented bonds passes sequentially ﬁxed sites i and j. The conformal ﬁeld theory for the Pottsmodel predicts the fractal dimension of the path to be 5/4. Using this result, we show that the probability in thepredecessors problem for two sites separated by large distance r decreases as P (r) ∼ r−3/4. If sites i and j are nearestneighbors on the square lattice, the probability P (1) = 5/16 can be found from the analytical theory developed for thesandpile model. The known equivalence between the loop erased random walk (LERW) and the directed path on thespanning tree states that P (1) is the probability for the LERW started at i to reach the neighboring site j. By analogywith the self-avoiding walk, P (1) can be called the return probability. Extensive Monte-Carlo simulations conﬁrm thetheoretical predictions.Keywords: loop erased random walk, spanning trees, Kirchhoﬀ theorem, Abelian sandpile model.1 Introduction and mainresultsIn graph theory, the spanning tree of connected graphG is a connected subgraph ofG containing all verticesof G and having no cycles. Numerous applicationsof spanning trees began with Kirchhoﬀ’s seminalproblem solved in 1847, and then spread out manybranches of mathematics and theoretical physics. Instatistical mechanics, spanning trees are related tothe Potts model [1], the dimer model [2], the sand-pile model [3] and many others. A relation betweenlattice models of statistical mechanics and spanningtrees via the Tutte polynomial has been establishedby Fortuin and Kasteleyn [4].The Kirchhoﬀ theorem claims that the number ofspanning trees of connected graph G is a cofactor ofthe Laplacian matrix Δ of graph G. If one deletesany row and any column from Δ, one obtains a ma-trix Δ∗ which gives the number of spanning trees asdetΔ∗. The determinantal structure allows easy cal-culation of local characteristics of the spanning trees,for instance, the average number of vertices with agiven number of adjacent bonds. A characterizationof non-local objects in the spanning tree is not sosimple. One such object is the chemical path deﬁnedas a path along two or more bonds of the tree. Thefractal dimension of a long chemical path on the two-dimensional lattice has been calculated by means ofa mapping of the spanning tree conﬁgurations ontothe Coulomb gas model.A closely related object is the loop erased randomwalk (LERW) on the two-dimensional lattice [11],which was proven to be equivalent to the directedchemical path of the spanning tree of the same lat-tice [7, 12]. In this paper, we consider a problem aris-ing in the theory of LERW and equally distributedspanning trees: given two lattice sites i and j, what isthe probability that the LERW or the directed chem-ical path passes i and j? If site i is passed ﬁrst, wesay that i is the predecessor of j and we refer to thisproblem as the predecessor problem. Surprisingly,the problem has no exact solution in the general case.Only two limiting cases are available: (a) If sites i andj are separated by large distance r, the asymptoticsof P (r) can be found from known results on the frac-tal dimension of the chemical path; (b) If points iand j are nearest neighbors of the square lattice, theseeking probability can be found from the theory ofsandpiles [9] (see also [16]).The asymptotic behavior of P (r) for large dis-tance r follows directly from the deﬁnition of thefractal dimension. Indeed, consider a large squarelattice L and the set of uniformly distributed span-ning trees on L. We assume that the root is situatedat the boundary of L. Consider site i in the bulk ofthe lattice and some circle contour C of radius R withthe center in i. Let Π be a directed chemical pathfrom i to the root along the oriented bonds of a tree.All points of the subset of Π inside C are descendantsof i. In accordance with the deﬁnition of the fractaldimension of the directed path on the spanning tree,the number of descendants inside C is proportionalto R5/4 (see Majumdar [7]). The probability thatpoint i is the predecessor of point j lying at distancer from i is the density of descendants ρ(r). Thus, wehave59Acta Polytechnica Vol. 51 No. 1/2011∫ R1ρ(r)r dr ∼ R5/4 (1.1)from where we conclude that ρ(R) ∼ R−3/4.As was mentioned above, the problem of pre-decessors for an arbitrary disposition of two latticepoints is not solved. In Section 2 we concentrate ona particular problem of probability P (1) when pointsi and j are nearest neighbors of the square lattice.An essential element of the theory of sandpiles is theprobability distribution of sites having 0, 1, 2 and 3predecessors among the nearest neighbors. The cor-responding probabilities are denoted by X0, X1, X2and X3. Having explicit expressions for these values,we obtain P (1) as their combination and get an un-expectedly simple result P (1) = 5/16. In Section 3,we relate this result to the return probability of theLERW. Section 4 contains the results of Monte-Carloveriﬁcations.2 The problem ofpredecessors for nearestneighborsThe spanning tree enumeration method, namely theKirchhoﬀ theorem is proved to be a powerful math-ematical instrument for the investigation of variouscombinatorial problems in theoretical physics. In thelast decade, it has been developed and adapted forcalculating the height probabilities of the Abeliansandpile model [5, 9, 16]. The Abelian sandpile modelis a stochastic dynamical system, which describes thephenomenon of self-organized criticality. During theevolution the system falls into a subset of all possi-ble states, called the subset of recurrent states. Theproblem is to calculate analytically various observ-able values in the recurrent state, such as heightprobabilities Pi, i = 1, 2, 3, 4 at a ﬁxed site and heightcorrelations between distinct ﬁxed sites [18].It was shown that the calculation of height prob-abilities in the Abelian sandpile can be reduced tothe calculation of X0, X1, X2 and X3 in the span-ning tree model. The exact relation between thesequantities is given byP1 =X04; P2 = P1 +X13;P3 = P2 +X22; P4 = P3 +X3. (2.2)Majumdar and Dhar [5] found in 1991 the prob-ability of height 1, constructing the correspondingdefect lattice for the situation when a site i0 has nopredecessors and calculating the determinant of thedefect matrix Δ∗. A technique for computing thenumbers X1, X2, X3 was devised in [9]. The resultsare (see also [16] for details)X0 =8(π − 2)π3;X1 =34+48π3− 15π2− 32π;X2 =14− 48π3+6π2+3π; (2.3)X3 =16π3+1π2− 32π,which giveP1 =2(π − 2)π3;P2 =14+12π3− 3π2− 12π;P3 =38− 12π3+1π; (2.4)P4 =38+4π3+1π2− 12π.Fig. 1: All possible situations with a ﬁxed vertex i0 (thecentral vertex in the diagrams) to have various nearestneighbouring predecessors on the square lattice. By blackcolor we denote vertices, which are predecessors of i0.White means that the corresponding vertex is not a pre-decessor of i0. On the k+1st row (k = 0, 1, 2, 3) situationswith k predecessors are presented. The conﬁgurations forwhich the right neighboring vertex is a predecessor of i0are circled by a dashed lineNow consider the problem of predecessors fornearest neighboring sites. First we ﬁx a site i0 inthe bulk of the square lattice. Denote its right near-est neighboring site by j0. Next consider four variouscases, when i0 has exactly k nearest neighboring pre-decessors (k = 0, 1, 2, 3) (see Fig. 1). For k = 0 thesite j0 trivially is not a predecessor of i0. For k = 1,we have 1 of 4 equivalent situations when j0 is thepredecessor of i0. For k = 3, we have 3 of 4 equiva-lent situations when j0 is the predecessor. The crucialcase is k = 2. Here we have 6 situations, but not allof them are equivalent. On the other hand, we canselect two groups of 4 and 2 elements (the ﬁrst fourand the last two in the third line of Fig. 1) so that60Acta Polytechnica Vol. 51 No. 1/2011the elements in each group are equivalent. We arelooking for situations where j0 is a predecessor of i0.There are 2 encircled elements from the ﬁrst groupand one from the second group, which correspondto the desired situations. Thus, if we take the lin-ear combination of X1, X2 and X3 with coeﬃcients1/4, 1/2 and 3/4 correspondingly, we get the desiredprobability P (1) that j0 is the predecessor of i0:P (1) =14X1 +12X2 +34X3 =516. (2.5)3 Return probability for theloop erased random walkConsider a ﬁnite square lattice L with vertex set Vand edge set E. Given P = [u0, u1, u2, . . . , un] a pathin L, its loop-erasure LE(P) = [γ0, γ1, γ2, . . . , γm] isdeﬁned by chronologically removing loops from P .Formally, this deﬁnition is as follows. We ﬁrst setγ0 = u0. Assuming γ0, . . . , γk have been deﬁned, welet sk = 1+max{j : uj = γk}. If sk = n+1, we stopand LE(P) = [γ0, γ1, γ2, . . . , γm] with m = k. Other-wise, we let γk+1 = usk . Note that the order in whichwe remove loops does matter, and it follows from thedeﬁnition that we remove loops as they are created,following the path. A walk obtained after applyingthe loop-erasure to a simple random walk path iscalled a Loop-Erased Random Walk (LERW). Sincethe inﬁnite simple random walk on ﬁnite connectedundirected graphs is recurrent, the inﬁnite LERW isnot deﬁned. On the other hand, we can ﬁx a subsetW ⊂ V of vertices and deﬁne LERW from a ﬁxedvertex u0 to W . To do this, we take the path of asimple random walk started at u0 and stopped uponhitting W , after which we apply the loop-erasure.Wilson established an algorithm to generate uni-form spanning trees, which uses LERW [13]. It turnsout to be extremely useful not only as a simulationtool, but also for theoretical analysis. It runs as fol-lows. Pick an arbitrary ordering V = {v0, v1, . . . , vN}for the vertices in L. Let S0 = v0. Inductively, fori = 1, 2, . . . , N deﬁne a graph Si to be the union ofSi−1 and a (conditionally independent) LERW pathfrom vi to Si−1. Note, if vi ∈ Si−1, then Si = Si−1.Then, regardless of the chosen order of the vertices,SN is a UST on L with root v0. If we take as an ini-tial condition S0 = W , with some W ⊂ V , then thegenerated structures will be spanning forests with aset of roots W . The spanning forest with a ﬁxed setof roots can be considered as a spanning tree, if weadd an auxiliary vertex and join it to all the roots.Now consider the Wilson algorithm on L with theset of boundary vertices ∂L and take S0 = ∂L. Whenthe size of the lattice tends to inﬁnity, the boundaryeﬀects will vanish, so we can neglect the details of theboundary. So we will not distinguish between span-ning forests and spanning trees. It follows from theWilson algorithm for a ﬁxed site i0, that if we start aLERW from i0 upon hitting the boundary ∂L, we willgenerate a path  of a spanning tree from i0 to theboundary (see also [7, 8]). All vertices on the path form the set of descendants of i0. So, if a ﬁxed vertexj0 belongs to , i0 is a predecessor of j0.Consequently, the probability P (j0− i0) that i0 isa predecessor of j0 in a randomly taken (from the uni-form distribution) spanning tree on the large squarelattice is equal to the probability that the LERWstarted from i0 passes j0. In the particular case|j0 − i0| = 1, the probability P (1) = 5/16 calculatedin the previous section is the return probability forthe LERW.4 Monte-Carlo simulationsConsider the ﬁnite 2N + 1 × 2N + 1 square latticeLN . Denote its central vertex by i0 and assumethat it is an origin of the coordinate system. Wedeliberately took an odd-odd lattice to provide thesymmetry which guarantees the most eﬃcient van-ishing of the boundary eﬀects for large lattices. Af-ter generating a large number of LERWs startingfrom i0, we get an approximation of the functionPN (j0− i0) ≡ PN (j0). Given ﬁxed j0, we can extrap-olate PN (j0), tending N to inﬁnity and get asymptot-ical function P (j0). Assume that the Euclidean dis-tance between the origin i0 and j0 is r and the coor-dinates of j0 are (r cosϕ, r sinϕ). The Monte-Carlosimulations and Coulomb gas arguments show thatthe asymptotical behaviour of the function P (j0) forlarge r (r  1) does not depend on ϕ. So, for r  1we have P (j0)  P (r). Fig. 2 shows the behaviourof P (r) for various j0 on the log-log scale, obtainedfrom Monte-Carlo simulations. From this result weconclude that P (r) decreases as a power function:P (r)  Crα, (4.6)Fig. 2: The results of Monte-Carlo simulations for theprobability P (r)61Acta Polytechnica Vol. 51 No. 1/2011with α ≈ 0.751 and C ≈ 0.305. During the sim-ulations, we took sizes up to N = 100 and num-ber of simulations 108. The obtained results are inagreement with α = 3/4, which follows from theCoulomb gas arguments mentioned above. The ef-fective Monte-Carlo algorithm allows the evaluationof probabilities P (j0 − i0) for arbitrary ﬁnite j0, i0.At the same time, the analytical calculation of P (r)for r > 1 remains a diﬃcult unsolved problem.AcknowledgementThis work was supported by Russian RFBR grantNo. 09-01-00271a, and by the Belgian Interuniver-sity Attraction Poles Program P6/02, through theNOSY (Nonlinear systems, stochastic processes andstatistical mechanics) network. 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B., Ruelle, P.:Symmetry, Integr. Geom.: Methods Appl. 3(2007), 001.[18] Poghosyan, V. S., Grigorev, S. Y., Priez-zhev, V. B., Ruelle, P.: Phys. Lett. B 659(2008), 768; J. Stat. Mech. (2010), P07025.[19] Azimi-Tafreshi, N., Dashti-Naserabadi, H., Mog-himi-Araghi, S., Ruelle, P.: J. Stat. Mech.(2010), P02004.[20] Grigorev, S. Y., Poghosyan, V. S., Priez-zhev, V. B.: J. Stat. Mech. (2009), P09008.V. S. PoghosyanE-mail: vahagn.poghosyan@uclouvain.beInstitute for Theoretical PhysicsThe Catholic University of LouvainB-1348 Louvain-La-Neuve, BelgiumV. B. PriezzhevE-mail: priezzvb@theor.jinr.ruBogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear Research141980 Dubna, Russia62

The problem of predecessors on spanning trees

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