Zero Range Process and Multi-Dimensional Random Walks

The special limit of the totally asymmetric zero range process of the low-dimensional non-equilibrium statistical mechanics described by the non-Hermitian Hamiltonian is considered. The calculation of the conditional probabilities of the model are based on the algebraic Bethe ansatz approach. We dem...

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Datum:	2017
Hauptverfasser:	Bogoliubov, N.M., Malyshev, C.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2017
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1485882019-02-19T01:31:30Z Zero Range Process and Multi-Dimensional Random Walks Bogoliubov, N.M. Malyshev, C. The special limit of the totally asymmetric zero range process of the low-dimensional non-equilibrium statistical mechanics described by the non-Hermitian Hamiltonian is considered. The calculation of the conditional probabilities of the model are based on the algebraic Bethe ansatz approach. We demonstrate that the conditional probabilities may be considered as the generating functions of the random multi-dimensional lattice walks bounded by a hyperplane. This type of walks we call the walks over the multi-dimensional simplicial lattices. The answers for the conditional probability and for the number of random walks in the multi-dimensional simplicial lattice are expressed through the symmetric functions. 2017 Article Zero Range Process and Multi-Dimensional Random Walks / N.M Bogoliubov, C. Malyshev // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05A19; 05E05; 82B23 DOI:10.3842/SIGMA.2017.056 http://dspace.nbuv.gov.ua/handle/123456789/148588 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The special limit of the totally asymmetric zero range process of the low-dimensional non-equilibrium statistical mechanics described by the non-Hermitian Hamiltonian is considered. The calculation of the conditional probabilities of the model are based on the algebraic Bethe ansatz approach. We demonstrate that the conditional probabilities may be considered as the generating functions of the random multi-dimensional lattice walks bounded by a hyperplane. This type of walks we call the walks over the multi-dimensional simplicial lattices. The answers for the conditional probability and for the number of random walks in the multi-dimensional simplicial lattice are expressed through the symmetric functions.
format	Article
author	Bogoliubov, N.M. Malyshev, C.
spellingShingle	Bogoliubov, N.M. Malyshev, C. Zero Range Process and Multi-Dimensional Random Walks Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Bogoliubov, N.M. Malyshev, C.
author_sort	Bogoliubov, N.M.
title	Zero Range Process and Multi-Dimensional Random Walks
title_short	Zero Range Process and Multi-Dimensional Random Walks
title_full	Zero Range Process and Multi-Dimensional Random Walks
title_fullStr	Zero Range Process and Multi-Dimensional Random Walks
title_full_unstemmed	Zero Range Process and Multi-Dimensional Random Walks
title_sort	zero range process and multi-dimensional random walks
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/148588
citation_txt	Zero Range Process and Multi-Dimensional Random Walks / N.M Bogoliubov, C. Malyshev // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 32 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT bogoliubovnm zerorangeprocessandmultidimensionalrandomwalks AT malyshevc zerorangeprocessandmultidimensionalrandomwalks
first_indexed	2025-07-12T19:43:47Z
last_indexed	2025-07-12T19:43:47Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 056, 14 pages Zero Range Process and Multi-Dimensional Random Walks Nicolay M. BOGOLIUBOV †‡ and Cyril MALYSHEV †‡ † St.-Petersburg Department of Steklov Institute of Mathematics of RAS, Fontanka 27, St.-Petersburg, Russia E-mail: bogoliub@yahoo.com, malyshev@pdmi.ras.ru ‡ ITMO University, Kronverksky 49, St.-Petersburg, Russia Received March 28, 2017, in final form July 14, 2017; Published online July 22, 2017 https://doi.org/10.3842/SIGMA.2017.056 Abstract. The special limit of the totally asymmetric zero range process of the low-dimen- sional non-equilibrium statistical mechanics described by the non-Hermitian Hamiltonian is considered. The calculation of the conditional probabilities of the model are based on the algebraic Bethe ansatz approach. We demonstrate that the conditional probabilities may be considered as the generating functions of the random multi-dimensional lattice walks bounded by a hyperplane. This type of walks we call the walks over the multi-dimensional simplicial lattices. The answers for the conditional probability and for the number of ran- dom walks in the multi-dimensional simplicial lattice are expressed through the symmetric functions. Key words: zero range process; conditional probability; multi-dimensional random walk; correlation function; symmetric functions 2010 Mathematics Subject Classification: 05A19; 05E05; 82B23 1 Introduction The zero-range process is a stochastic lattice gas where the particles hop randomly with an on-site interaction that makes the jump rate dependent only on the local particle number. The zero range processes (ZRPs) belong to a class of minimal statistical-mechanics models of the low-dimensional non-equilibrium physics [12, 30]. Being exactly solvable the model and its other variations are intensively studied both by mathematicians and physicists [9, 15, 16, 21, 25, 26]. In this paper we consider the totally asymmetric simple zero range process (TASZRP) [13, 17], which describes a system of indistinguishable particles placed on a one-dimensional lattice, moving randomly in one direction from right to left with the equal hopping rate on a periodic ring. The dynamical variables of the model are the phase operators [11] which can be regarded as a special limit of q-bosons [5, 19]. The application of the quantum inverse method (QIM) [13, 17, 20] allows to calculate the scalar products and form-factors of the model and represent them in the determinantal form [3, 8]. The relation of the considered model and the totaly asymmetric simple exclusion process (TASEP) was discussed in [8, 27]. Certain quantum integrable models solvable by the QIM demonstrate close relationship [2, 6, 7] with the different objects of the enumerative combinatorics [31, 32] and the theory of the symmetric functions [22]. It appeared that the correlation functions of some integrable models may be regarded as the generating functions of plane partitions and random walks. Different types of random walks [14, 18, 31, 32] are of considerable recent interest due to their role in quantum information processing [10, 29]. The walks on multi-dimensional lattices were studied by many authors [4, 23, 24, 28]. This paper is a contribution to the Special Issue on Recent Advances in Quantum Integrable Systems. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2016.html mailto:bogoliub@yahoo.com mailto:malyshev@pdmi.ras.ru https://doi.org/10.3842/SIGMA.2017.056 http://www.emis.de/journals/SIGMA/RAQIS2016.html 2 N.M. Bogoliubov and C. Malyshev In this paper we shall calculate the conditional probability of the model and reveal its con- nection with the generating function of the lattice paths on multi-dimensional oriented lattices bounded by a hyperplane. The layout of the paper is as follows. In the introductory Section 2 we give the definition of the TASZRP. The solution of the model by QIM is presented in Section 3. In Section 4 condi- tional probability is calculated. The random walks over the M -dimensional oriented simplicial lattice with retaining boundary conditions are introduced and the connection of their generating function with the conditional probability is established. 2 Totally asymmetric simple zero range hopping model Consider a system with N particles on a periodic one-dimensional lattice of length M + 1, i.e., on a ring, where sites m and m+M + 1 are identical. Each site of a lattice contains arbitrary number of particles. The particles evolve with the following dynamics rule: during the time interval [t, t+ dt] a particle on a site i jumps with probability dt to the neighbouring site i+ 1. There are no restrictions on the number of particles on a lattice site. m m+1 m-1 Figure 1. Totally asymmetric simple zero range process. A configuration C of the system is characterized by the list of all possible arrangements of N particles amongst the M + 1 available sites. The total number of configurations is, therefore, Ω = (N +M)! (N)! (M)! . (2.1) The probability Pt(C) of finding the system in configuration C at time t satisfies the master equation dPt(C) dt =MPt(C), (2.2) whereM is the Markov matrix of the size Ω×Ω. For C 6= C ′ the entryM(C ′, C) is the transition rate from C to C ′. It is equal to unity, if the transition is allowed, and is zero otherwise. The diagonal entry −M(C,C) is equal to the number of occupied sites in the configuration C. The elements of the columns and rows of M add up to zero and total probability is conserved∑ C Pt(C) = 1. The process described is stochastic, the unique stationary state of this system being the one, in which all Ω different configurations C have equal weight. Zero Range Process and Multi-Dimensional Random Walks 3 A configuration C is represented by the sequence (n0, n1, . . . , nM ) of occupation numbers nj , satisfying the condition 0 ≤ n0, n1, . . . , nM ≤ N , n0 + n1 + · · · + nM = N . We can rewrite the master equation (2.2) for the TAZRP in the form d dt Pt(n0, . . . , nM ) = M∑ m=0 Pt(n0, . . . , nm + 1, nm+1 − 1, . . . , nM ) −KPt(n0, . . . , nM ), (2.3) where K is the number of occupied sites in the configuration (n0, n1, . . . , nM ), i.e., the number of nj 6= 0. This equation has to be supplemented by the condition that Pt(n0, . . . , nM ) = 0, if at least one of the nj < 0. To formulate the dynamic of the system in terms of a quantum mechanical model we denote a particle configuration as a Fock vector \|n0, . . . , nM 〉 and define a probability vector \|Pt〉 = ∑ 0≤n0,n1,...,nM≤N n0+n1+···+nM=N Pt(n0, . . . , nM )\|n0, . . . , nM 〉, where Pt(n0, . . . , nM ) are the probabilities of configuration (n0, . . . , nM ). The generator of the master equation (2.3) may be written as H = M∑ m=0 ( φmφ † m+1 − φ † mφm ) = M∑ m=0 ( φ†m+1 − φ † m ) φm. (2.4) Here, the phase operators φn, φ † n were introduced. They satisfy commutation relations [N̂i, φj ] = −φiδij , [N̂i, φ † j ] = φ†iδij , [φi, φ † j ] = πiδij , where N̂j is the number operator and πi = 1−φ†iφi is vacuum projector: φjπj = πjφ † j = 0. This algebra has a representation on the Fock space: φj \|n0, . . . , 0j , . . . , nM 〉 = 0, φ†j \|n0, . . . , nj , . . . , nM 〉 = \|n0, . . . , nj + 1, . . . , nM 〉, φj \|n0, . . . , nj , . . . , nM 〉 = \|n0, . . . , nj − 1, . . . , nM 〉, (2.5) N̂j \|n0, . . . , nj , . . . , nM 〉 = nj \|n0, . . . , nj , . . . , nM 〉, πj \|n0, . . . , 0j , . . . , nM 〉 = \|n0, . . . , 0j , . . . , nM 〉. The states \|n0, . . . , nM 〉 are orthogonal, 〈p0, . . . , pM \|n0, . . . , nM 〉 = M∏ l=0 δplnl . The Fock vectors are generated from the vacuum state \|0〉 ≡ M∏ j=0 \|0〉j (2.6) by action of rising operators φ†j \|n〉 ≡ \|n0, . . . , nM 〉 = M∏ j=0 ( φ†j )nj \|0〉, 0 ≤ n ≤ N, M∑ j=0 nj = N. (2.7) 4 N.M. Bogoliubov and C. Malyshev The total number operator N̂ = M∑ j=0 Nj commutes with the Hamiltonian (2.4): [H, N̂ ] = 0. (2.8) The hopping term φ†m+1φm of the Hamiltonian (2.4) annihilates the particle in the site m and creates it in the site m+ 1: M∑ m=0 φmφ † m+1\|n0, . . . , nM 〉 = M∑ m=0 \|n0, . . . , nm − 1, nm+1 + 1, . . . , nM 〉. The vacuum projector πm due to the definition (2.5) acts on the Fock vector in the following way πm\|n0, . . . , nM 〉 = \|n0, . . . , nM 〉, if nm = 0, (2.9) and πm\|n0, . . . , nM 〉 = 0, if nm 6= 0. The operator K̂ = M∑ m=0 φ†mφm = M∑ m=0 (1− πm) counts the number of occupied sites in the configuration (n1, n2, . . . , nM ) K̂\|n0, . . . , nM 〉 = K\|n0, . . . , nM 〉. The evolution of the quantum system \|Pt〉 = etH \|P0〉 is governed by the imaginary time Schrödinger equation d dt \|Pt〉 = H\|Pt〉, which is equivalent to the master equation (2.3) for the probabilities Pt(n0, . . . , nM ) equal to the matrix elements Pt(n0, . . . , nM ) = 〈n0, . . . , nM \|Pt〉. The initial (t = 0) probability distribution defines the state \|P0〉 \|P0〉 = ∑ 0≤n0,n1,...,nM≤N n0+n1+···+nM=N P0(n0, . . . , nM )\|n0, . . . , nM 〉. The state 〈S\| = ∑ 0≤n0,n1,...,nM≤N n0+n1+···+nM=N 〈n0, . . . , nM \| = 〈0\|  M∑ j=0 φjπj−1 · · ·π1 N Zero Range Process and Multi-Dimensional Random Walks 5 is a left eigenvector of the model, which obeys 〈S\|H = 0. (2.10) Correspondingly, H has one right eigenvector with eigenvalue zero, which is associated with the state \|Ω〉 = 1 Ω \|S〉: H\|Ω〉 = 0, where Ω is (2.1), 〈S\|Ω〉 = 1. The vector 〈S\| does not evolve in time and, therefore, corresponds up to normalization factor to a steady state distribution of the system 〈S\|Pt〉 = 〈S\|P0〉 = 1. In this paper we shall calculate the conditional probability Pt(n\|m) = 1 Ω 〈n0, . . . , nM \|etH \|m0, . . . ,mM 〉, (2.11) which is equal to probability, that in a time t the system will be in a pure state defined by the occupation numbers (n0, . . . , nM ) provided that initially the system was prepared in a pure state \|P0〉 = \|m0, . . . ,mM 〉. 3 Solution of the TASZRP To apply the scheme of the QIM to the solution of the Hamiltonian (2.4) we define L-opera- tor [8] which is 2×2 matrix with the operator-valued entries acting on the Fock states according to (2.5): L(n\|u) ≡ ( u−1 + uπn φ†n φn u ) , (3.1) where u ∈ C is a parameter. This L-operator satisfies the intertwining relation R(u, v) (L(n\|u)⊗ L(n\|v)) = (L(n\|v)⊗ L(n\|u))R(u, v), in which R(u, v) is the R-matrix R(u, v) =  f(v, u) 0 0 0 0 g(v, u) 1 0 0 0 g(v, u) 0 0 0 0 f(v, u)  , (3.2) where f(v, u) = u2 u2 − v2 , g(v, u) = uv u2 − v2 , u, v ∈ C. (3.3) The monodromy matrix is the matrix product of L-operators T (u) = L(M \|u)L(M − 1\|u) · · ·L(0\|u) = ( A(u) B(u) C(u) D(u) ) . (3.4) The commutation relations of the matrix elements of the monodromy matrix are given by the same R-matrix (3.2) R(u, v) (T (u)⊗ T (v)) = (T (v)⊗ T (u))R(u, v). (3.5) 6 N.M. Bogoliubov and C. Malyshev The transfer matrix τ(u) is the trace of the monodromy matrix in the auxiliary space τ(u) = trT (u) = A(u) +D(u). (3.6) The relation (3.5) means that [τ(u), τ(v)] = 0 for arbitrary u, v ∈ C. From the definitions (3.1) and (3.4) one finds by direct calculation that the entries of the monodromy matrix are polynomials in u2. For A(u) and D(u) one has uM+1A(u) = 1 + u2 ( M−1∑ m=0 φmφ † m+1 + M∑ m=0 πm ) + · · ·+ u2(M+1) M∏ m=0 πm, uM+1D(u) = u2φ†0φM + · · ·+ u2(M+1), (3.7) where the dots stand for the terms not important for further consideration. We also find that lim u→0 B̃(u) ≡ lim u→0 uMB(u) = φ†0, (3.8) lim u→0 C̃(u) ≡ lim u→0 uMC(u) = φM . (3.9) The representation (3.7) allows to express the Hamiltonian (2.4) through the transfer mat- rix (3.6) H = ∂ ∂u2 uM+1τ(u) ∣∣∣ u=0 −(M + 1) = ∂ ∂u2 uM+1(A(u) +D(u)) ∣∣∣ u=0 −(M + 1). (3.10) By construction this Hamiltonian commutes with the transfer matrix [H, τ(u)] = 0. Since the Hamiltonian (2.4) is non-Hermitian we have to distinguish between its right and left eigenvectors. The N -particle right state-vectors are taken in the form \|ΨN (u)〉 =  N∏ j=1 B̃(uj)  \|0〉, (3.11) where B̃(u) is defined in (3.8), and u implies a collection of arbitrary complex parameters uj ∈ C: u = (u0, u1, . . . , uN ). The left state-vectors are equal to 〈ΨN (u)\| = 〈0\|  N∏ j=1 C̃(uj)  , (3.12) where C̃(u) is given by (3.9). The vacuum state (2.6) is an eigenvector of A(u) and D(u), A(u)\|0〉 = α(u)\|0〉, D(u)\|0〉 = δ(u)\|0〉 with the eigen-values α(u) = ( u−1 + u )M+1 , δ(u) = uM+1. (3.13) The state-vectors (3.11) and (3.12) are the eigenvectors both of the Hamiltonian (2.4) and of the transfer matrix τ(u) (3.6), if, and only if, the variables uj satisfy the Bethe equations α(un) δ(un) = N∏ m 6=n f(um, un) f(un, um) , Zero Range Process and Multi-Dimensional Random Walks 7 where f are the elements of the R-matrix (3.3). In the explicit form the Bethe equations are given by u−2N n ( 1 + u−2 n )M+1 = (−1)N−1 U2 , U2 ≡ N∏ j=1 u2 j . (3.14) There are Ω equation (2.1) sets of solutions of these equations. The eigenvalues ΘN (v,u) of the transfer matrix (3.6) in the general form are equal to ΘN (v;u) = α(v) N∏ j=1 f(v, uj) + δ(v) N∏ j=1 f(uj , v). For the model under consideration vM+1ΘN (v;u) = ( 1 + v2 )M+1 N∏ m=1 u2 m u2 m − v2 + v2(M+1) N∏ m=1 v2 v2 − u2 m = (( 1 + v2 )M+1 + (−1)Nv2(M+N+1)U−2 ) H ( v2;u−2 ) . Here, the generating function of complete symmetric functions hl ( u−2 ) ≡ hl ( u−2 1 , u−2 2 , . . ., u−2 N ) [22] is introduced H ( v2;u−2 ) ≡ N∏ m=1 1 1− v2/u2 m = ∑ l≥0 hl ( u−2 ) v2l. Equation (3.10) enables to obtain the spectrum of the Hamiltonian (2.4). The N -particle eigenenergies H\|ΨN (u)〉 = EN \|ΨN (u)〉 are equal to EN (u) = ∂ ∂v2 vM+1ΘN (v;u) ∣∣∣ v=0 = h1 ( u−2 ) = N∑ k=1 u−2 k . (3.15) The steady state (2.10) corresponds to a special solution of Bethe equations (3.14) when all uj =∞. 4 The calculation of conditional probability For the models associated with the R-matrix (3.2) the scalar product of the state-vectors (3.11) and (3.12) is given by the formula [1]: 〈ΨN (v)\|ΨN (u)〉 = detQ VN ( v2 ) VN ( u−2 ) N∏ j=1 ( vj uj )M+N−1 , (4.1) where VN (x) is the Vandermonde determinant, VN (x) ≡ VN (x1, x2, . . . , xN ) = ∏ 1≤i<k≤N (xk − xi), (4.2) 8 N.M. Bogoliubov and C. Malyshev and the matrix Q is characterized by the entries Qjk, 1 ≤ j, k ≤ N , Qjk = α(vj)δ(uk) ( uk vj )N−1 − α(uk)δ(vj) ( vj uk )N−1 uk vj − vj uk , with α(u) and δ(u) given by (3.13). The norm of the state-vector N 2(u) ≡ 〈ΨN (u)\|ΨN (u)〉 is defined by the scalar product (4.1) when the arguments v and u satisfy the Bethe equations (3.14). For the present case of the generalized phase model we substitute vk = uk, ∀ k, respecting the Bethe equations (3.14) into the entries of the matrix Q. The resulting matrix is denoted as Q̃, and its entries at j 6= k are equal to Q̃jk = (−1)N (ukuj) N+M+1 U2 , where U2 is given by (3.14). L’Hôspital rule gives the diagonal entries of Q̃ Q̃jj = (N − 1)α(uj)δ(uj) + ( α(uj)δ ′(uj)− α′(uj)δ(uj) )uj 2 = ( 1−N −Gj )(−1)Nu 2(N+M+1) j U2 , where Gj ≡ M + 1 u2 j + 1 . As a result, the squared norm N 2(u) on the Bethe solution takes the form N 2(u) = det Q̃ VN ( u2 ) VN ( u−2 ) , (4.3) det Q̃ = U2(M+1) ( 1− N∑ l=1 1 N + Gl ) N∏ j=1 (N + Gj). (4.4) The state-vectors belonging to the different sets of solutions of the Bethe equations (3.14) are orthogonal. The eigenvectors (3.11) and (3.12) provide the resolution of the identity operator I = ∑ {u} \|ΨN (u)〉〈ΨN (u)\| N 2(u) , (4.5) where the summation ∑ {u} is over all independent solutions of the Bethe equations (3.14). Inserting the resolution of the identity operator (4.5) into (2.11), one obtains the general answer for the conditional probability Pt(n \|m) = 1 Ω ∑ {u} etEN (u) 〈n\|ΨN (u)〉〈ΨN (u)\|m〉 N 2(u) . (4.6) For the simplicity let us consider the initial state equal to \|N, 0, . . . , 0〉 and the final one respectively to 〈0, 0, . . . , N \|. The conditional probability (2.11) of this process is specified as follows Pt ≡ 1 Ω 〈 0, 0, . . . , N ∣∣etH ∣∣N, 0, . . . , 0〉 = 1 Ω 〈 0 ∣∣(φM )NetH(φ†0)N ∣∣0〉, (4.7) Zero Range Process and Multi-Dimensional Random Walks 9 where equation (2.7) has been used. Inserting the resolution of the identity operator (4.5) into (4.7), we obtain Pt = 1 Ω ∑ {u} etEN (u) N 2(u) 〈 0\|(φM )N \|ΨN (u)〉〈ΨN (u)\|(φ†0)N \|0 〉 , where the summation is over all independent solutions of equations (3.14). The decomposition (3.9) for B(u) and C(u) gives that〈 0\|(φM )N \|ΨN (u) 〉 = lim v→0 〈ΨN (v)\|ΨN (u)〉 = 1,〈 ΨN (u)\|(φ†0)N \|0 〉 = lim v→0 〈ΨN (u)\|ΨN (v)〉 = 1, and eventually the answer is Pt = 1 Ω ∑ {u} etEN (u) N 2(u) , where N 2(u) is given by (4.3), (4.4). To obtain the explicit answer for the conditional probability in the general case (4.6) we shall express state vectors (3.11) and (3.12) in the coordinate form. The state-vector (3.11) has the representation \|ΨN (u)〉 = ∑ λ⊆{MN} χRλ(u)  M∏ j=0 (φ†j) nj  \|0〉, (4.8) where the symmetric function χRλ is equal, up to a multiplicative pre-factor, to χRλ(x) = χRλ(x1, x2, . . . , xN ) = 1 VN (x) det ( x 2(N−j) i (1 + x−2 i )λj ) 1≤i,j≤N . (4.9) Here λ denotes the partition (λ1, . . . , λN ) of non-increasing non-negative integers, M ≥ λ1 ≥ λ2 ≥ · · · ≥ λN ≥ 0, and VN (x) is the Vandermonde determinant (4.2). There is a one-to-one correspondence between a sequence of the occupation numbers (n0, n1, . . . , nM ), n0+n1+· · ·+nM = N , and the partition λ = ( MnM , (M − 1)nM−1 , . . . , 1n1 , 0n0 ) , where each number S appears nS times (see Fig. 2). The sum in equation (4.8) is taken over all partitions λ into at most N parts with N ≤M . Acting by the Hamiltonian (2.4) on the state-vector (4.8), we find that the wave function (4.9) satisfies the equation N∑ k=1 χRλ1,...,λk+1,...,λN (u) = EN (u)χRλ1,...,λN (u), (4.10) together with the exclusion condition χRλ1,...,λl−1=λl−1,λl,...,λN (u) = χRλ1,...,λl−1=λl,λl,...,λN (u), 1 ≤ l ≤ N. (4.11) 10 N.M. Bogoliubov and C. Malyshev 0 6 Figure 2. A configuration of particles (N = 4) on a lattice (M = 6), the corresponding partition λ = (61, 50, 40, 32, 20, 11, 00) ≡ (6, 3, 3, 1) and its Young diagram. The energy EN is given by (3.15). The state-vector (4.8) is the eigenvector of the Hamilto- nian (2.4) with the periodic boundary conditions if the parameters uj satisfy the Bethe equa- tions (3.14). The relations (4.8), (4.10) and (4.11) can be viewed as an implementation of the coordinate Bethe ansatz [17], which is an alternative to the approach of the algebraic Bethe ansatz consid- ered in Section 3. Although the model is solved by the algebraic Bethe ansatz, representations of the type of (4.8) are especially useful in discussing the combinatorial properties of the quantum integrable models [2, 6, 7]. Expanding the left state-vector (3.12), we obtain 〈ΨN (u)\| = ∑ λ⊆{MN} χLλ(u)〈0 ∣∣( M∏ i=0 φni i ) , (4.12) where the wave function is given by the symmetric function χLλ(x) = det (( 1 + x−2 i )λjx2(N−j) i ) 1≤i,j≤N VN (x) . It satisfies the equations N∑ k=1 χLλ1,...,λk−1,...,λN (u) = EN (u)χLλ1,...,λN (u), χLλ1,...,λl,λl+1=λl+1,...,λN (u) = χLλ1,...,λl,λl+1=λl,...,λN (u), 1 ≤ l ≤ N. From equations (4.8), (4.12) one obtains 〈n0, n1, . . . , nM \|ΨN (u)〉 = χRλR (u), 〈ΨN (u)\|m0,m1, . . . ,mM 〉 = χLλL (u), where λR = ( MnM , (M − 1)nM−1 , . . . , 1n1 , 0n0 ) , λL = ( MmM , (M − 1)mM−1 , . . . , 1m1 , 0m0 ) . Finally, the expression for the conditional probability (4.6) has the form Pt(n \|m) = 1 Ω ∑ {u} etEN (u) N 2(u) χRλR (u)χLλL (u). (4.13) Here N 2(u) is the squared norm (4.3). Zero Range Process and Multi-Dimensional Random Walks 11 5 Multi-dimensional lattice walks bounded by a hyperplane Starting from (M+1)-dimensional hypercubical lattice with unit spacing ZM+1 3m ≡ (m0,m1, . . . ,mM ), let us define the non-negative orthant NM+1 0 ≡ {m \| 0 ≤ mi, i ∈ M} as a subset of ZM+1 (hereafter M ≡ {0, 1, . . .M}). Consider a subset of NM+1 0 consisting of sites with coordinates constrained by the requirement m0 +m1 + · · ·+mM = N : Simp(N)(ZM+1) ≡ { m ∈ NM+1 0 ∣∣∣ ∑ i∈M mi = N } . The set Simp(N) ( ZM+1 ) is compact M -dimensional, and we shall call it simplicial lattice. A two- dimensional triangular simplicial lattice is presented in Fig. 3. A sequence of K + 1 points in ZM+1 is called lattice path of K steps [18]. 2 1 N N N 0 0 Figure 3. A two-dimensional triangular simplicial lattice. Random walks over sites of Simp(N) ( ZM+1 ) are defined by a set of admissible steps ΩM (step set ΩM ) so that at each step an ith coordinate mi increases by unity, while the nearest neighbour- ing one decreases by unity. Namely, each element of ΩM is given by sequence (e0, e2, . . . , eM ) so that ei = ±1, ei+1 = ∓1 for all pairs (i, i + 1) with i ∈ M and M + 1 = 0 (mod 2), and ej = 0 for all j ∈ M and j 6= i, i + 1. The step set ΩM ≡ ΩM (m0) ensures that trajectory of a random walk (lattice path) determined by the starting point m0 lies in M -dimensional set Simp(N) ( ZM+1 ) . Directed random walks on M -dimensional oriented simplicial lattice are defined by a step set ΓM = (k0, k1, . . . , kM ) so that ki = −1, ki+1 = 1 for all pairs (i, i + 1) with i ∈ M and M + 1 = 0 (mod 2), and kj = 0 for all j ∈ M\{i, i + 1}. It may occur that some points on the boundary of the simplicial lattice also belong to a random walk trajectory. Therefore, the walker’s movements should be supplied with appropriate boundary conditions. The boundary of the simplicial lattice consists of M + 1 faces of highest dimensionality M − 1. Under the retaining boundary conditions the walker comes to a node of the boundary, and either continues to move in accordance with the elements of ΓM , or keeps staying in the node. An oriented two-dimensional simplicial lattice with the retaining boundary conditions is presented in Fig. 4. To establish the connection of the exponential generation function of lattice paths and the con- ditional probability (2.11) we shall interpret the coordinates nj of a walker n = (n0, n1, . . . , nM ) ∈ ZM+1 in a simplicial lattice Simp(N) ( ZM+1 ) as the occupation numbers of (M+1)-component Fock space and describe the steps of a walker with the help of the Fock state-vectors \|n〉 ≡ \|n0, n1, . . . , nM 〉. Operator φj shifts the value of the jth coordinate of the walker downwards nj → 12 N.M. Bogoliubov and C. Malyshev a b Figure 4. The oriented two-dimensional simplicial lattice defined by the step set Γ2 with retaining boundary conditions (a). The round trip lattice path of the 14 steps from the node with coordinates (0, 5, 0) is shown in (b). nj−1, while φ†j upwards nj → nj+1. If the walker arrived at an arbitrary node of sth component of the boundary, it can make either allowed step or stay on it due the action of operator πs (2.9). The number operator Nj acts as the coordinate operator Nj \|n0, n1, . . . , nj , . . . , nD〉 = nj \|n0, n1, . . . , nj , . . . , nD〉. Since the occupation numbers are non-negative integers and their sum is con- served (2.8), we can regard operator Hrw = M∑ m=0 ( φmφ † m+1 + πm ) (5.1) as a generator of steps of a walker in Simp(N) ( ZM+1 ) with the retaining boundary conditions. The operators (5.1) and (2.4) are related Hrw = H +M + 1. The number of lattice paths made by a walker over the M -dimensional simplicial lattice in k steps with the ending nodes (j0, j1, . . . , jM ) and (l0, l1, . . . , lM ) is given by the expression Gk(l \| j) = 〈 l0, l1, . . . , lM ∣∣Hk rw ∣∣j0, j1, . . . , jM〉. (5.2) It is straightforward to verify that Gk(l \| j) = M∑ s=0 Gk−1(l \| j0, j1, . . . , js + 1, js+1 − 1, . . . , jM ) + (M + 1−K)Gk−1(l \| j0, . . . , jM ), where k ≥ 1, and it is natural to impose the condition G0(l \| j) = M∏ l=0 δlljl . The exponential generating function of lattice paths in Simp(N)(ZM+1) is defined as a formal series Ft(l \| j) = ∞∑ k=0 tk k! Gk(l \| j). (5.3) Due to definition (5.2) it may be expressed as Ft(l \| j) = 〈 l ∣∣ ∞∑ k=0 tk k! Hk rw ∣∣j〉 = 〈 l ∣∣etHrw ∣∣j〉. Zero Range Process and Multi-Dimensional Random Walks 13 Taking into account the connection (5.2) we obtain the desired relation of the conditional prob- ability (2.11) and generation function of lattice paths (5.3): Pt(l \| j) = e−t(M+1) Ω Ft(l \| j). From the expression (4.13) it follows that the number of lattice paths made by a walker in the M -dimensional simplicial lattice in k steps with the ending nodes (j0, j1, . . . , jM ) and (l0, l1, . . . , lM ) is equal to Gk(l \| j) = ∑ {u} hk1 ( u−2 ) N 2(u) χRλR (u)χLλL (u), where the function h1 ( u−2 ) is introduced in (3.15). Acknowledgements This work was supported by RFBR grant 16-01-00296. 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[32] Stanley R.P., Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1999. https://doi.org/10.1088/1751-8113/40/26/001 https://arxiv.org/abs/cond-mat/0701190 https://doi.org/10.1017/CBO9780511628832 https://arxiv.org/abs/1503.05930 https://doi.org/10.1088/0305-4470/23/9/003 https://doi.org/10.1088/0305-4470/23/9/003 https://doi.org/10.1007/3-540-11190-5_8 https://doi.org/10.1007/s10955-016-1555-3 https://arxiv.org/abs/1602.00764 https://doi.org/10.1088/0305-4470/35/12/304 https://arxiv.org/abs/quant-ph/0108004 https://arxiv.org/abs/1402.4448 https://doi.org/10.1103/PhysRevE.69.061109 https://arxiv.org/abs/cond-mat/0401249 https://doi.org/10.1088/1751-8113/46/46/465205 https://arxiv.org/abs/1308.3250 https://doi.org/10.1007/s10955-006-9023-0 https://arxiv.org/abs/cond-mat/0411558 https://doi.org/10.1103/PhysRevA.90.022329 https://arxiv.org/abs/1408.5300 https://arxiv.org/abs/1605.03842 https://doi.org/10.1016/0001-8708(70)90034-4 https://doi.org/10.1017/CBO9781139058520 https://doi.org/10.1017/CBO9780511609589 1 Introduction 2 Totally asymmetric simple zero range hopping model 3 Solution of the TASZRP 4 The calculation of conditional probability 5 Multi-dimensional lattice walks bounded by a hyperplane References

Zero Range Process and Multi-Dimensional Random Walks

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