Finite cycle Gibbs measures on permutations of $\mathbb Z^d$
We consider Gibbs distributions on the set of permutations of associated to the Hamiltonian , where is a permutation and is a strictly convex potential. Call finite-cycle those permutations composed by finite cycles only. We give conditions on ensuring that for large enough temperature there exists a unique infinite volume ergodic Gibbs measure concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We construct as the unique invariant measure of a Markov process on the set of finite-cycle permutations that can be seen as a loss-network, a continuous-time birth and death process of cycles interacting by exclusion, an approach proposed by Fernández, Ferrari and Garcia. Define as the shift permutation . In the Gaussian case , we show that for each , given by is an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with boundary conditions. For a general potential , we prove the existence of Gibbs measures when is bigger than some -dependent value.
The Feynman-Kac representation of the Bose gas consists of trajectories of interacting Brownian motions in a fixed time interval, which start and finish at the points of a spatial point process  . In order to attempt a rigorous analysis of the model, several simplifications have been proposed over the years [5, 6, 12, 13]. In the resulting model, the starting and ending points belong to the -dimensional lattice, and the interaction is reduced to an exclusion condition on the paths at the beginning and the end of the time interval. The state space is therefore the set of permutations or bijections .
For a finite set , denote by the set of permutations that reduce to the identity outside , i.e.,
A function such that is called a potential. We assume is strictly convex and define the Hamiltonian
and associated measure ,
where is a normalizing constant. The nonnegative parameter is called the temperature; we omit the dependence of on . We refer to the condition if as an identity boundary condition, and the finite volume measure associated to a finite set is called a specification.
When the potential is Gaussian, , the value is proportional to the density at the site of a Gaussian distribution with mean and variance . Hence, is proportional to the joint density of the arrival points at time of a family of independent Brownian motions started at each point in , which are conditioned to arrive at distinct points of at that time. This is the case arising from the Feynmann-Kac representation of the Bose gas.
Given permutations , define the composed permutation by and let be the law of when is distributed according to , that is
for continuous real functions . For any vector denote by the shift permutation given by
A permutation is called a ground state if is a local minimum of the Hamiltonian . Since is strictly convex, the shift permutation is a ground state for any vector .
Our main results are the following.
Identity boundary conditions. In Theorem 2.1 we define a function such that when it is finite, for any , there exists an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with identity boundary conditions at temperature . The measure concentrates on finite-cycle permutations.
Shift boundary conditions. In Theorem 2.3 we fix and extend the results of Theorem 2.1 to -boundary conditions. That is, we define , and assuming that it is finite, we show that for any temperature there exists an ergodic Gibbs measure associated to boundary conditions such that concentrates on finite-cycle permutations.
Gaussian potential. The physically relevant Gaussian potential is covered by Theorem 2.1; in this case the results for -boundary conditions follow directly from the observation that the specifications matching at the boundary satisfy , a relation that extends to the limit . In particular, here is the same for all , .
The statements of these results establish the existence of Gibbs measures as a weak limit of specifications. In fact, we obtain pointwise limits. For instance, in the proof of Theorem 2.1 we construct a coupled family of permutations , each distributed according to (), such that for the random variables converge almost surely to , as .
The proofs follow the approach of Fernández, Ferrari and Garcia , relying on the fact that the Peierls-contour representation of the low temperature Gibbs measure for the Ising model is reversible for a loss network of contours interacting by exclusion. In the case of identity boundary conditions, instead of contours, we consider the finite cycles that compose a permutation. Let be the set of finite cycles on with length larger than 1. A finite-cycle permutation is represented as a “gas” of finite cycles in , and the Gibbs measure can be described as a product of independent Poisson random variables in the space , conditioned to non overlapping of cycles, that is, each site belongs to at most one cycle. This is automatically well defined in finite volume. We explicitly construct an infinite volume random configuration with non overlapping cycles, means that the cycle is present in the configuration . This configuration is naturally associated to the permutation composed by the cycles indicated by . We then show that is the almost sure limit as of permutations in distributed according to the specifications .
The loss network is a continuous-time Markov process , having as unique invariant measure the target Gibbs measure. In this process, each cycle attempts to appear independently at a rate defined later in (2.4), and is allowed to join the existing configuration only if it does not overlap with the already present cycles. Cycles also die, independently, at rate 1. If is sufficiently large this process is well defined in infinite volume, and a realization of the stationary process running for all can be constructed as a function of a family of space-time Poisson processes, the usually called Harris graphical construction. The condition for the existence of the stationary process is related to the absence of oriented percolation of cycles in the space–time realization of a free process in , where all cycles are allowed to be born, regardless whether they overlap with pre-existing cycles or not. The no-percolation condition follows from dominating the percolation cluster by a subcritical multitype branching process, a standard technique, see for instance . The subcriticality condition for the branching process leads to the condition .
Background and further prospects
The existence of a Gibbs measure concentrating on finite-cycle permutations of was first obtained by Gandolfo, Ruiz and Ueltschi  in the large temperature regime for the Gaussian potential. Recently, Betz  gave a condition yielding tightness of the specifications for a more general family of potentials, for any value of , his results imply that thermodynamic limits of specifications with identity boundary conditions exist for any and dimension . However, the problem of identifying these limits and their typical cycle length remains open.
Biskup and Richthammer  consider the one dimensional case and strictly convex potentials satisfying some additional growth conditions. They prove that the set of all ground states associated to in (1.2) is , as in (1.5), and that for each ground state and temperature there is a Gibbs measure . Furthermore, they show that the set of extremal Gibbs measures is , that is, each extremal Gibbs measure is associated to a ground state. The measure is translation invariant and supported on configurations having exactly infinite cycles. They also prove that for any , the measure has a regeneration property, which in the case entails the convergence as of the specifications with identity boundary conditions to . In particular, this implies that for , identity boundary conditions lead to finite cycles, for all temperatures.
Infinite cycles. In –dimensions our results say that under identity boundary conditions, for large enough, the Gibbs measures concentrate on finite-cycle permutations. On the other hand, for the Gaussian potential and small , Gandolfo, Ruiz and Ueltschi  performed numerical simulations of the 3-dimension specification associated to a box yielding cycles with macroscopic length, i.e., length that grows proportionally to the size of . More recent numerical results by Grosskinsky, Lovisolo and Ueltschi  suggest that the scaled down size of these macroscopic cycles converges to a Poisson-Dirichlet distribution. See also Goldschmidt, Ueltschi and Windridge  for a discussion relating cycle representations and fragmentation-coagulation models, where the Poisson-Dirichlet distributions appear naturally. The authors in  argue that the situation should be similar in higher dimensions, in contrast to the case . In 2–dimensions it is expected that the size of the cycles grows as , but in this case the length would not be macroscopic, a conjecture that is supported by numerical simulations in [1, 7]. The question remains whether a positive fraction of sites belongs to these mesoscopic cycles. Betz  provides numerical evidence that for long cycles are fractals in the thermodynamic limit, and conjectures a connection to Schramm-Loewner evolution.
Domain of attraction of Gibbs measures. Let and denote the first vector in the canonical basis. In a forthcoming paper, Yuhjtman considers the Gaussian potential with ground state defined by
(see Figure 1) and shows that the thermodynamic limit of , the specifications with boundary conditions given by , is equal to the measure associated to identity boundary conditions. In particular, in dimensions higher than 1, the one-to-one correspondence between ground states and extremal Gibbs measures fails to hold. It would be interesting to find the domain of attraction of each Gibbs state. That is, if is a Gibbs measure, one would like to characterize the set .
Further translation invariant Gibbs measures. Set and consider the ground states given by
Our approach requires translation invariance of the boundary conditions, which are satisfied neither by nor by (see Figure 2). The conjecture is that the thermodynamic limit arising from any of these boundary conditions should lead to a Gibbs measure with - density of paths crossing the hyperplane from left to right. In connection to these ground states, it would be interesting to describe the macroscopic shape determined by these left-right crossing paths.
Permutations of point processes. When the points are distributed according to a point process there are two possibilities. In the quenched case one studies the random permutation of a fixed point configuration. In this case we expect that our approach would be useful to show that for almost all point configuration there is a unique Gibbs measure when the temperature is high enough in relation to the point density . The 1–dimensional quenched case is studied by Biskup and Richthammer , who prove that there are no infinite cycles for any value of the temperature. Süto [14, 15] investigates the annealed case, where one jointly averages point positions and permutations. By integrating over the former, it is then possible to explicitly identify the temperature below which infinite cycles appear, Süto points out that this is equivalent to Bose-Einstein condensation in the Bose gas. These results are generalized by Betz and Ueltschi in .
Organization of the article
2 Notation and Results
Denote by the set of permutations of , that is
equipped with the product topology generated by the sets , , and the associated Borel sigma-algebra . Given a permutation and a finite set , let
be the set of permutations that match outside of . Let be the identity permutation, for all , and denote . Let be a strictly convex potential with and recall the definition (1.2) of the Hamiltonian .
Fix . The Hamiltonian determines a family of probability measures called specifications, indexed by the set of finite and permutations , defined by
where is the normalizing constant . Denote .
A measure on is said to be Gibbs at temperature for the family of specifications if the conditional distribution of on given outside coincides with the specification . That is, for finite and ,
We denote the set of Gibbs measures at temperature by , and let .
Take . A finite cycle of length associated to the set of distinct sites is a permutation such that for all , for all , with the convention . An infinite cycle associated to a doubly infinite sequence of distinct sites is a permutation such that if for any and for all . The support of a cycle associated to is . Denote the set of finite cycles by
the set of cycles with support contained in . We say that two permutations are disjoint if their supports are so.
Denote the composition of the permutations :
Any permutation can be written as a finite or countable composition of disjoint cycles:
note that the order of the cycles in this composition does not matter. The identity has no cycle decomposition. A permutation is called finite-cycle if all cycles in its decomposition are finite. In this case we identify with the “gas of cycles” , , while the identity is identified with the empty set. We denote when is one of the cycles in the decomposition of .
For a finite cycle , define the weight of by
Since is a cycle and is strictly convex, the sum in (2.4) is strictly positive, which in turn implies for all . Define
If is finite for some , then is decreasing in and for all defined by
If for all we set .
In our first theorem we give sufficient conditions on for the existence of a Gibbs measure as limit of specifications with identity boundary conditions. The proof follows the lines proposed in  to construct the infinite volume limit of the contour representation for the Ising model at low temperature. We include the proof for the convenience of the reader.
Identity boundary conditions.
Fix a strictly convex potential satisfying . Assume . Then, for each there exists a random process on such that
(i) for finite , is distributed according to , the specification with identity boundary conditions,
(ii) almost surely, for each . Call the distribution of . Then, weakly.
(iii) is an ergodic Gibbs measure at temperature with mean jump .
(iv) is the unique Gibbs measure for the specifications , supported on the set of finite-cycle permutations of .
We next consider more general boundary conditions.
We will say that the permutation is a local perturbation of if the set is finite; in this case, the energy difference between and is defined by
A ground state is a permutation such that for any local perturbation of , . For , the shift permutation defined in (1.5) is a ground state: given a finite cycle , the permutation is a local perturbation of with energy difference
by the strict convexity of .
The next lemma says that a local perturbation of is a composition of a finite number of finite cycles with . We leave the proof to the reader. See Figure 3.
If is a local perturbation of , then there exist disjoint finite cycles in such that . If is strictly convex, then .
In the following theorem we establish conditions on that allow to extend the results of Theorem 2.1 to -boundary conditions. For a finite cycle , denote the -weight of by
Given a measure and a permutation recall the definition of the shifted measure from (1.4). In order to obtain the result, we first consider the composition of a configuration with boundary conditions with the permutation to produce a finite cycle permutation with cycles weighted by . We then apply Theorem 2.1 to this random permutation, take limits in , and as a last step compose the resulting measure with the permutation to recover the initial boundary conditions.
Fix a vector and a strictly convex potential such that . If is finite, then for any there exists a random process on such that
(i) For finite , is distributed according to , the specification with boundary conditions.
(ii) almost surely, for all . Calling the law of , we get weakly.
(iii) is an ergodic Gibbs measure at temperature with mean jump .
(iv) is the only measure with cycle weights supported on the set of finite-cycle permutations of .
We finally consider separately the Gaussian potential. Although it is in principle covered by the previous results, it is worth pointing out that in this case the associated -weights do not actually depend on , for all , with the consequence that the shift boundary condition measures are just the composition of the identity boundary conditions Gibbs measure with , , and the value of is the same for all , . We also compute an explicit bound on .
The Gaussian case.
Let , then .
Fix , let be the process constructed in Theorem 2.1, and let be the distribution of . Then, for each ,
(i) for finite , , the specification with boundary conditions. In particular, has distribution .
(ii) almost surely, for all . As a consequence weakly.
(iii) is an ergodic Gibbs measure at temperature and mean jump .
2.1 Sketch of the proofs
Identity boundary conditions. Consider a finite and recall is the set of permutations that equal the identity outside of .
A finite-cycle permutation can be identified with the configuration defined by . Thus can be described as a subset of :
We interpret the measure as the distribution of the gas of cycles with weights and interacting by exclusion. This is the setup proposed in  to study the contour representation of the low temperature Ising model.
Let now . Note that in cycles may have intersecting support; indeed, the same cycle may have multiplicity larger than 1. Given a configuration , counts the number of times the cycle is present in . Let be the product measure on with marginal Poisson for each . If has law , then the random variable is Poisson with mean , and the random variables , are independent. For finite , is just the law conditioned to :
We claim that for large enough we can construct a Poisson measure on conditioned to the event that each cycle is present at most once, and present cycle supports are disjoint. That is, the measure is supported on the set of configurations associated to finite-cycle permutations of . Since this set has zero -probability, an argument is required to give a proper sense to this notion. For large we construct as the invariant measure for a continuous-time birth and death process of cycles interacting by exclusion, and show that it concentrates on finite-cycle permutations. We also prove that is the limit as of given by (2.12).
Given a cycle , consider the rates of a continuous-time birth and death process on defined by
We construct birth and death processes with the above rates as a function of a Poisson process. Let be a Poisson process on with rate measure . If the point , we say that a cycle is born at time and lives until . Define as the number of cycles alive at time . By construction is a time-stationary continuous-time birth and death process with rates given in (2.13); that is, at any time a new copy of a cycle is born at rate , whereas existing copies die independently at rate . The marginal distribution of is Poisson with mean , for each . Letting , the process is a family of stationary independent birth and death processes with marginal distribution at any time .
Our goal is to perform such a graphical construction for a birth and death process with the same rates, subject to an exclusion rule as follows. Now
the point represents a birth attempt of a cycle at time (see Figure 4), but the cycle will be effectively born only if its support does not intersect the support of any of the cycles already present at that time . When the process is restricted to a finite set , the points in , can be ordered by their birth time . Since the free process is empty infinitely often: for all for infinitely many positive and negative times, it is possible to iteratively decide for each if it actually produces a birth of in the model with exclusion, or not. We so construct a stationary birth and death process on with rates subjected to the exclusion condition on cycles in . The marginal distribution of is .
In infinite volume the above argument does not work because the configuration is never empty. Instead, for each point one can look for the points of born prior to that could interfere with the birth of the cycle at time . This set is called the clan of ancestors of . If the clan of ancestors of any point is finite with probability one, then it is possible to construct the stationary loss network of finite cycles in . We call the resulting Markov process, obtained as a deterministic function of . Let us suggestively denote by , the notation previously used to name the Gibbs measure, the distribution of the permutation with cycles for a given time . Since the construction is time-stationary, the measure does not depend on : it is an invariant measure for the process. In fact one can check that is reversible for the process. We show that is the thermodynamic limit of and the unique invariant measure for the process .
In order to prove that is the thermodynamic limit of , we construct a stationary family of processes for any as a function of a unique realization of the Poisson process; a coupling. For finite , the marginal distribution of is . We use the finiteness of the clan of ancestors to show that for each finite-cycle , converges to as , for almost all realizations of the point process . In particular, this proves that converges weakly to and yields several properties of the limit.
To show that the clan of ancestors of a point is finite we dominate it by a multitype branching process and then show that the condition is a sufficient condition for the branching process to die out. We give more details of these processes in Section 3.
For any fixed , the process satisfies the properties attributed to the process in Theorem 2.1.
-jump boundary conditions. The specifications associated to the potential
with identity boundary conditions are given by
on , for any finite . We prove that for , satisfies the conditions of Theorem 2.1 to obtain a process on with the properties stated in that theorem. We then use the fact that has law to obtain Theorem 2.3.
3 Loss networks of finite cycles
We here construct the invariant measure of a loss network of cycles and show that it is the Gibbs measure related to the specifications with identity boundary conditions. The section consists of a review of  described in terms of cycles instead of contours.
Take a potential and a set . Recall the definition (2.10) of . We introduce a continuous-time Markov process in called loss network of finite cycles. We say that two cycles are compatible if their supports are disjoint. Given a configuration of the process, we add a new cycle at rate , if it is compatible with , that is, if is compatible with all cycles with . If and are not compatible, then the cycle is not added and the attempt is lost, hence the name loss network. Finally, any cycle in is deleted at rate one. Loss networks were introduced as stochastic models of a telecommunication network in which calls are routed between nodes around a network. In our case the nodes are the -dimensional integers and a call uses the nodes in the support of a non-identity cycle. Each node has capacity to support at most one call and hence arriving calls that would occupy an already busy node are lost. An account of the properties of loss networks can be found in Kelly .
Denote if is compatible with ; in particular implies . The loss network process on has formal generator
where is a test function, and if and only if . When is finite, the loss network is a well defined, irreducible Markov process on a finite state space, with a unique invariant measure.
The next lemma shows that defined in (2.11) is reversible for the loss network ; the proof is left to the reader.
Let be finite. The measure is reversible for the dynamics (3.1). In particular, this is the unique invariant measure, and the weak limit of the distribution of the process starting from any initial permutation as .
In the following we show that when given in (2.6) there exists a stationary process with generator (3.1) for any . The proof relies on a coupling argument applying the Harris graphical construction of the process: to each configuration of an appropriate Poisson process we associate a realization of the loss network, , for any . We now introduce the basic elements of the argument.
The Poisson process. Let be a Poisson process on with intensity measure
This process can be thought of as a product of independent Poisson processes on , indexed by .
The free process. Given the Poisson process , define the free process on by
If a point , we say that a cycle is born at time and lives time units. We represent it as a cylinder with base , height with its higher point located at . See Figure 4 where the basis is represented by a segment.
The construction implies that cycles of type are born independently at rate , and each of them lives for an exponential time of parameter 1; there may be more than one cycle of type present at any given time. The process is thus obtained as the product of independent birth and death processes , with birth rates and death rate 1. The generator of is given by
where is any local test function in the domain of . It is easy to see that the product measure on with Poisson marginals
is reversible for the free process. Indeed, this is the law of the configuration defined in (3.2), for any fixed .
The clan of ancestors. We will construct a stationary version of the loss network in infinite volume starting from the stationary free process, by simply erasing those cycles that violate the exclusion condition at birth. In order to make sense of this construction we need to consider the clan of ancestors of each point , as follows.