Transition Density of an Infinite-dimensional diffusion with the Jack Parameter
aa r X i v : . [ m a t h . P R ] F e b TRANSITION DENSITY OF AN INFINITE-DIMENSIONALDIFFUSION WITH THE JACK PARAMETER
YOUZHOU ZHOU
Abstract.
An infinite-dimensional diffusion with the Jack parameter is constructedby Olshanski in [16], and an explicit transition density is also obtained by Korotkikhthrough eigen expansion in [13]. We will rearrange the transition density in [13] to geta new expression, which has nice probabilistic interpretations. Introduction
Integer partitions are almost ubiquitous in mathematics. An integer partition λ =( λ , · · · , λ l ) is a descending vector with positive integer components(see [14] for moredetailed information). In probability theory, random integers can be used to describeclusters of a random sampling, such as the Ewens sampling formula [3] in populationgenetics. In group representation theory, integer partitions and its geometric counterpartYoung tableaux are important tools. Through careful study of the Markov chains onthe branching graph, Vershik and Kerov [8] have made fundamental contributions inrepresentation theory. The branching graph can be regarded as a weighted infinite tree.Its root is an empty integer partition, all nonempty integer partitions are lattices ofthe tree. In particular, integer partitions of n will be the lattices of the n − th layer.Moreover, there will be weights assigned to each edge. These weights are usually chosento be the coefficients or multiplicities in the Pieri formula of symmetric functions. Clearlythese weights will uniquely determine the structure of the branching graph. Moreover,we define the weight of each path as the multiplications of all edge weights in the path.Symmetric functions has countably many variables x i , i ≥
1, and they are defined asindirect limit of symmetric polynomials (see [14] for more detailed explanation). Underaddition, scalar multiplication and product, the totality of symmetric functions becomean algebra. There are various linearly independent base vectors in this algebra, such asthe monomial symmetric function m η ( x ), the Shur functions s η ( x ), the Jack functions J η ( x ; α ) and the MacDonald functions P η ( x ; q, t ). Especially the monomial symmetricfunctions m η ( x ) are sampling probabilities. However, there are no such probabilityinterpretations for the Jack functions and the MacDonald functions. The Pieri formulaof these symmetric functions describes the product of these base vectors, i.e.( ∞ X i =1 x i ) f λ ( x ) = X | Λ | = n +1 ,λ ⊂ Λ χ ( λ, Λ) f Λ ( x ) , | Λ | = Λ + · · · + Λ l , Date : February 26, 2021.2010
Mathematics Subject Classification.
Primary 60J60; secondary 60C05.
Key words and phrases.
Jack graph, Transition density, Symmetric functions, Ergodic Inequality.This research is supported by NSFC: 11701570. where λ ⊂ Λ means the young diagram of λ is a subset of the young diagram of Λ.Then the coefficients χ ( λ, Λ) are weights of the tree. If the branching graph has weightsdetermined by the Pieri formula of the monomial symmetric functions, then we callit Kingman graph. Similarly, we will have the Young graph and the Jack graph if theweights are determined by the Pieri formula of the Shur functions and the Jack functionsrespectively.One may also construct a Markov chain onΓ n = { λ = ( λ , · · · , λ l ) | λ ≥ · · · λ l > , l X i =1 λ i = n } by introducing down transition probability and upper transition probability as follows: p ↓ (Λ , λ ) = χ ( λ, Λ)dim( λ )dim(Λ)(1) p ↑ ( λ, Λ) = χ ( λ, Λ) ϕ (Λ) ϕ ( λ )(2)where dim( λ ) is summation of weights on each path from the root ∅ to lattice λ and ϕ ( λ ) is a harmonic positive function on the graph satisfying ϕ ( λ ) = X | Λ | = n +1 ,λ ⊂ Λ χ ( λ, Λ) ϕ (Λ) . As you may see that down transition probability p ↓ (Λ , λ ) depends only on the graphstructure, while upper transition probability p ↑ ( λ, Λ) depends on some exterior quantity ϕ ( λ ). Usually the function ϕ ( λ ) comes from some sampling formula(or partition struc-ture coined by Kingman in [12]) M n ( λ ) = dim( λ ) ϕ ( λ ) . For the Jack graph, this statementwill be guaranteed by the characterization theorem of positive harmonic function in [10].Then p ↑ ( λ, Λ) = M n +1 (Λ) M n ( λ ) p ↓ (Λ , λ ) . The transition matrix of Markov chain on Γ n can be defined as T ( λ, ˜ λ ) = X | Λ | = n +1 ,λ, ˜ λ ⊂ Λ p ↑ ( λ, Λ) p ↓ (Λ , ˜ λ ) . This Markov chain is reversible and ergodic, and its stationary distribution is M n ( λ ).This idea was first conceived by Kerov, and rediscovered by Fulman in [6]. Rescaling ofthis Markov chain yields diffusions on either Kingman simplex ∇ ∞ = ( x ∈ [0 , ∞ | x ≥ x ≥ · · · ≥ , ∞ X i =1 x i = 1 ) or Thoma simplexΩ = ( α, β ) ∈ [0 , ∞ × [0 , ∞ | α ≥ α ≥ · · · ≥ , β ≥ β ≥ · · · ≥ , ∞ X i =1 α i + ∞ X j =1 β j ≤ , see [2],[17],[16]. RANSITION DENSITY OF AN INFINITE-DIMENSIONAL DIFFUSION WITH THE JACK PARAMETER3
Surprisingly, one can derive the explicit transition probability of these diffusions byeigen expansion initiated by Either in 1992, see[4],[5],[13]. However, these explicit ex-pressions have no clear probabilistic interpretation. In [19], a rearrangement of eigenexpansion in [13] provides a new expression(3) p ( t, x, y ) = d θ ( t ) + d θ ( t ) + ∞ X n =2 d θn ( t ) p n ( x, y ) , where p n ( x, y ) = X | λ | = n p λ ( x ) p λ ( y ) E p λ , p λ ( x ) = n ! λ ! · · · λ l ! α ! · · · α n ! X i , ··· ,i l = x λ i · · · x λ l i l , and α i = { j | λ j = i } . The expectation E p λ is calculated under the two-parameterPoisson-Dirichlet distribution PD( α, θ ). d θn ( t ) , n ≥ Diffusion with the Jack Parameter and Its Transition Density
The Jack graph is a branching graph with weights χ α ( λ, Λ) = Y b ∈ Ver ï a ( b ) α + l ( b ) + 2 a ( b ) α + l ( b ) + 1 ò · ï ( a ( b ) + 1) α + l ( b )( a ( b ) + 1) α + l ( b ) + 1 ò , where Ver is the set of boxes above box b = Λ − λ in the Young diagram of Λ, and a ( i, j ) = λ i − j and l ( i, j ) = λ ′ j − i are called arm length and leg length. λ ′ j is the lengthof j th column in the Young diagram of λ . A candidate partition structure in the Jackgraph is(4) M n ( λ ) = n ! H ( λ ) H ′ ( λ ) z λ,α z ′ λ,α ( θ ) n , where(i) θ = zz ′ α, α >
0, and the parameters z, z ′ ∈ C is either in principal domain z ′ = z or the degenerate domain ( z, z ′ ) = ( N/α, c + ( N − /α ) , N ∈ Z ≥ , c > z λ,α = Q b ∈ λ ( z + c ( b )) , z ′ λ,α = Q b ∈ λ ( z ′ + c ( b )), c ( b ) = j − − α ( i −
1) for box b = ( i, j ). YOUZHOU ZHOU (iii) H ( λ ) = Q b ∈ λ h ( b ) , H ′ ( λ ) = Q b ∈ λ h ′ ( b ), h ( b ) = ( a ( b ) + 1) α + l ( b ) and h ′ ( b ) = a ( b ) α + l ( b ) + 1 . ( θ ) n = Γ( θ + n )Γ( θ ) .The partition structure (4) was first proposed in [11] and are thoroughly studied byBorodin and Olshanski in [1]. Then one can construct a Markov chain X nz,z ′ ,α ( t ) on Γ n through up-down transition probabilities. Rescale the time variables as [ tn ] and thespace variables as α i = a i n , β j = b j n , ≤ i, j ≤ r, where λ = ( a , · · · , a r | b , · · · , b r ) is the Frobenius coordinates of integer partition λ ,i.e. r is the length of diagonal line in the Young diagram of λ and a i = ® λ i − i + , i ≤ r, , i > r and b i = ® λ ′ i − i + , i ≤ r, , i > r . As n → ∞ , the rescaled Markov chain will converge to a diffusion X z,z ′ ,α ( t ) on theThoma simplex Ω with generator A z,z ′ ,α = 12 X i,j ≥ ij ( p oi + j − − p oi p oj ) ∂ ∂p oi ∂p oj + 12 α X i,j ≥ ( i + j + 1) p oi p oj ∂∂p oi + j +1 + 12 X i ≥ [(1 − α ) i ( i − p oi − + ( z + z ′ ) ip oi − − i ( i − p oi − iαzz ′ p oi ] ∂∂p oi . One may notice that 2 A z,z ′ ,α is the generator in [16] and [13]. This convention is mostlyadopted in population genetics. The core of A z,z ′ ,α is spanned by { , p ok | k ≥ } , where p ok ( ω ) = ∞ X i =1 α ki + ( − α ) k − ∞ X j =1 β kj , ω = ( α, β ) ∈ Ω . Here p ok ( ω ) can be regarded as a specialization of power symmetric function P ∞ i =1 x ki .Correspondingly, the Jack functions J λ ( x ; α ) will also has a specialization J oλ ( ω ; α ) whichis a continuous function on Thoma simplex Ω. We denote j λ ( ω ; α ) = dim α ( λ ) J oλ ( ω ; α ),where dim α ( λ ) is the summation of weights on each path from the root ∅ to lattice λ inthe Jack graph.The diffusion X z,z ′ ,α ( t ) is reversible and its stationary distribution is Z -measure M z,z ′ ,α on Thoma simplex. By eigen expansion of A z,z ′ ,α in Hilbert space L (Ω , M z,z ′ ,α ),Korotkikh obtained the following explicit transition density of X z,z ′ ,α ( t ) in [13]. Theorem 2.1.
The transition density of X z,z ′ ,α ( t ) is p ( t, σ, ω ) = 1 + ∞ X m =2 e − tλ m G m ( σ, ω ) , where λ m = m ( m − θ )2 , G m ( σ, ω ) = P mn =0 ( − m − n ( θ +2 m − θ ) m + n − ( m − n )! K on ( σ, ω ) and (5) K on = X | λ | = n dim α ( λ ) J o ( σ ; α ) J o ( ω ; α ) n !( θ ) n M n ( λ ) . RANSITION DENSITY OF AN INFINITE-DIMENSIONAL DIFFUSION WITH THE JACK PARAMETER5
Remark 2.1. (1) In [13] , both M n ( λ ) and K on are defined as M n ( λ ) = dim α ( λ ) ( z ) λ,α ( z ′ ) λ,α n !( θ ) n b αλ , K on = X | λ | = n b αλ J o ( σ ; α ) J o ( ω ; α )( z ) λ,α ( z ′ ) λ,α then one can easily see that (5) is equivalent to the definition in [13] . (2) The expression of G m can be seen in Proposition 15 in [13] . If we consider a new reproducing kernel K n ( σ, ω ) = X | λ | = n j λ ( σ ; α ) j λ ( ω ; α ) M n ( λ ) , j λ ( ω ; α ) = dim α ( λ ) J oλ ( ω ; α ) , then K n ( σ, ω ) = n !( θ ) n K on ( σ, ω ). Thus, G m ( σ, ω ) = m X n =0 ( − m − n ( θ + 2 m − θ ) m + n − ( m − n )! n !( θ ) n K n ( σ, ω )= m X n =0 ( − m − n Ç mn å ( θ + 2 m − θ + n ) m − m ! K n ( σ, ω ) . If we switch the summation order, then we have the following new representation
Theorem 2.2.
The transition density of X z,z ′ ,α ( t ) is (6) p ( t, σ, ω ) = d θ ( t ) + d θ ( t ) + ∞ X n =2 d θn ( t ) K n ( σ, ω ) where d θ ( t ) =1 − ∞ X k =1 k − θk ! ( − k − θ ( k − e − λ k t d θn ( t ) = ∞ X k =1 k − θk ! ( − k − n Ç kn å ( n + θ ) ( k − e − λ k t , n ≥ . Due to the estimation of G m in [13], the proof of Theorem 2.2 is the same as the proofof Theorem 2.1 in [19].3. Ergodic Inequality and Some Discussions
Ergodic Inequality.
Making use of the expression in Theorem 2.2, one can easilyobtain the following ergodic inequality.
Proposition 3.1.
The diffusion X z,z ′ ,α ( t ) satisfies the following ergodic inequality sup ω ∈ Ω k p ( t, ω, · ) − M z,z ′ ,α ( · ) k Var ≤ ( θ + 1)( θ + 2)2 e − ( θ +1) t . This proposition can be easily derived from an inequality of tail probabilities(see [18]) ∞ X n =2 d θn ( t ) ≤ ( θ + 1)( θ + 2)2 e − ( θ +1) t . YOUZHOU ZHOU
Some Discussion.
As a matter of fact, Petrov’s diffusion is also the degenerationof the diffusion X z,z ′ ,α ( t ). It is surprising that the convex combination structure in (6) istrue for all these models. It would be nice to verify that the dual process of X z,z ′ ,α ( t ) isalso the Kingman coalescent. To this end, one need to show the following dual equationfor normalized test function g η ( ω ) = j η ( ω ; α ) M n ( η ) ,(7) A z,z ′ ,α g η ( ω ) = − n ( n − θ )2 g η ( ω ) + 12 l X i =1 η i ( n − θ ) g η − e i ( ω )In [7], the dual equation (7) of the Petrov’s diffusion has already been obtained. Butdue to the complexity of the Jack functions, the dual equation (7) still remains open.An even deeper question is why the transition structure (6) is true for all these models.This question is certainly worthy of study. Moreover, one may easily see the support ofthe marginal distribution of X z,z ′ ,α ( t ) is the same as the Z measure M z,z , α as in [15].4. Acknowledgement
The author would like to thank Stewart N. Ethier for introducing the reference [13] tohim. Moreover, the gratitude should also go to Robert C. Griffiths for many stimulatingdiscussions on dual processes.
References [1] A.Borodin, G.Olshanski. Z-measures on Partitions and Their Scaling Limits.
European Journal ofCombinatorics. (2005):795-834.[2] A.Borodin, G.Olshanski. Markov Processes on Partitions. Probability Theory and Related Fields. (2006):84-152.[3] W.J.Ewens. The sampling theory of selectively neutral alleles.
Theoret.Population Biology. (1972):87-112.[4] S.N.Ethier. Eigenstructure of the infinitely-many-neutral-alleles diffusion model. Journal of AppliedProbability. (2006):487-498.[5] S.Feng,W.Sun,F.Y.Wang and F.Xu. Functional Inequalities for the two-parameter extension of theinfinitely-many-neutral-alleles diffusion. J.Funct.Anal. (2011):39-413.[6] J.Fulman. Stein’s method and random character ratios.
Transactions of the American MathematicalSociety (2008):687-730.[7] R.Griffiths,D.Span´o,M. Ruggiero and Y.Zhou. Dual process in the Two-parameter Poisson-Dirichletdiffusion. arXiv 2102.08520 .[8] S. Kerov. Asymptotic Representation Theory of the Symmetric Group and its Applications in Anal-ysis.
Amer.Math.Soc.,Providence, RI, (2003).[9] S. Kerov. Aniostropic Young Diagrams and Jack Symmetric Functions.
Functional Analysis and ItsApplication (2000):45-51.[10] S. Kerov, A.Okounkov, and G. Olshanski. The boundary of Young graph with Jack edge multiplic-ities.
International Mathematics Research Notices. (1998):173-199.[11] S. Kerov, G. Olshanski, and A. Vershik. Harmonic Analysis on the Infinite Symmetric Group.
Inventiones Mathematicae. (2004):551-642.[12] J.F.C.Kingman. The Representation of Partition Structures
Journal of the London MathematicalSociety. (1978):374-380.[13] S. Yu. Korotkikh. Transition Functions of Diffusion Processes on the Thoma Simplex.
FunctionalAnalysis and Its Application. (2020):118-134.[14] I. MacDonald. Symmetric Functions and Hall Polynomials.
New York:Cambridge University Press. (1995).
RANSITION DENSITY OF AN INFINITE-DIMENSIONAL DIFFUSION WITH THE JACK PARAMETER7 [15] G.I.Olshanski. The Topological Support of the z-Measures on the Thoma Simplex.
Functional Anal-ysis and Its Application. (2018):308-311.[16] G.I.Olshanski. Anisotropic Young Diagrams and Infinite-Dimensional Diffusion Processes with theJack Parameter.
International Mathematics Research Notices. (2009):1102-1166.[17] L. Petrov. A two-parameter family of infinite-dimensional diffusions in the Kingman simplex.
Func-tional Analysis and Its Application. (2009):45-66.[18] S. Tavar´e. Line-of-descent and genealogical processes, and their application in population geneticsmodels. Theor. Popul.Biol. (1984):119-164.[19] Y. Zhou. Ergodic Inequality of a two-parameter infinitely-many-alleles diffusion model. J. Appl.Prob. (2015):238-246. Department of Pure Mathematics, Xi’an Jiaotong-Liverpool University, 111 Renai Road,Suzhou, China 215 123
Email address ::