ABSTRACT
In the previous chapter, we showed that for any parameter t = (α, β) of the Thoma simplex T, if λ (n) is a random partition chosen under the central measure ℙ t , n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq8093.tif"/> , then λ i ( n ) ≃ n α i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq8094.tif"/> , and λ i ( n ) ≃ n β j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq8095.tif"/> as n goes to infinity. Consider the particular case when t = 0 = (0, 0) is the parameter of the Thoma simplex corresponding to the regular traces of the symmetric group algebras and to the Plancherel measures: τ 0 ( σ ) = 1 σ = id ℕ * ; ℙ 0 , n [ λ ] = d λ s λ ( E ) = ( dim λ ) 2 n ! . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq8096.tif"/>
