Manon Defosseux
email: manon.defosseux@parisdescartes.fr

AN INTERACTING PARTICLE MODEL AND A PIERI-TYPE FORMULA FOR THE ORTHOGONAL GROUP

.

introduction

In [START_REF] Borodin | Random surface growth with a wall and Plancherel measures for O(∞)[END_REF] A. Borodin and J. Kuan consider a random tiling model with a wall which is related to the Plancherel measure for the orthogonal group and thus to representation theory of this group. Similar connection holds for the interacting particle model and the random matrix model considered in [START_REF] Defosseux | Generalized Laguerre Unitary Ensembles and an interacting particles model with a wall[END_REF]. The aim of this paper is to establish a direct link between the random tiling model on one side and the interacting particle model or the random matrix model on the other side. For this we consider an interacting particle model depending on a parameter and show that these models correspond to different parameter values.

The paper is organized as follows. Definition of the set of Gelfand-Tsetlin patterns for the orthogonal group is recalled in section 2. Section 3 is devoted to the description of the particle model. We recall in section 4 the description of an interacting particle model equivalent to the random tiling model studied in [START_REF] Borodin | Random surface growth with a wall and Plancherel measures for O(∞)[END_REF]. Models considered in that paper involve Markov kernels which can be obtained with the help of a Pieri-type formula for the orthogonal group. These Markov kernels are constructed in section 5 after recalling some elements of representation theory. We describe the matrix model related to the particle model in section 6. Results are stated in section 7 and proved in section 8.

Acknowlegments: The author would like to thank Alexei Borodin for his suggestions and helpful explanations.

Gelfand-Tsetlin patterns

Let n be a positive integer. For x, y ∈ R n such that x n ≤ • • • ≤ x 1 and y n ≤ • • • ≤ y 1 , we write x y if x and y are interlaced, i.e.

x n ≤ y n ≤ x n-1 ≤ • • • ≤ x 1 ≤ y 1 .
When x ∈ R n and y ∈ R n+1 we add the relation y n+1 ≤ x n . We denote by |x| the vector of R n whose components are the absolute values of those of x. Definition 2.1. Let k be a positive integer.

(1) We denote by GT k the set of Gelfand-Tsetlin patterns defined by

GT k = {(x 1 , • • • , x k ) : x i ∈ N j-1 × Z when i = 2j -1,
x i ∈ N j when i = 2j, and |x i-1 | |x i |, 1 ≤ i ≤ k}.

(2) If x = (x 1 , . . . , x k ) is a Gelfand-Tsetlin pattern, x i is called the i th row of

x for i ∈ {1, . . . , k}.

(3) For λ ∈ Z [ k+1 2 ] the subset of Gelfand-Tsetlin patterns having a k th row equal to λ is denoted by GT k (λ) and its cardinality is denoted by s k (λ).

Usually, a Gelfand Tsetlin pattern is represented by a triangular array as indicated at figure 1 for k = 2r.

-x 2r 1 • • • -x 2r r x 2r r • • • x 2r 1 -x 2r-1 1 • • • -x 2r-1 r-1 x 2r-1 r x 2r-1 r-1 • • • x 2r-1 1 • • • • • • -x 4 1 -x 4 2 x 4 2 x 4 1 -x 3 1 x 3 2 x 3 1 -x 2 1 x 2 1 x 1 1 Figure 1. A Gelfand-Tsetlin pattern of GT 2r

An interacting particle model with exponential jumps

In this section we construct a random process (X(t)) t≥0 evolving on the subset of GT k of Gelfand-Tsetlin patterns with non negative valued components. This process can be viewed as an interacting particle model. For this, we associate to a Gelfand-Tsetlin pattern x = (x 1 , . . . , x k ), a configuration of particles on the integer lattice Z 2 putting one particle labeled by (i, j) at point (k -i, x i j ) of Z 2 for i ∈ {1, . . . , k}, j ∈ {1, . . . , [ i+1 2 ]}. Several particles can be located at the same point. In the sequel we will say "particle x i j " instead of saying "particle labeled by (i, j) located at point (x i j , k -i)". Let q ∈ (0, 1). Consider two independent families

(ξ i j (n + 1 2 )) i=1,...,k,j=1,...,[ i+1 2
]; n≥0 , and (ξ i j (n)) i=1,...k,j=1,...,[ i+1 2 ]; n≥1 , of identically distributed independent random variables such that

P(ξ 1 1 ( 1 2 ) = x) = P(ξ 1 1 (1) = x) = q x (1 -q), x ∈ N, and the Markov kernel R on N defined by R(x, y) =    1-q 1+q (q |x-y| + q x+y ) if y ∈ N * 1-q 1+q q x otherwise,
for x ∈ N. Actually the probability measure R(x, .) on N is the law of the random variable |x + ξ 1 1 (1) -ξ 1 1 ( 1 2 )|. Particles evolve as follows. At time 0 all particles are at zero, i.e. X(0) = 0. All particles, except those labeled by (2l -1, l) for l ∈ {1, . . . , [ k+1 2 ]} (i.e. particles near the wall), try to jump to the left at times n + 1 2 and to the right at times n, n ∈ N. Particles labeled by (2l -1, l), for l ∈ {1, . . . , [ k+1 2 ]}, jump on their own volition at integer times only. Notice that these particles can eventually move at half-integer times if they are pushed by another particle. Suppose that at time n there is one particle at point (k

-i, X i j (n)) of Z 2 , for i = 1, . . . , k, j = 1, . . . , [ i+1 2 ]
. Positions of particles are updated downward as follows. Figure 2 gives an example of an evolution of a pattern between times n and n + 1. In that example, the particles (3, 2) and (4, 2) are respectively pushed by the particles (2, 1) and (3, 1) and the particles (3, 1) and (4, 2) are respectively blocked by the particles (2, 1) and [START_REF] Defosseux | Orbit measures and interlaced determinantal point processes[END_REF][START_REF] Borodin | Anisotropic growth of random surfaces in 2+1 dimensions[END_REF] between times n and n + 1 2 . The particle (3, 1) is pushed by the particle (2, 1) and the particles (3, 2) and (4, 2) are respectively blocked by the particles (2, 1) and [START_REF] Defosseux | Orbit measures and interlaced determinantal point processes[END_REF][START_REF] Borodin | Random surface growth with a wall and Plancherel measures for O(∞)[END_REF] between times n + 1 2 and n + 1.

At time n + 1/2 : All particles except particles X 2l-1 l (n) for l ∈ {1, . . . , [ k+1 2 ]
}, try to jump to the left one after another in the lexicographic order pushing the particles in order to stay in the set of Gelfand-Tsetlin patterns and being blocked by the initial configuration X(n) of the particles. Let us indicate how the first three rows of a pattern are updated at time n + 1 2 . • Particle X 1 1 (n) doesn't move. We let

X 1 1 (n + 1 2 ) = X 1 1 (n).
• Particle X 2 1 (n) tries to jump to the left according to a geometric jump. It is blocked by

X 1 1 (n). If it is necessary it pushes X 3 2 (n) to an intermediate position denoted by X3 2 (n), i.e. X 2 1 (n + 1 2 ) = max X 1 1 (n), X 2 1 (n) -ξ 2 1 (n + 1 2 ) , X3 2 (n) = min X 3 2 (n), X 2 1 (n + 1 2
) .

• Particle X 3 1 (n) tries to move to the left according to a geometric jump being blocked by X 2 1 (n) :

X 3 1 (n + 1 2 ) = max X 2 1 (n), X 3 1 (n) -ξ 3 1 (n + 1 2 ) .
Particle X3 2 (n) doesn't move. We let

X 3 2 (n + 1 2 ) = X3 2 (n).
Suppose now that rows 1 through l -1 have been updated for some l > 1. Then the particles X l 1 (n), . . . , X l

[ l+1 2 ] (n) of row l are pushed to intermediate positions Xl i (n) = min X l i (n), X l-1 i-1 (n + 1 2 ) , i ∈ {1, . . . , [ l + 1 2 ]}, whit the convention X l-1 0 (n + 1 2 ) = +∞. Then particles Xl 1 (n), . . . , Xl [ l 2 ]
(n) try to jump to the left according to geometric jump being blocked as follows by the initial position X(n) of the particles. For i = 1, . . . , [ l 2 ],

X l i (n + 1 2 ) = max X l-1 i (n), Xl i (n) -ξ l i (n + 1 2 ) .
When l is odd, particle Xl l+1 2

(n) doesn't move and we let

X l l+1 2 (n + 1 2 ) = Xl l+1 2 (n).
At time n + 1 : All particles except particles X 2l-1 l (n+ 1 2 ) for l ∈ {1, . . . , [ k+1 2 ]}, try to jump to the right one after another in the lexicographic order pushing particles in order to stay in the set of Gelfand-Tsetlin patterns and being blocked by the initial configuration X(n + 1 2 ) of the particles. Particles X 2l-1

Suppose rows 1 through l -1 have been updated for some l > 1. Then particles of row l are pushed to intermediate positions

Xl i (n + 1 2 ) = max X l-1 i (n + 1), X l i (n + 1 2 ) , i ∈ {1, . . . , [ l + 1 2 ]},
with the convention X l-1 l+1 2

(n + 1) = 0 when l is odd. Then particles Xl

1 (n + 1 2 ), . . . , Xl [ l 2 ] (n + 1
2 ) try to jump to the right according to geometric jump being blocked by the initial position of the particles as follows. For i = 1, . . . , [ l 2 ],

X l i (n + 1) = min X l-1 i-1 (n + 1 2 ), Xl i (n + 1 2 ) + ξ l i (n + 1) .
When l is odd, particle X l l+1 2

(n + 1 2 ) is updated as follows.

X l l+1 2 (n + 1) = min( X l l+1 2 (n + 1 2 ) + ξ l l+1 2 (n + 1) -ξ l l+1 2 (n + 1 2 ) , X l-1 l-1 2 (n + 1 2
)).

An interacting particle model with exponential waiting times

In this section we describe an interacting particle model on Z 2 where particles try to jump by one rightwards or leftwards after exponentially distributed waiting times. The evolution of the particles is described by a random process (Y (t)) t≥0 on the subset of GT k of Gelfand-Tsetlin patterns with non negative valued components. As in the previous model, at time t ≥ 0 there is one particle labeled by (i, j) at point (k -i, Y i j (t)) of the integer lattice, for i = 1, . . . , k, j = 1, . . . , [ i+1 2 ]. Every particle tries to jump to the left or to the right by one after independent exponentially distributed waiting time with mean 1. Particles are pushed and blocked according to the same rules as previously. That is when particle labeled by (i, j) wants to jump to the right at time t ≥ 0 then

(1

) if i, j ≥ 2 and Y i j (t -) = Y i-1 j-1 (t -) then the particles don't move and Y (t) = Y (t -), (2)
else particles (i, j), (i + 1, j), . . . , (i + l, j) jump to the right by one for l the largest integer such that Y i+l j

(t -) = Y i j (t -) i.e. Y i j (t) = Y i j (t -) + 1, . . . , Y i+l j (t) = Y i+l j (t -) + 1.
When particle labeled by (i, j) wants to jump to the left at time t ≥ 0 then (1) if i is odd, j = (i + 1)/2 and Y i j (t -) = 0 then particle labeled by (i, j) is reflected by 0 and everything happens as described above when this particle try to jump to the right by one, (2

) if i is odd, j = (i + 1)/2 and Y i j (t -) ≥ 1 then Y i j (t) = Y i j (t -) -1, ( 3
) if i is even or j = (i + 1)/2, and Y i j (t -) = Y i-1 j (t -
) then particles don't move, (4) if i is even or j = (i + 1)/2, and Y i j (t -) > Y i-1 j (t -) then particles (i, j), (i + 1, j + 1), . . . , (i + l, j + l) jump to the left by one for l the largest integer such that Y i+l j+l (t -) = Y i j (t -). Thus

Y i j (t) = Y i j (t -) -1, . . . , Y i+l j+l (t) = Y i+l j+l (t -) -1.
This random particle model is equivalent to a random tiling model with a wall, as it is explained in detail in [START_REF] Borodin | Random surface growth with a wall and Plancherel measures for O(∞)[END_REF].

Markov Kernel on the set of irreducible representations of the orthogonal group

When a finite dimensionnal representation V of a group G is completely reducible, there is a natural way that we'll recall later in our particular case to associate to this decomposition a probability measure on the set of irreducible representations of G. The transition probabilities of the random process (X k (t), t ≥ 0) which will be proved to be Markovian are obtained in that manner. Actually we recover them considering decomposition into irreducible components of tensor products of particular irreducible representations of the special orthogonal group.

Let d be an integer greater than 2. Let us recall some usual properties of the finite dimensional representations of the compact group SO(d) of d × d orthogonal matrices with determinant equal to 1 (see for instance [START_REF] Knapp | Lie groups, beyond an introduction[END_REF] for more details). The set of finite dimensional representations of SO(d) is indexed by the set

{λ ∈ R r : 2λ r ∈ N, λ i -λ i+1 ∈ N, i = 1, . . . , r -1}, when d = 2r + 1 and by the set {λ ∈ R r : λ r-1 + λ r ∈ N, λ i -λ i+1 ∈ N, i = 1, . . . , r -1},
when d = 2r. Actually we are only interested with representations indexed by a subset W d of these sets defined by

W d = {λ ∈ R r : λ r ∈ N, λ i -λ i+1 ∈ N, i = 1, . . . , r -1}, when d = 2r + 1 and
W d = {λ ∈ R r : λ r ∈ Z, λ r-1 + λ r ∈ N, λ i -λ i+1 ∈ N, i = 1, . . . , r -1},
when d = 2r. For λ ∈ W d , using standard notations, we denote by V λ the so called irreducible representation with highest weight λ of SO(d). The subset of W d of elements having non-negative components is denoted by W + d . Let m be an integer and λ an element of W d . Consider the irreducible representations V λ and V γm of SO(d), with γ m = (m, 0, • • • , 0). The decomposition of the tensor product V λ ⊗ V γm into irreducible components is given by a Pieri-type formula for the orthogonal group. As it has been explained in [START_REF] Defosseux | Orbit measures and interlaced determinantal point processes[END_REF], it can be deduced from [START_REF] Nakashima | Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras[END_REF]. We have

V λ ⊗ V γm = ⊕ β M λ,γm (β)V β , (1)
where the direct sum is over all β ∈ W d such that

• when d = 2r + 1, there exists an integer s ∈ {0, 1} and c ∈ N r which satisfy

   c λ, c β r i=1 (λ i -c i + β i -c i ) + s = m, s being equal to 0 if c r = 0.
In addition, the multiplicity M λ,γm (β) of the irreducible representation with highest weight β is the number of (c, s) ∈ N r × {0, 1} satisfying these relations.

• when d = 2r, there exists c ∈ N r-1 which verifies

   c |λ|, c |β| r-1 k=1 (λ k -c k + β k -c k ) + |λ r -µ r | = m.
In addition, the multiplicity M λ,γm (β) of the irreducible representation with highest weight β is the number of c ∈ N r-1 satisfying these relations. Let us consider a family (µ m ) m≥0 of Markov kernels on W d defined by

µ m (λ, β) = dim(V β ) dim(V λ ) dim(V γm ) M λ,γm (β), for λ, β ∈ W d and m ≥ 0. It is known that for λ ∈ W d the dimension of V λ is given by the number s d-1 (λ) defined in definition 2.1. Thus µ m (λ, β) = s d-1 (β) s d-1 (λ)s d-1 (γ m ) M λ,γm (β).
Let ξ 1 , . . . , ξ d be independent geometric random variables with parameter q and ǫ a Bernoulli random variable such that

P(ǫ = 1) = 1 -P(ǫ = 0) = q 1 + q .
Consider a random variable T on N defined by

T = d-1 i=1 ξ i + ǫ,
when d = 2r + 1 and

T = |ξ 1 -ξ 2 | + d i=3 ξ i , when d = 2r.
Lemma 5.1. The law of T is a measure ν on N defined by

ν(m) = 1 1 + q (1 -q) d-1 q m s d-1 (γ m ), m ∈ N.
Proof. When d = 2r + 1, for m = 0 the property is true. For m ≥ 1

P(T = m) = q 1 + q P( d-1 i=1 ξ i = m -1) + 1 1 + q P( d-1 i=1 ξ i = m) = 1 1 + q (1 -q) d-1 q m Card{(k 1 , . . . , k d-1 ) ∈ N d-1 : d-1 i=1 k i ∈ {m -1, m}} = 1 1 + q (1 -q) d-1 q m (k1,...,k d-1 )∈N d-1 : d-1 i=1 ki=m (21 k1≥1 + 1 k1=0 ) = 1 1 + q (1 -q) d-1 q m s d-1 (γ m ).
So the lemma is proved in the odd case. Moreover

P(|ξ 1 -ξ 2 | = k) =    2 1-q 1+q q k if k ≥ 1,

1-q 1+q

otherwise.

Thus when d = 2r,

P(T = m) = 1 1 + q (1 -q) d-1 q m (k1,...,k d-1 )∈N d-1 : d-1 i=1 ki=m (21 k1≥1 + 1 k1=0 ) = 1 1 + q (1 -q) d-1 q m s d-1 (γ m ).
Lemma 5.1 implies in particular that the measure ν is a probability measure. Thus one defines a Markov kernel P d on W d by letting

P d (λ, β) = +∞ m=0 µ m (λ, β)ν(m), (2)
for λ, β ∈ W d . We'll see that the kernel P d describes the evolution of the (d -1) th row of the random process on the set of Gelfand-Tsetlin patterns observed at integer times.

Proposition 5.2. For λ, β ∈ W d ,

P d (λ, β) = c∈N r :c λ,β (1 -q) d-1 s d-1 (β) s d-1 (λ) q r i=1 (λi+βi-2ci) (1 cr>0 + 1 cr=0 1 + q )
when d = 2r + 1 and

P d (λ, β) = c∈N r-1 :c |λ|,|β| (1 -q) d-1 q + 1 s d-1 (β) s d-1 (λ) q r-1 i=1 (λi+βi-2ci)+|λr-βr| when d = 2r.
Proof. The proposition follows immediately from the tensor product rules recalled for the decomposition (1).

M ∈ M d,d ′ → 1 dd ′ √ 2π exp(- 1 2 tr(M M * )).
We

λ 1 ≥ λ 2 ≥ • • • ≥ λ d . Morever, antisymmetry implies that λ d-i+1 = -λ i , for i = 1, • • • , [d/2] + 1, in particular λ [d/2]+1 = 0 when d is odd. Consider the subset C d of R [ d 2 ]
+ defined by

C d = {x ∈ R [ d 2 ] : x 1 > • • • > x [ d 2 ] > 0}, and its closure Cd = {x ∈ R [ d 2 ] : x 1 ≥ • • • ≥ x [ d 2 ]
≥ 0}. Definition 6.1. We define the function h d on C d by

h d (λ) = c d (λ) -1 V d (λ), λ ∈ C d ,
where the functions V d and c d are given by :

V n (λ) = 1≤i<j≤[ d 2 ] (λ i -λ j ) 1≤i<j≤[ d 2 ] (λ i + λ j ) 1≤i≤[ d 2 ] λ ε i , c n (λ) = 1≤i<j≤[ d 2 ] (j -i) 1≤i<j≤[ d 2 ] (d -j -i) 1≤i≤[ d 2 ] ([ d 2 ] + 1 2 -i) ε ,
whit ε equal to 1 when d / ∈ 2N and 0 otherwise.

The next proposition is a consequence of Propositions 4.8 and 5.1 of [START_REF] Defosseux | Orbit measures and interlaced determinantal point processes[END_REF]. Proposition 6.2. Let (M (n), n ≥ 0), be a random process on iA d defined by

M (n) = n l=1 Y l 0 i -i 0 Y * l ,
where the Y l 's are independent standard Gaussian variables on M d,2 . If Λ(n) is the vector of Cd whose components are the

[ d 2 ] largest eigenvalues of M (n), n ∈ N,

Results

The main result of our paper states in particular that if only one row of the patterns (X(t), t ≥ 0) is considered by itself, it found to be a Markov process too. Actually we state the result for the process observed at integer times, even if the process observed at the whole time is also Markovian as we'll see in section 8.

Theorem 7.1. The random process (X k (n)) n≥0 is a Markov process on W + k+1 . If we denote by R k its transition kernel then

• R 1 = R. • when k is even R k = P k+1 ,
• when k is an odd integer greater than 2 . The random process (Y k (t), t ≥ 0) is a Markov process with infinitesimal generator defined by

R k (x, y) =    P k+1 (x, y) + P k+1 (x, ỹ) if y k+1 2 = 0 P k+1 (x,
A k (λ, β) =      2 s k (β) s k (λ) 1 β∈W k if k is odd, λ k+1 2 = 0 and β k+1 2 = 1 s k (β) s k (λ) 1 β∈W k otherwise, for λ ∈ W k , and β ∈ {λ + e 1 , . . . , λ + e [ k+1 2 ] , λ -e 1 , . . . , λ -e [ k+1 2 ] }. If (Λ(n), n ≥ 0)
is the process of eigenvalues considered in Proposition 6.2 with d = k + 1 then the following theorem holds. Theorem 7.3. Letting q = 1 -1 N , the process ( X k (n) N , n ≥ 1) converges in distribution towards the process of eigenvalues (Λ(n), n ≥ 1) as N goes to infinity.

proofs

8.1. Proof of Theorem 7.1. For k = 1, Theorem 7.1 is clearly true. The proof of the theorem for k ≥ 2 rests on an intertwining property and an application of a Pitman and Rogers criterion given in [START_REF] Pitman | Markov functions[END_REF].

Notation 8.1. Let ξ 1 and ξ 2 be two independent geometric random variables. For x, a ∈ N such that x ≥ a, the law of the random variable max(a, x -ξ 1 ), is denoted by For x, y ∈ R 2 such that x ≤ y we let P (x, y) = (1 -q)q y-x .

The two following lemmas are proved by straightforward computations.

Lemma 8.2. For a, x, y ∈ N such that a ≤ y ≤

x a← P (x, y) = (1 -q)q x-y if a + 1 ≤ y q x-a if y = a. For b, x, y ∈ N such that b ≥ y ≥ x →b P (x, y) = (1 -q)q y-x if y ≤ b -1 q b-x if y = b. For b, x, y ∈ N such that b ≥ y, x →b R (x, y) =                    1-q 1+q (q |y-x| + q x+y ) if y ≤ b -1, y > 0 1-q 1+q q x if y ≤ b -1, y = 0 1 1+q q b (q -x + q x ) if y = b, y > 0 1 if y = b, y = 0. Lemma 8.3. For (x, y, z) ∈ N 3 such that 0 < z ≤ y z u=0 (1 u=0 + 2 1 u>0 )R(u, x) u← P (y, z) = (1 -q)(1 x=0 + 2 1 x>0 )q x∨z+y-2z . (3)
For (x, y, a) ∈ N 3 such that a ≤ y and y ≤ x y u=a q u u← P (x, y) = q x-y q a . (4)

For (x, y, a) ∈ N 3 such that y ≤ a and x ≤ y a v=y q -v →v P (x, y) = q y-x q -a . (5)

For y ∈ N, y ′ ∈ N * such that y ′ ≤ a a v=y ′ q v∨y-2v →v R (y ∧ v, y ′ ) = 1 1 -q q -a R(y, y ′ ). ( 6
)
Let us first prove Theorem 7.1 for k = 2. Consider the set

W + 2,3 = {(z, y) ∈ N 2 : z ≤ y}. Define a Markov kernel S 2 on W + 2,3 by letting S 2 ((z 0 , y 0 ), (z, y)) =      (1 -q) 2 s2(y) s2(y0) q y0+y-2z 1 z≤y0∧y if z > 0 (1-q) 2 1+q s2(y) s2(y0) q y0+y if z = 0.
for (z 0 , y 0 ), (z, y) ∈ W + 2,3 and another one L 2 from W + 2,3 to N × W + 2,3 by letting

L 2 ((z 0 , y 0 ), (x, z, y)) = (1 x=0 + 2 1 x>0 ) 1 s 2 (y) 1 x≤y 1 (z0,y0)=(z,y) ,
for (z 0 , y 0 ), (z, y) ∈ W + 2,3 and x ∈ N. The fact that S 2 is a Markov kernel follows from Proposition 5.2 with d = 3. The random process

(X 1 1 (n), X 2 1 (n - 1 2 ), X 2 1 (n)) n≥1 ,
is clearly Markovian. Let us denote by Q 2 its transition kernel. Then Q 2 , L 2 and S 2 satisfy the following intertwining property.

Lemma 8.4.

L 2 Q 2 = S 2 L 2 .
Proof. For (x, z, y), (x ′ , z ′ , y ′ ) ∈ N × W + 2,3 such that x ≤ y and

x ′ ≤ y ′ Q 2 ((x, z, y), (x ′ , z ′ , y ′ )) = R(x, x ′ ) x← P (y, z ′ )P (x ′ ∨ z ′ , y ′ ).
Thus

L 2 Q 2 ((z, y), (x ′ , z ′ , y ′ )) = z ′ x=0 s 1 (x) s 2 (y) R(x, x ′ ) x← P (y, z ′ )P (x ′ ∨ z ′ , y ′ )
As L 2 , S 2 and Q 2 are Markov kernels, it is sufficient to prove the identity for z ′ > 0.

In that case the identity (3) of Lemma 8.3 implies that

z ′ x=0 (1 x=0 + 2 1 x>0 ) s 2 (y) R(x, x ′ ) x← P (y, z ′ ) = (1 -q)(1 x ′ =0 + 2 1 x ′ >0 )q x ′ ∨z ′ +y-2z ′ . Thus L 2 Q 2 ((z, y), (x ′ , z ′ , y ′ )) = (1 x ′ =0 + 2 1 x ′ >0 ) s 2 (y) (1 -q) 2 q y+y ′ -2z ′ , which proves that L 2 Q 2 = S 2 L 2 .
Since the random process

(X 1 1 (n), X 2 1 (n - 1 2 ), X 2 1 (n)) n≥1
is Markovian with transition kernel Q 2 , the intertwining property stated in Lemma 8.4 and the criterion of Pitman and Rogers given in [START_REF] Pitman | Markov functions[END_REF] imply the following proposition. It states that the second row of the random process on the set of Gelfand-Tsetlin patterns is Markovian and gives its transition kernel.

Proposition 8.5. We let

X 2 1 (-1 2 ) = X 2 1 (1) = 0. The random process (X 2 1 (n - 1 2 ), X 2 1 (n)) n≥0
is a Markov process on W + 2,3 with transition kernel S 2 .

As for (z, y) ∈ W + 2,3 the probability S 2 ((z, y), .) doesn't depend on z, Theorem 7.1 easily follows from Proposition 8.5 when k = 2.

For the general case one defines the random process

(Z k (n), Y k (n)) n≥1 by letting Z k (n) = (X k 1 (n - 1 2 ), . . . , X k [ k 2 ] (n - 1 2
)),

Y k (n) = X k (n), for n ≥ 1 and Z k (0) = Y k (0) = 0. Let us notice that Z k is equal to X k when k is even, whereas it is obtained from X k by deleting its smallest component when k is odd. We consider the subset W + k,k+1 of W + k × W + k+1 defined by W + k,k+1 = {(z, y) ∈ W + k × W + k+1 :
z y}, and define a Markov kernel S k on W + k,k+1 by letting for every (z, y), (z ′ , y

′ ) ∈ W + k,k+1 S k ((z, y), (z ′ , y ′ )) = (1 -q) k s k (y ′ ) s k (y) q r i=1 (yi+y ′ i -2zi) (1 zr>0 + 1 zr=0 1 + q )1 z ′ y,y ′ (7)
when k = 2r, and

S k ((z, y), (z ′ , y ′ )) = (1 -q) k-1 s k (y ′ ) s k (y) R(y r , y ′ r )q r-1 i=1 (yi+y ′ i -2zi) 1 z ′ y,y ′ (8)
when k = 2r -1. The fact that for (z, y) ∈ W + k,k+1 the measure S k ((z, y), .) is a probability measure is a consequence of Proposition 5.2 with d = k + 1.

Notation. Since for (z, y) ∈ W + k,k+1 the probability S k ((z, y), .) doesn't depend on z, it is denoted by S k (y, .) when there is no ambiguity.

Even if the Markov kernel P k+1 is relevant for our purpose, we'll prove Theorem 7.1 showing that the Markov kernel S k describes the evolution of the k th row of the process (X(t), t ≥ 0) observed at the whole time. We'll prove it by induction on k. Lemma 8.6. If the random process

(Z k-1 (n), Y k-1 (n)) n≥1 ,
is a Markov process on W + k-1,k with transition kernel S k-1 then the random process

(Y k-1 (n), Z k (n), Y k (n)) n≥1
is a Markov process on the set {(x, (z, y)) ∈ W + k × W + k,k+1 : x y}. If we denote its transition kernel by Q k then for (u, z, y), (x, z ′ , y

′ ) ∈ W + k × W + k,k+1
such that u y and x y ′

Q k ((u, z, y), (x, z ′ , y ′ )) = v∈N r-1 S k-1 (u, (v, x)) →vr-1 R (y r ∧ v r-1 , y ′ r ) × r-1 i=1 ui← P (y i ∧ v i-1 , z ′ i ) r-1 i=1 →vi-1 P (z ′ i ∨ x i , y ′ i ), (9)
when k = 2r -1 and

Q k ((u, z, y), (x, z ′ , y ′ )) = v∈N r-1 S k-1 (u, (v, x)) ur ← P (y r ∧ v r-1 , z ′ r ) × r-1 i=1 ui← P (y i ∧ v i-1 , z ′ i ) r i=1 →vi-1 P (z ′ i ∨ x i , y ′ i ), ( 10
)
when k = 2r. In the odd and the even cases v 0 = +∞ and the sum runs over = (v 1 , . . . , v r-1 ) ∈ N r-1 such that v i ∈ {y ′ i+1 , . . . , x i ∧ z ′ i }, for i ∈ {1, . . . , r -1} Proof. The dynamic of the model implies that the process

(Z k-1 (n), Y k-1 (n), Z k (n), Y k (n), n ≥ 0)
is Markovian. Since for (z, y) ∈ W + k-1,k the transition probability S k-1 ((z, y), .) doesn't depend on z, the Markovianity of the process

(Y k-1 (n), Z k (n), Y k (n), n ≥ 0)
follows. Identities (9) and ( 10 L k ((z 0 , y 0 ), (x, y, z)) = 1 (z0,y0)=(z,y)

s k-1 (x) s k (y) 1 x y , (11)
when k is odd and

L k ((z 0 , y 0 ), (x, y, z)) = (1 {0} (x k 2 ) + 2 1 N * (x k 2 ))1 (z0,y0)=(z,y) s k-1 (x) s k (y) 1 x y , ( 12
)
when k is even. The following proposition generalizes Lemma 8.4.

Proposition 8.7. The Markov kernels S k , L k and Q k defined as in the identities [START_REF] Pitman | Markov functions[END_REF], ( 8) and ( 11), ( 12) and ( 9), (10) satsify the intertwining

L k Q k = S k L k . Proof. For (z, y) ∈ W + k,k+1 , (x, z ′ , y ′ ) ∈ W + k × W + k,k+1 such that x y ′ , L k Q k ((z, y), (x, z ′ , y ′ )) = u∈W + k L k ((z, y), (u, z, y))Q k ((u, z, y), (x, z ′ , y ′ )).
We prove separately the even and the odd cases. When k = 2r, the sum is equal to

(u,v)∈N r ×N r-1 s k-1 (x) s k (y) (1 {0} (u r ) + 2 1 N * (u r ))(1 -q) 2r-2 R(u r , x r )q r-1 i=1 (xi+ui-2vi) × P (z ′ 1 ∨ x 1 , y ′ 1 ) r i=1 ui← P (y i ∧ v i-1 , z ′ i ) r i=2 →vi-1 P (z ′ i ∨ x i , y ′ i ).
where the sum runs over (u, v)

∈ N r × N r-1 such that u r ∈ {0, . . . , z ′ r }, v i ∈ {y ′ i+1 , . . . , x i ∧ z ′ i }, u i ∈ {v i ∨ y i+1 , . . . , z ′ i }, for i ∈ {1, . . . , r -1}. Thus the sum equals v∈N r-1 s k-1 (x) s k (y) (1 -q) 2r-2 q r-1 i=1 xi P (z ′ 1 ∨ x 1 , y ′ 1 ) r i=2 q -2vi-1 →vi-1 P (z ′ i ∨ x i , y ′ i ) × u∈N r (1 {0} (u r ) + 2 1 N * (u r ))(1 -q) 2r-2 R(u r , x r ) r i=1 q ui ui← P (y i ∧ v i-1 , z ′ i ).
For a fixed v the sum over u is equal to

z ′ r ur=0 (1 {0} (u r )+2 1 N * (u r ))R(u r , x r ) ur ← P (y r ∧v r-1 , z ′ r ) r-1 i=1 z ′ i ui=vi∨yi+1 q ui ui← P (y i ∧v i-1 , z ′ i ).
Since L k and Q k are Markov kernels it is sufficient to consider the case when z r > 0.

In that case, identities (3) and ( 4) of Lemma 8.3 imply that the sum over u equals

(1 {0} (x r ) + 2 1 N * (x r ))q xr∨z ′ r +yr∧vr-1-2z ′ r (1 -q) r-1 i=1 q yi∧vi-1-z ′ i +vi∨yi+1 , i.e. ( 1
{0} (x r ) + 2 1 N * (x r ))q xr ∨z ′ r +yr -2z ′ r + r-1 i=1 yi+vi-z ′ i (1 -q). Thus L 2r Q 2r ((z, y), (x, z ′ , y ′ )) equals s k-1 (x) s k (y) (1 -q) 2r-1 (1 {0} (x r ) + 2 1 N * (x r ))q xr ∨z ′ r +yr-2z ′ r + r-1 i=1 yi-z ′ i q r-1 i=1 xi × P (z ′ 1 ∨ x 1 , y ′ 1 ) r i=2 xi-1∨z ′ i-1 vi-1=yi q -vi-1 →vi-1 P (z ′ i ∨ x i , y ′ i ). Identity (5) of Lemma 8.3 gives that r i=2 xi-1∨z ′ i-1 vi-1=yi q -vi-1 →vi-1 P (z ′ i ∨ x i , y ′ i ) = r i=2 q y ′ i -z ′ i ∨xi-xi-1∧z ′ i-1 = q y ′ r -z ′ r ∨xr-x1∧z ′ 1 q r-1 i=2 y ′ i -xi-z ′ i , which implies L 2r Q 2r ((z, y), (x, z ′ , y ′ )) = s k-1 (x) s k (y) (1 -q) 2r (1 {0} (x r ) + 2 1 N * (x r ))q r i=1 yi+y ′ i -2z ′ i ,
and achieves the proof for the even case. Similarly when k = 2r -1

L 2r-1 Q 2r-1 ((z, y), (x, z ′ , y ′ )) = u,v∈N r-1 s k-1 (x) s k (y) q r-1 i=1 xi-2vi →vr-1 R (y r ∧ v r-1 , y ′ r ) × r-1 i=1 q ui ui← P (y i ∧ v i-1 , z ′ i ) r-1 i=1 →vi-1 P (z ′ i ∨ x i , y ′ i ),
where the sum runs over (u, v)

∈ N r-1 × N r-1 such that v i ∈ {y ′ i+1 , . . . , x i ∧ z ′ i }, u i ∈ {v i ∨ y i+1 , . . . , z ′ i }, for i ∈ {1, .
. . , r -1}. We obtain the intertwining in a quite similar way as in the even case, using identities (4), ( 5) and ( 6) of Lemma 8.3.

The following proposition generalizes Proposition 8.5.

Proposition 8.8. The random process (Z k (n), Y k (n)) n≥1 , is Markovian with transition kernel S k defined in [START_REF] Pitman | Markov functions[END_REF] and (8).

Proof. The random process (X k (t), t ≥ 0) is conditionally independent of the processes (X l (t), t ≥ 0), for l = 1, . . . , k -2, given the process (X k-1 (t), t ≥ 0). So the property can be proved by induction on k. Proposition 8.5 claims that Proposition 8.8 is true for k = 2. Suppose that the proposition is true for a fixed interger k -1 greater that 1. Lemma 8.6 implies that the process

(Y k-1 (n), Z k (n), Y k (n)) n≥1
is Markovian with transition kernel Q k . The intertwining property of Proposition 8.7 implies, by using the Pitman and Rogers criterion given in [START_REF] Pitman | Markov functions[END_REF], that the process Lemma 8.9. Let T 1 (q) and T 2 (q) be two (possibly infinite) lower and upper triangular matrices, whose matrix coefficients are polynomials in an indeterminate q > 0:    T 1 (q) = A 0 + qA 1 + q 2 A 2 + . . . , T 2 (q) = B 0 + qB 1 + q 2 B 2 + . . . , and assume that A 0 = B 0 = I. Then for t ∈ R + , lim q→0 (T 1 (q)T 2 (q)) [t/q] = exp(t(A 1 + B 1 )).

(Z k (n), Y k (n)) n≥1
Proof. Because of the triangularity assumption, the lemma follows, as in the proof of Lemma 2.21 of [START_REF] Borodin | Anisotropic growth of random surfaces in 2+1 dimensions[END_REF], from the claim for finite size matrices, which is standard. Lemma 8.9 implies immediately the following proposition.

Proposition 8.10. Letting q = 1 N , the process (X([N t]), t ≥ 0) converges in distribution towards the process (Y (t), t ≥ 0) as N goes to infinity.

Proof. It follows from Lemma 8.9 by taking    T 1 (q)(x, y) = P(X(n + 1 2 ) = y|X(n) = x), T 2 (q)(x, y) = P(X(n + 1) = y|X(n + 1 2 ) = x), for x, y ∈ GT k .

With the help of the identities of Proposition 5.2, Theorem 7.1 and Lemma 2.21 of [START_REF] Borodin | Anisotropic growth of random surfaces in 2+1 dimensions[END_REF] imply that the process (X k ([N t]), t ≥ 0) converges towards a Markov process with infinitesimal generator equal to A k as N goes to infinity. The convergence stated in Proposition 8.10 achieves the proof of Corollary 7.2. 8.3. Proof of Theorem 7.3. Let (x N ) N ≥1 be a sequence of elements of W + k+1 such that xN N converges to x ∈ C k+1 as N goes to infinity and (ν N ) N ≥1 be a sequence of probability measures on W + k+1 defined by

ν N = y∈W + k+1 R k (x N , y)δ 1 N y .
Propositions 5.2 and 6.2 imply that the measure ν N converges to the measure p k+1 defined in Proposition 6.2 as N goes to infinity. Theorem 7.3 follows.

  write A d for the set {M ∈ M d,d : M + M * = 0} of antisymmetric d × d real matrices, and iA d for the set {iM : M ∈ A d }. Since a matrix in iA d is Hermitian, it has real eigenvalues

+ 1 R r- 1 + 1

 111 then the random process (Λ(n), n ≥ 0) is a Markov chain on Cd with transition probabilitiesp d (x, dy) = h d (y) h d (x) m d (x,y) dy, for x, y ∈ C d , where dy is the Lebesgue measure on R {z x,y} e -m i=1 (yi+xi-2zi) dz when d = 2r + 1 and m d (x, y) = {z |x|,|y|} e -r-1 i=1 (xi+yi-2zi) (e -|xr-yr| + e -(xr+yr) ) dz when d = 2r.

a←P

  (x, .). For x, b ∈ N such that x ≤ b we denote by →b P (x, .) and →b R (x, .) the laws of the random variables min(b, x + ξ 1 ) and min(b, |x + ξ 1 -ξ 2 |).

  ) are deduced from the blocking and pushing interactions of the model. Proof of Theorem 7.1 for every integer k follows as in the case when k = 2 from an intertwining property. Let us define Markov Kernel L k from W + k,k+1 to W + k × W + k,k+1 by letting for x ∈ W + k and (z, y), (z 0 , y 0 ) ∈ W + k,k+1

is 2 .

 2 Markovian with probability S k .The Markov kernels P k+1 and S k satisfyP k+1 (y, y ′ ) = z ′ ∈W + k S k (y, (z ′ , y ′ )), y, y ′ ∈ W + k+1 .Thus Theorem 7.1 is an immediate corollary of Proposition 8.8.8.Proof of Corollary 7.2. The proof of Corollary 7.2 rests on a similar argument as in section 2.7 of[START_REF] Borodin | Anisotropic growth of random surfaces in 2+1 dimensions[END_REF].

6 .

 6 Random matrices Let us denote by M d,d ′ the set of d × d ′ real matrices. A standard Gaussian variable on M d,d ′ is a random variable having a density with respect to the Lebesgue measure on M d,d ′ equal to

Interactions between times n and n + 1 2

Interactions between times n + 1 2 and n + 1

X 2 1 (n + 1)

X 4 2 (n + 1)

An example of blocking and pushing interactions between times n and n+1 for k = 4. Different kinds of dots represent different particles.


Manon Defosseux

INTERACTING PARTICLE MODELS AND THE PIERI-TYPE FORMULAS : THE SYMPLECTIC CASE WITH NON EQUAL WEIGHTS

.

introduction

Let us recall that the Pieri's formula describes the product of a Schur polynomial by a complete symmetric function. It is a specific case of the Littlewood-Richardson rules for decomposing the tensor products of representations of the unitary group into irreducible components. In that specific case, irreducible components of the tensor products have a multiplicity equal to one.

In [START_REF] Defosseux | An interacting particles model and a Pieri-type formula for the orthogonal group[END_REF] an interacting particle model has been introduced and proved to be related to a Pieri-type formula for the orthogonal group. Particles of the model can move to the left or to the right and are submitted to blocking and pushing interactions. Moreover, they are constrained to stay on the non-negative real axis. Such a model is said to be with a wall. One can find for instance in [START_REF] Borodin | Random surface growth with a wall and Plancherel measures for O(∞)[END_REF] and [START_REF] Warren | Some Examples of Dynamics for Gelfand Tsetlin patterns[END_REF] other examples of models with a wall. In these last two references, models only differ by the behavior of the particles near the wall : in the first one, these particles are reflected by the wall, in the second one, they are blocked by the wall. Actually the first one, connected to models of [START_REF] Defosseux | An interacting particles model and a Pieri-type formula for the orthogonal group[END_REF], is strongly related to representations of the orthogonal group whereas the second one involves representations of the symplectic compact group. Pursuing the study of models with a wall in the same way as in [START_REF] Defosseux | An interacting particles model and a Pieri-type formula for the orthogonal group[END_REF] we construct here a new interacting particle model with a wall depending on parameters. Two particular values of the parameters lead on the one hand to a model studied in [START_REF] Warren | Some Examples of Dynamics for Gelfand Tsetlin patterns[END_REF] on the other hand to a random matrix model of [START_REF] Defosseux | Orbit measures and interlaced determinantal point processes[END_REF].

A fundamental difference between models considered here and models usually considered in the literature is that they are related to tensor products of representations which aren't decomposed into irreducible components with multiplicity one. Besides, the case with non equal weights studied here is of particular interest since it exposes connections between symplectic Schur functions and the random particle models. For these reasons, this paper could hopefully provide a step towards a generalization.

The paper is organized as follows. In the second section, I shall recall the definitions of the symplectic Gelfand-Tsetlin patterns and Schur functions. In the third one I will describe the interacting particle model studied in the paper. In the fourth section I'll recall some properties about tensor product of particular representations of the symplectic compact group, which naturally leads to some Markov kernels involved in the interacting particle model. These Markov kernels are defined in the fifth section. Section six is devoted to a random matrix model related to the particle model. Results of the paper are stated in section seven. I will sketch the proofs in the last section.

Symplectic Gelfand-Tsetlin patterns, symplectic Schur functions

Here and elsewhere N stands for the set of nonnegative integers. For n ∈ N * and x, y ∈ R n such that x n ≤ • • • ≤ x 1 and y n ≤ • • • ≤ y 1 , we write x y if x and y are interlaced, i.e.

x n ≤

y n ≤ x n-1 ≤ • • • ≤ x 1 ≤ y 1 .
When x ∈ R n and y ∈ R n+1 we add the relation y n+1 ≤ x n . We denote by |x| the sum of the coordinates n i=1 x i . Definition 2.1. Let k be a positive integer.

(1) We denote by GT k the set of symplectic Gelfand-Tsetlin patterns defined by

GT k = {(x 1 , • • • , x k ) : x i ∈ N [ i+1 2 ] and x i-1 x i , 1 ≤ i ≤ k}.
(

) If x = (x 1 , . . . , x k ) is a symplectic Gelfand-Tsetlin pattern, x i is called the i th row of x for i ∈ {1, . . . , k}. (3) For λ ∈ N [ k+1 2 ] such that λ [ k+1 2 ] ≤ • • • ≤ λ 1 , the subset of GT k of symplectic Gelfand-Tsetlin patterns having a k th row equal to λ is denoted by GT k (λ). 2
Usually, a symplectic Gelfand-Tsetlin pattern is represented by a triangular array as indicated at figure 1 for k = 2r. Actually the more right an entry is, the largest it is. Thus the interlacing property satisfied by the coordinates of a symplectic Gelfand-Tsetlin pattern is graphically represented. Note that the zero and the minus indicate that the coordinates are positive. For x a symplectic Gelfand-Tsetlin pattern of GT k and (q i ) i≥1 a sequence of positive real numbers one defines w k

x (q 1 , . . . , q [ k+1 2 ] ) by recursion by letting w 1

x (q 1 ) = q

|x 1 |
1 , and w 2i

x (q 1 , . . . , q i ) = w 2i-1

x (q 1 , . . . , q i ) q

|x 2i-1 |-|x 2i | i , w 2i+1
x (q 1 , . . . , q i , q i+1 ) = w 2i x (q 1 , . . . , q i ) q

|x 2i+1 |-|x 2i | i+1 , for i ∈ N * . Definition 2.2. For λ ∈ N [ k+1 2 ] such that λ 1 ≥ • • • ≥ λ [ k+1 2 ]
, we denote by s k λ the symplectic Schur function defined by

s k λ (q) = x∈GT k (λ) w k x (q), for q = (q 1 , . . . , q [ k+1 2 ] ) ∈ R [ k+1 2 ] + . Notice that the cardinality of GT k (λ) is equal to s k λ (1), with 1 = (1, . . . , 1) ∈ R [ k+1 2 ] . -x 2r 1 • • • -x 2r r 0 x 2r r • • • x 2r 1 -x 2r-1 1 • • • -x 2r-1 r x 2r-1 r • • • x 2r-1 1 • • • • • • -x 4 1 -x 4 2 0 x 4 2 x 4 1 -x 3 1 -x 3 2 x 3 2 x 3 1 -x 2 1 0 x 2 1 -x 1 1 x 1 1 0 Figure 1.
A symplectic Gelfand-Tsetlin pattern of GT 2r

Interacting particle models

In this section we construct two processes evolving on the set GT k of symplectic Gelfand-Tsetlin patterns. These processes can be viewed as interacting particle models. For this, we associate to a symplectic Gelfand-Tsetlin pattern x = (x 1 , . . . , x k ), a configuration of particles on the integer lattice Z 2 putting one particle labeled by (i, j) at point (x i j , k-i) of Z 2 for i ∈ {1, . . . , k}, j ∈ {1, . . . , [ i+1 2 ]}. Several particles can be located at the same point. In the sequel we will say "particle x i j " instead of saying "particle labeled by (i, j) located at point (x i j , k -i)".

3.1. Geometric jumps. Let q = (q 1 , . . . , q r ) ∈ R r + and α ∈ (0, 1) such that αq i ∈ (0, 1) and αq -1 i ∈ (0, 1) for i = 1, . . . , r, with r = [ k+1 2 ]. Consider two independent families

(ξ i j (n + 1 2
)) i=1,...,k,j=1,...,[ i+1 2 ]; n≥0 , and (ξ i j (n)) i=1,...k,j=1,...,[ i+1 2 ]; n≥1 , of independent geometric random variables such that

P(ξ 2i-1 j (n + 1 2 ) = x) = P(ξ 2i j (n) = x) = (αq -1 i ) x (1 -αq -1 i ), x ∈ N, and
P(ξ 2i-1 j (n) = x) = P(ξ 2i j (n + 1 2 ) = x) = (αq i ) x (1 -αq i ), x ∈ N.
The evolution of the particles is given by a process (X(t)) t≥0 on the set GT k of symplectic Gelfand-Tsetlin patterns. At each time t ≥ 0, a particle labeled by (i, j) is at point (X i j (t), k -i) of Z 2 . Particles evolve as follows. At time 0 all particles are at zero, i.e. X(0) = 0. All particles try to jump to the left at times n + 1 2 and to the right at times n, n ∈ N. Suppose that at time n, after all particles have jumped, there is one particle at point (X

i j (n), k -i) of Z 2 , for i = 1, . . . , k, j = 1, . . . , [ i+1 2
]. Positions of particles are updated recursively as follows (see figure 2).

At time n + 1/2 : All particles try to jump to the left one after another in the lexicographic order pushing the other particles in order to stay in the set of symplectic Gelfand-Tsetlin patterns and being blocked by the initial configuration X(n) of the particles:

• Particle X 1 1 (n) tries to move to the left being blocked by 0, i.e.

X 1 1 (n + 1 2 ) = max(X 1 1 (n) -ξ 1 1 (n + 1 2 ), 0). • Particle X 2 1 (n) tries to jump to the left. It is blocked by X 1 1 (n). If it is necessary it pushes X 3 2 (n) to an intermediate position denoted by X3 2 (n), i.e. X 2 1 (n + 1 2 ) = max X 1 1 (n), X 2 1 (n) -ξ 2 1 (n + 1 2 ) X3 2 (n) = min X 3 2 (n), X 2 1 (n + 1
2 )

• Particle X 3 1 (n) tries to move to the left being blocked by X 2 1 (n) :

X 3 1 (n + 1 2 ) = max X 2 1 (n), X 3 1 (n) -ξ 3 1 (n + 1 2 ) .
Particle X3 2 (n) tries to move to the left being blocked by 0, i.e

X 3 2 (n + 1 2 ) = max( X3 2 (n) -ξ 3 2 (n + 1 2 ), 0).
Suppose now that rows 1 through l -1 have been updated for some l > 1. Particles

X l 2 (n), . . . , X l [ l+1 2 ] (n) of row l are pushed to intermediate positions Xl i (n) = min X l i (n), X l-1 i-1 (n + 1 2 ) , i ∈ {2, . . . , [ l + 1 2 ]}. Then particles X l 1 (n), Xl 2 (n), . . . , Xl [ l+1 2 ]
(n) try to jump to the left being blocked as follows by the initial position X(n) of the particles. For i = 1, . . . , [ l+1 2 ],

X l i (n + 1 2 ) = max X l-1 i (n), Xl i (n) -ξ l i (n + 1 2 ) ,
with the convention that X l-1 l+1 2

(n) = 0 when l is odd.

At time n + 1 : All particles try to jump to the right one after another in the lexicographic order pushing particles in order to stay in the set of symplectic Gelfand-Tsetlin patterns and being blocked by the initial configuration X(n + 1 2 ) of the particles. The first three rows are updated as follows.

• Particle X 1 1 (n+ 1 2 ) moves to the right pushing X 2 1 (n+ 1 2 ) to an intermediate position X2

1 (n + 1 2 ) :

X 1 1 (n + 1) = X 1 1 (n + 1 2 ) + ξ 1 1 (n + 1) X2 1 (n + 1 2 ) = max X 2 1 (n + 1 2 ), X 1 1 (n + 1) • Particle X2 1 (n+ 1
2 ) jumps to the right pushing X 3 1 (n+ 1 2 ) to an intermediate position X3

1 (n + 1 2 ), i.e.

X 2 1 (n + 1) = X2 1 (n + 1 2 ) + ξ 2 1 (n + 1) X3 1 (n + 1 2 ) = max X 3 1 (n + 1 2 ), X 2 1 (n + 1)
• Particle X 3 2 (n + 1 2 ) tries to move to the right being blocked by X 2 1 (n + 1 2 ). Particle X3

1 (n + 1 2 ) moves to the right. That is

X 3 2 (n + 1) = max(X 3 2 (n + 1 2 ) + ξ 3 2 (n + 1)) , X 2 1 (n + 1 2
))

X 3 1 (n + 1) = X3 1 (n + 1 2 ) + ξ 3 1 (n + 1)
Suppose rows 1 through l -1 have been updated for some l > 1. Then particles of row l are pushed to intermediate positions

Xl i (n + 1 2 ) = max X l-1 i (n + 1), X l i (n + 1 2 ) , i ∈ {1, . . . , [ l + 1 2 ]},
with the convention X l-1 l+1 2

(n + 1) = 0 when l is odd. Then particles Xl

1 (n + 1 2 ), . . . , Xl [ l+1 2 ] (n + 1
2 ) try to jump to the right being blocked by the initial position of the particles as follows. For i = 1, . . . , [ l+1 2 ],

X l i (n + 1) = min X l-1 i-1 (n + 1 2 ), Xl i (n + 1 2 ) + ξ l i (n + 1) .
3.2. Exponential waiting times. The interacting particle model described now has been introduced in [START_REF] Warren | Some Examples of Dynamics for Gelfand Tsetlin patterns[END_REF]. In this model particles evolve on Z 2 and jump on their own volition by one rightwards or leftwards after an exponentially distributed waiting time. The evolution of the particles is described by a random process (Y (t)) t≥0 on GT k . As in the previous model, at time t ≥ 0 there is one particle labeled by (i, j) at point (Y i j (t), k -i) of the integer lattice, for i = 1, . . . , k, j = 1, . . . , [ i+1 2 ]. Particle labeled by (2i, j) tries to jump to the left by one after an exponentially distributed waiting time with mean q i or to the right by one after an exponentially distributed waiting time with mean q -1 i . Particle labeled by (2i-1, j) tries to jump to the left by one after an exponentially distributed waiting time with mean q -1 i or to the right by one after an exponentially distributed waiting time with mean q i . Waiting times are all independent. When a particle tries to jump, all particles are pushed and blocked according to the same rules as previously: particles above push and block particles below. That is if particle labeled by (i, j) wants to jump to the right at time t ≥ 0 then (1) if i, j ≥ 2 and Y i j (t -) = Y i-1 j-1 (t -) then particles don't move and Y (t) = Y (t -).

(2) else particles (i, j), (i + 1, j), . . . , (i + l, j) jump to the right by one for l the largest integer such that Y i+l j

(t -) = Y i j (t -) i.e. Y i j (t) = Y i j (t -) + 1, . . . , Y i+l j (t) = Y i+l j (t -) + 1.
If particle labeled by (i, j) wants to jump to the left at time t ≥ 0 then (1) if i is odd, j = (i + 1)/2 and Y i j (t -) = 0 then particle labeled by (i, j) doesn't move.

(

) if i is odd, j = (i + 1)/2 and Y i j (t -) ≥ 1 then Y i j (t) = Y i j (t -) -1. ( 2
) if i is even or j = (i + 1)/2, and Y i j (t -) = Y i-1 j (t -) then particles don't move.

(4) if i is even or j = (i + 1)/2, and Y i j (t -) > Y i-1 j (t -) then particles (i, j), (i + 1, j + 1), . . . , (i + l, j + l) jump to the left by one for l the largest integer such that

Y i+l j+l (t -) = Y i j (t -). Thus Y i j (t) = Y i j (t -) -1, . . . , Y i+l j+l (t) = Y i+l j+l (t -) -1.
Actually, process (Y (t), t ≥ 0) is obtained by letting α go to zero in the previous model. More precisely we get the following proposition.

Proposition 3.1. The process (X([α -1 t]), t ≥ 0) converges in the sense of finitedimensional distributions towards the process (Y (t), t ≥ 0) as α goes to zero.

Proof. The proposition is obtained by replacing q by α in Lemma 8.9 of [START_REF] Defosseux | An interacting particles model and a Pieri-type formula for the orthogonal group[END_REF].

A Pieri type formula for the symplectic group

Let r be a positive integer. One recalls some usual properties of the finite dimensional representations of the compact symplectic group Sp 2r (see for instance [START_REF] Knapp | Lie groups, beyond an introduction[END_REF] for more details). The set of finite dimensional representations of Sp 2r is indexed by the set

W 2r = {λ ∈ R r : λ r ∈ N, λ i -λ i+1 ∈ N, i = 1, . . . , r -1}.
For λ ∈ W 2r , using standard notations, we denote by V λ the so called irreducible representation with highest weight λ of Sp 2r .

Let m be an integer and λ an element of W 2r . Consider the irreducible representations V λ and V γm of Sp 2r , with γ m = (m, 0, • • • , 0). The decomposition of the tensor product V λ ⊗ V γm into irreducible components is given by a Pieri-type formula for the symplectic group. It has been recalled in [START_REF] Defosseux | Orbit measures and interlaced determinantal point processes[END_REF]. We have

V λ ⊗ V γm = ⊕ β M λ,γm (β)V β , (1)
where the direct sum is over all β ∈ W 2r such that there exists c ∈ W 2r which satisfies

   c λ, c β r i=1 (λ i -c i + β i -c i ) = m.
In addition, the multiplicity M λ,γm (β) of the irreducible representation with highest weight β is the number of c ∈ W 2r satisfying these relations. Note that V λ ⊗ V γm is not free multiplicity if m / ∈ {0, 1}.

Markov kernels

Since for λ ∈ W 2r the Schur function s 2r λ is the character of the irreducible representation V λ , decomposition (1) implies

s 2r λ (q)s 2r γm (q) = β∈W2r M λ,γm (β)s 2r β (q).
Thus one defines a family (µ m ) m≥0 of Markov kernels on W 2r by letting

µ m (λ, β) = s 2r β (q) s 2r λ (q)s 2r γm (q) M λ,γm (β),
for λ, β ∈ W 2r and m ≥ 0. Let ξ 1 , . . . , ξ 2r be independent random variable such that ξ 1 , ξ 2 , . . . , ξ r are geometric random variables with respective parameters αq 1 , . . . , αq r and ξ r , ξ r+1 , . . . , ξ 2r are geometric random variables with respective parameters αq -1 1 , . . . , αq -1 r . Consider a random variable T on N defined by

T = 2r i=1 ξ i .
Lemma 5.1. The law of T is a measure ν on N defined by

ν(m) = α m a(q)s 2r γm (q), m ∈ N,
where

a(q) = r i=1 (1 -αq i )(1 -αq -1 i ).
Proof. The lemma follows from straightforward computations.

Lemma 5.1 implies in particular that the measure ν is a probability measure. Thus one defines a Markov kernel P 2r on W 2r by letting

P 2r (λ, β) = +∞ m=0 µ m (λ, β)ν(m), for λ, β ∈ W 2r . Proposition 5.2. For λ, β ∈ W 2r , P 2r (λ, β) = c∈W2r:c λ,β a(q) s 2r β (q) s 2r λ (q) α r i=1 (λi+βi-2ci) . (2)
Proof. The proposition follows immediately from the tensor product rules recalled for the decomposition (1).

We let W 2r-1 = W 2r . For c 0 , λ, c, β ∈ W k , we let

S k ((c 0 , λ), (c, β)) = a(q) s k β (q) s k λ (q) α r i=1 (λi+βi-2ci) 1 c λ,β , (3)
when k = 2r and

S k ((c 0 , λ), (c, β)) = ã(q) s k β (q) s k λ (q) α r i=1 (λi+βi-2ci) ((1 -αq -1 r )1 cr>0 + 1 cr=0 )1 c λ,β , (4)
when k = 2r -1, with ã(q) = (1 -αq r ) r-1 i=1 (1 -αq i )(1 -αq -1 i ).
The main purpose of this paper is to show that S k describes the evolution of the k th row of the random symplectic Gelfand-Tsetlin patterns (X(t), t ≥ 0). Note that Proposition 5.2 ensures that S 2r defines a Markov kernel on W 2r × W 2r . There isn't such an argument for S 2r-1 . Anyway as Λ k and Q k defined in section 8 are Markov kernels, Proposition 8.1 ensures that S k is a Markov kernel in both the odd and the even cases. Thus one defines also a Markov kernel P 2r-1 on W 2r-1 by letting

P 2r-1 (λ, β) = c∈W2r: c λ,β ã(q) s 2r-1 β (q) s 2r-1 λ (q) α r i=1 (λi+βi-2ci) ((1 -αq -1 r )1 cr>0 + 1 cr=0 ). (5)
Actually P 2r-1 (λ, .) is the image of the measure S 2r-1 ((c 0 , λ), (., .)), for any arbitrary c 0 ∈ W 2r , by the map

(x, y) ∈ W 2r × W 2r → y ∈ W 2r .
We'll see that the image measure P k describes the evolution of the k th row of the random symplectic Gelfand-Tsetlin patterns observed at integer times. This Markov kernel is relevant for the understanding of the relation between the particle model and the random matrix model of the next section.

Random matrices

We denote by H the set of quaternions. For us, a quaternion is just a 2 × 2 matrix Z with complex entries which can be written as

Z = a b -b ā ,
where a, b ∈ C. Its conjugate Z * is the usual adjoint of the complex matrix Z. Let us denote by M r,m the real vector space of r × m matrices with entries in H and by P r the set of r × r Hermitian matrices with entries in iH. Since a matrix in P r is a 2r × 2r Hermitian complex matrix, it has real eigenvalues

λ 1 ≥ λ 2 ≥ • • • ≥ λ 2r . Moreover λ 2r-i+1 = -λ i , for i = 1, • • • , 2r.
We put on M r,m the Euclidean structure defined by the scalar product,

M, N = tr(M N * ), M, N ∈ M r,m .
Let C r be the subset of R r defined by

C r = {x ∈ R r : x 1 > • • • > x r > 0}.
Theorem 4.5 of [START_REF] Defosseux | Orbit measures and interlaced determinantal point processes[END_REF] and Proposition 4.8 of [START_REF] Defosseux | Orbit measures and interlaced determinantal point processes[END_REF] imply the following proposition.

Proposition 6.1. Let r be a positive integer and (M (n), n ≥ 0), be a discrete process on P r defined by

M (n) = n l=1 Y l 1 0 0 -1 Y * l ,
where the Y l 's are independent standard Gaussian variables in M r,1 . For n ∈ N,

let Λ 1 (n), • • • , Λ r (n) be the r largest eigenvalues of M (n) such that Λ 1 (n) ≥ • • • ≥ Λ r (n).
Then the process (Λ(n), n ≥ 0), is a Markov process with a transition densities p r defined by x i , x, y ∈ C r .

results

We have introduced a Markov process (X(t), t ≥ 0) on the set of symplectic Gelfand-Tsetlin patterns. We will show that if only one row of the pattern is considered by itself, it found to be a Markov process too. Similar results have been proved in [START_REF] Warren | Some Examples of Dynamics for Gelfand Tsetlin patterns[END_REF]. The main specificity of our model is that coordinates of particles are not upgraded in the same way at every times. We are mainly interested in the process (X k (n), n ≥ 0) but intermediate states (X k (n + 1 2 ), n ≥ 0) are considered for the proofs. Actually these intermediate states come from the fact that the tensor product of ( 1) is not free-multiplicity. We let

Z k (n) = (X k (n -1 2 ), X k (n)), for n ∈ N, with X k (- 1
2 ) = 0. Theorem 7.1. The process (Z k (n), n ≥ 0) is a Markov process on W k × W k with transition kernel S k .

If P k is the Markov kernel defined in ( 2) and ( 5) then Theorem 7.1 implies immediately the following theorem which is our main result. [START_REF] Borodin | Anisotropic growth of random surfaces in 2+1 dimensions[END_REF] implies that the process

(X k ([α -1 t]), t ≥ 0)
converges towards a Markov process with infinitesimal generator equal to A as α goes to zero. The convergence stated in Proposition 3.1 achieves the proof.

If (Λ(n), n ≥ 0) is the process of eigenvalues considered in Proposition 6.1 then the following corollary holds.

Corollary 7.4. Letting q i = 1 for i = 1, . . . , r, the process ((1 -α)X 2r (n), n ≥ 1) converges in distribution towards the process of eigenvalues (Λ(n), n ≥ 1) as α goes to one.

Proof. The Weyl dimension formula (see Knapp [START_REF] Knapp | Lie groups, beyond an introduction[END_REF], Thm V.5.84) for the symplectic groups gives

s 2r λ (1) = 1≤i<j≤r (λ i -λ j + j -i)(λ i + λ j + 2n + 2 -j -i) (j -i)(2n + 2 -j -i) r i=1 λ i + n + 1 -i n + 1 -i .
Thus the corollary follows immediately from Theorem 7.2 and Proposition 6.1.

proofs

The proof of Theorem 7.1 rests on the same ingredients as the proof of Proposition 8.8 of [START_REF] Defosseux | An interacting particles model and a Pieri-type formula for the orthogonal group[END_REF]. It will follow from an intertwining property stated in Proposition 8.1 and an application of a criterion established in [START_REF] Pitman | Markov functions[END_REF] by Pitman and Rogers who give a simple condition sufficient to ensure that a function of a Markov process is again a Markov process. For λ ∈ W k we consider the measure M λ on GT k (λ) defined by

M λ = x∈GT k (λ) w k (x) s k λ (q) δ x ,
where δ x is the dirac measure at x, and the measure m λ defined as the image of the measure M λ by the map

x ∈ GT k (λ) → x k-1 ∈ W k-1 , i.e m λ = β∈W k-1 :β λ q |β|-|λ| r s k-1 β (q) s k λ (q) δ β ,
when k = 2r, and

m λ = β∈W k-1 :β λ q |λ|-|β| r s k-1 β (q) s k λ (q) δ β , when k = 2r -1, with q = (q 1 , . . . , q r-1 ). One defines a kernel Λ k form W k × W k to W k-1 × W k × W k by letting Λ k ((c, λ), (β, c ′ , λ ′ )) = m λ (β)1 c λ 1 c=c ′ ,λ=λ ′ , for c, λ, c ′ , λ ′ ∈ W k , β ∈ W k-1 .
As (Z k (n), n ≥ 0) is conditionally independent of (Z i (t), t ≥ 0, i = 1, . . . , k -2) given (Z k-1 (t), t ≥ 0), Theorem 7.1 may be proved by induction on k. The theorem is true for k = 1. Suppose that the process (Z k-1 (n), n ≥ 0) is Markovian with transition kernel S k-1 . The dynamic of the model implies that (Z k-1 (n), Z k (n), n ≥ 0) is also Markovian. As for any λ ∈ W k-1 , S k-1 ((c 0 , λ), (., .)) doesn't depend on c 0 , it implies that the process ((X k-1 (n), Z k (n)), n ≥ 0) is also Markovian. Let us denote by Q k its transition kernel. Proposition 8.1 claims that it satisfies an intertwining property, which implies, using the Rogers and Pitman criterion of [START_REF] Pitman | Markov functions[END_REF], that the process (Z k (n), n ≥ 0) is Markovian with transition kernel S k . Thus the theorem is true for the integer k. the law of min(a + ξ -, b) when k is even and the law of min(a + ξ + , b) when k is odd. We have for (u, z, y), (x, z ′ , y ′ ) ∈ W k-1 × W k × W k such that u, z y and x, z ′ y ′ Q k ((u, z, y), (x, z ′ , y ′ )) = v∈W k-1 S k-1 (u, (v, x))

[ k+1 2 ] i=1 ui← P k (y i ∧ v i-1 , z ′ i ) × [ k+1 2 ]-1 i=0 →vi P k (z ′ i+1 ∨ x i+1 , y ′ i+1 ), (6)
with the conventions that x [ k+1 2 ] = u [ k+1 2 ] = 0 when k is odd, v 0 = +∞ in the odd and the even cases and the sum runing over v ∈ W k-1 such that v i ∈ {y ′ i+1 , . . . , x i ∧ z ′ i }, for i ∈ {1, . . . , [ k+1 2 ] -1}. Then we write

L k Q k ((z, y), (x, z ′ , y ′ )) = u∈W k ,v∈W k-1
L k ((z, y), (u, z, y))S k-1 (u, (v, x))

× [ k+1 2 ] i=1 ui← P k (y i ∧ v i-1 , z ′ i ) × [ k+1 2 ]-1 i=0 →vi P k (z ′ i+1 ∨ x i+1 , y ′ i+1 ).
We get the intertwining summing over u first and over v after, using respectively identities (4) and ( 5) of Lemma 8.3 of [START_REF] Defosseux | An interacting particles model and a Pieri-type formula for the orthogonal group[END_REF].

p

  r (x, y) = d r (y) d r (x) I(x, y), x, y ∈ C r ,

Proposition 8. 1 .

 1 Λ k Q k = S k Λ k Proof. For x, y, z ∈ W k , we let S k (x, (y, z)) = S k ((c, x), (y, z)),where c is any vector of W k such that cx. Let ξ + and ξ -be two geometric random variables with respective parameters αq k and αq -1 k . For a, b ∈ R + we denote by b← P k (a, .) the law of max(a -ξ + , b) when k is even and the law of max(a -ξ -, b) when k is odd. We denote by →b P k (a, .)

  Theorem 7.2. The process (X k (n)) n≥0 is a Markov process on W k with transition kernel P k . Convergence stated in Proposition 3.1 and Theorem 7.2 lead to the following corollary, which is exactly Theorem 2.3 of [8]. Let us denote e 1 , . . . , e [ k+1 2 ] the standard basis of R [ k+1 2 ] . for x ∈ W k , y ∈ {x + e 1 , . . . , x + e [ k+1 2 ] , x -e 1 , . . . , x -e [ k+1 2 ] }. Proof. Theorem 7.2 and Lemma 2.21 of

	Corollary 7.3. The process (Y k (t), t ≥ 0) is a Markov process with infinitesimal
	generator defined by
	A(x, y) =	s k y (q) s k x (q)	1 y∈W k ,

Interactions between times n and n + 1 2

Interactions between times n + 1 2 and n + 1

X 2 1 (n + 1)

X 4 2 (n + 1)


Fabien Balezeau
email: fabien.balezeau@gmail.com

Pierre-Antoine Eliat

Alejandro Bordelois Cayamo

Hervé Saint-Jalmes

Mapping of low flip angles in magnetic resonance

Errors in the flip angle have to be corrected in many MRI applications, especially for T1 quantification. However, existing method of B1 mapping fail to measure lower values of flip angle despite the fact that these are extensively used in dynamic acquisition and 3D imaging. In this study, the nonlinearity of the radiofrequency transmit chain, especially for very low flip angles, is investigated and a simple method is proposed to determine accurately both the gain of the RF transmitter and the B1 field map for low flip angles. The method makes use of the Spoiled Gradient Echo sequence with long TR, such as applied in the double-angle method. It uses an image acquired with a flip angle of 90° as a reference image that is robust to B1 inhomogeneity. The ratio of the image at flip angle alpha to the image at flip angle 90° enables us to calculate the actual value of alpha. The present study was carried out at 1.5T and 4.7T, showing that the linearity of the radiofrequency supply system is highly dependent on the hardware. The method proposed here allows us to measure the flip angle from 1° to 60° with a maximal uncertainty of 10% and to correct T1 maps based on the variable flip angle method.

Introduction

Knowledge of the actual flip angle and its spatial distribution is necessary in many MR applications, especially for T1 quantification [START_REF] Venkatesan | Accurate Determination of Spin-Density and T1 in the Presence of RF-Field Inhomogeneities and Flip-Angle Miscalibration Magnetic Resonance in[END_REF][START_REF] Cheng | Rapid High-Resolution T1 Mapping by Variable Flip Angles: Accurate and Precise Measurements in the Presence of Radiofrequency Field Inhomogeneity Magnetic Resonance in[END_REF][START_REF] Deoni | High-Resolution T1 Mapping of the Brain at 3T with Driven Equilibrium Single Pulse Observation of T1 with High-Speed Incorporation of RF Field Inhomogeneities (DESPOT1-HIFI)[END_REF][START_REF] Treier | Optimized and combined T1 and B1 mapping technique for fast and accurate T1 quantification in contrast enhanced abdominal MRI Magnetic Resonance in[END_REF]Schabel and Morrell, 2008). The problem has been well known since the early quantitative work on MRI, and many B1 mapping methods have subsequently been developed [START_REF] Pelnar | Measurements of the distribution of B1 field in the presence of linear magnetic field gradient[END_REF][START_REF] Murphy-Boesch | Precision mapping of the B1 field using the rotating-frame experiment[END_REF][START_REF] Akoka | Radiofrequency map of an NMR coil by imaging Magnetic[END_REF][START_REF] Insko | Mapping of the radiofrequency fied[END_REF][START_REF] Stollberger | Imaging of the Active B1 Field In Vivo Magnetic Resonance in[END_REF][START_REF] Cunningham | Saturated Double-Angle Method for Rapid B1+ Mapping Magnetic Resonance in[END_REF][START_REF] Dowell | Fast, Accurate, and Precise Mapping of the RF Field In Vivo Using the 180[END_REF]Yarnykh, 2007;[START_REF] Morrell | A Phase-Sensitive Method of Flip Angle Mapping Magnetic Resonance in[END_REF]Wang et al, 2009;[START_REF] Schär | Simultaneous B0-and B1+-Map Acquisition for Fast Localized Shim, Frequency, and RF Power Determination in the Heart at 3 T Magnetic Resonance in[END_REF]Weber et al, 2010;[START_REF] Sacolick | B1 Mapping by Boch-Siegert Shift Magnetic Resonance[END_REF][START_REF] Chang | Rapid B1 Mapping Using Orthogonal, Equal-Amplitude Radio-Frequency Pulses Magnetic Resonance in Medicine[END_REF]. Latest developments have mainly focused on in vivo imaging as well as accelerated sequences to enable 3D acquisition. The current trend is to perform rapid 3D flip angle mapping to correct 3D T1 maps calculated with the variable flip angle method (VFA). Studies on the optimization of the T1 measurement [START_REF] Imran | Two-points method for T1 estimation with optimized gradient-echo sequence[END_REF][START_REF] Cheng | Rapid High-Resolution T1 Mapping by Variable Flip Angles: Accurate and Precise Measurements in the Presence of Radiofrequency Field Inhomogeneity Magnetic Resonance in[END_REF][START_REF] Deoni | High-Resolution T1 Mapping of the Brain at 3T with Driven Equilibrium Single Pulse Observation of T1 with High-Speed Incorporation of RF Field Inhomogeneities (DESPOT1-HIFI)[END_REF][START_REF] Fleysher | Optimizing the Precision-Per-Unit-Time of Quantitative MR Metrics: Examples for T1, T2[END_REF] have shown that low flip angles (20°) or even very low flip angles (<4°) are required when using VFA using the Spoiled Gradient (SPGR) echo sequence and short TR. However, none of the existing B1 mapping methods really lays emphasis on obtaining accuracy for low flip angle values. An analysis of existing methods [START_REF] Morrell | An analysis of the accuracy of magnetic resonance flip angle measurement methods[END_REF] has shown that the performance of three of the most common methods is degraded, to different degrees, with decreasing flip angle. An additional restriction concerns all approaches based on a pair or set of proportional flip angles, such as applied in double-angle methods [START_REF] Insko | Mapping of the radiofrequency fied[END_REF][START_REF] Cunningham | Saturated Double-Angle Method for Rapid B1+ Mapping Magnetic Resonance in[END_REF]Wang et al, 2009), the phase-sensitive method [START_REF] Morrell | An analysis of the accuracy of magnetic resonance flip angle measurement methods[END_REF] or the method of [START_REF] Akoka | Radiofrequency map of an NMR coil by imaging Magnetic[END_REF] based on a stimulated echo: the radiofrequency supply system must be linear. If this condition is not satisfied, methods based on proportional flip angles become fundamentally flawed. Our study shows that the RF transmitters do not always provide the appropriate B1 field for a nominal flip angle, especially for the lowest flip angles. As a result, we propose a simple approach to determine both the gain of the RF transmitter and the spatial variations of the B1 field with a precision that does not decrease for low flip angles.

Materials and method

Characterization of the output RF pulse magnitude

The length, the magnitude and the shape of a radiofrequency pulse determine the flip angle of the magnetization. Generally, for a given sequence, the flip angle is proportional to the magnitude of the pulse. However, if the amplifier or other components of the RF supply system are used outside their range of linearity, this leads to significant bias in the resulting flip angle. Some effects of RF amplifier distortion on slice selection were studied by [START_REF] Chan | Effects of RF amplifier distortion on selective excitation and their selection by prewarping Magnetic Resonance in[END_REF]. To quantify the effect of this nonlinearity in commonly used pulses, we carried out direct measurements of the resulting pulse magnitude as a function of the nominal flip angle, at constant pulse length. A radiofrequency probe and oscilloscope (DSO6032A, Agilent Technologies) were used to register the waveform of the RF pulses and measure their magnitude. Measurements were performed using two MR imaging systems: a whole-body Avanto 1.5T scanner (Siemens, Erlangen, Germany) using the body coil for transmission and a Bruker Biospec 4.7T scanner (Bruker Biospin, Rheinstetten, Germany) with a 36-mm linear volume coil for transmission. At 1.5T, 1-ms hermite pulses (Blümich, 2000) were registered for nominal flip angles of 1, 2, 3, 4, 8, 16, 32, 64 and 90°. At 4.7T, 1-ms sine cardinal pulses (Blümich, 2000) were registered for nominal flip angles of 1,2,3,4,5,6,8,10,12,15,20,25,30,40,50,60,70, 80 and 90°. The probe was placed at the centre of the receiver coil and all measurements falling outside the linearity range of the RF probe were discarded.

The Low Angle Mapping method

Our method is based on the same sequence used in the double-angle method: a SPGR sequence with a long TR (ideally TR>5T1), to obtain a flip angle weighting. The general expression of the SPGR sequence [START_REF] Haase | Snapshot FLASH MRI: applications to T, T, and chemical-shift imaging Magnetic Resonance[END_REF]) is given by equation (1). With sequence parameters satisfying TE<5T2* and TR>5T1, the expression of the signal can be approximated by the equation (2). For low flip angles, the signal is poorly T1 weighted, thus the TR can be reduced. For example, at 10°, TR can be equal to T1 and at 1°, TR can be equal to 0.1T1.

€ S = M 0 sinα (2)
Under the conditions of equation (2), the signal variations due to B1 inhomogeneity will be negligible if the flip angle is equal or close to 90°. Indeed, the signal variations between 82° and 98° do not exceed 1% of the 90° signal. In our approach, the signal of the image acquired with a 90° flip angle is considered independent of the flip angle, and is used as a reference value of M0. In other words, the 90° image provides a map of the other sources of inhomogeneity in the signal, arising from B0, M0 and the sensitivity of the receiver coil. By dividing the image at the nominal flip angle alpha by the 90° image, we obtain the sine of the actual flip angle alpha.

€ α = arcsin S α M 0 = arcsin S α S 90°(
3)

The theoretical range of the method includes flip angles from 0° (excluded) to 90° (excluded), with a decreasing precision for flip angles close to 90°. However, the method is most advantageous when the flip angles are in the range from 1° to 20°, to the extent that the flip angle is equivalent to the ratio Sα/S90°. Considering that Sα<<S90°, the Angle to Noise Ratio of the flip angle map is equivalent to the Signal to Noise Ratio (SNR) of the image at flip angle alpha. Consequently, the method is able to map extremely low flip angles, despite the fact that because of the use of magnitude in signal processing, a non-zero mean value of the noise will appear and may be wrongly assigned to a signal value. Practically, this problem can be avoided by using a short TR at low flip angles and increasing the number of averages of the signal for the same total acquisition time. Thereby, even for very low flip angles, a good precision can be achieved without increasing the acquisition time.

Accuracy analysis

We used MATLAB (Mathworks,USA) to simulate the accuracy of the Low Angle Mapping (LAM) method and establish the main guidelines for using this approach. The three most important sources of errors were investigated: the actual effect of B1 inhomogeneity on the 90° reference image, the effect of the T1 weighting and the noise on the signal. Since the approximation of negligible effects of B1 on the 90° signal is a key assumption of the flip angle estimation, the effects of a bias on the 90° flip angle needs to be studied to determine the limits of the LAM. The method is symmetrical around 90°, so biases of +10° or -10° will result in the same flip angle map. We simulated actual flip angles of 70°, 75°, 80°, and 85° for the reference signal, instead of a nominal flip angle of 90°. Like the DAM, the residual T1 weighting is a source of error and cannot be totally avoided in conditions of in vivo acquisitions. For example, accurate B1 mapping in brain, while satisfying the condition TR>5T1, would need a TR of 10s. Finally, we characterized the precision of our method within its valid range by means of Monte Carlo simulations. The performances of the double-angle method and our method were simulated using an identical SNR. For both methods, we simulated 30 000 samples of the signal with an added Gaussian noise for 90 flip angle values from 1° to 90°. Then, the resulting flip angle samples and their probability density functions were calculated. The cross-comparison with other flip angle measurement methods is based on two studies (Wade and Rutt, 2007;[START_REF] Morrell | An analysis of the accuracy of magnetic resonance flip angle measurement methods[END_REF] analysing the efficiency of the best known methods: the Double-Angle Method [START_REF] Insko | Mapping of the radiofrequency fied[END_REF], the phase sensitive method [START_REF] Morrell | A Phase-Sensitive Method of Flip Angle Mapping Magnetic Resonance in[END_REF] and Actual Flip-angle Imaging.

Experimental validation: B1 maps and application to T1 correction

Experimental validation was performed using the TO4 test-object (Spinsafety Ltd, Rennes, France). TO4 is a cylinder containing a CuSO 4 solution, which surrounds an assembly of 12 glass tubes filled with MnCl 2 solutions having T1 values ranging from 100 ms to 1000 ms. Both at 1.5T and 4.7T, T1 reference values of the test objects were obtained using several SE measurements, with chosen values of TR optimized for each tube [START_REF] Spandonis | High resolution MRI relaxation measurements of water in the articularcartilage of the meniscectomized rat knee at 4[END_REF]. On the 1.5 T scanner, a set of SPGR images was acquired for B1 mapping with TR=2000 ms, TE=2.4 ms, alpha=[1 2 3 4 5 9 32 90]°, matrix size of 128x128, FOV 230 mm x 230 mm, 12 slices, slice thickness=5 mm; bandwidth= 55 000 Hz, 4 min 16 s imaging time per flip angle. Then, a set of SPGR 3D images for T1 mapping was acquired using the parameters: TR=6 ms, TE=2.4 ms, matrix size 256x256x48, alpha= [2 4 8 16 32 64]°. A 16channels head coil was used as a receiver. On the 4.7T scanner, a set of SPGR images was acquired from one MnCl 2 tube of the TO4 assembly, with TR=800 ms, TE=2.4 ms, alpha=[3 4 5 8 9 10 12 16 17 25 30 60 90]°, bandwidth=50 000 Hz, matrix size=64x64, FOV=56 mm x 40 mm. T1 maps were drawn up with a SPGR sequence at TR=6ms, TE=2.4 ms alpha=[4 6 8 10 30 60 90]°, using the same geometry and bandwidth as the B1 maps. The 36mm volume coil was used for both transmitting and receiving. Images were processed using ImageJ (National Institute of Health, USA). Flip angles maps were calculated with LAM, and T1 maps were calculated with VFA firstly using the nominal flip angles and then the measured flip angles.

Results

RF measurement

Figure 1a shows the normalized output RF pulse magnitude as a function of the nominal flip angle for the 1.5T Siemens system, while Figure 1b shows this same function for the Bruker 4.7T system.

Values are normalized as a fraction of the 90° magnitude. A constant value of 1 is expected for a perfectly linear RF transmitter. The transmitter of the Siemens system is linear in the range 5° to 90°. The transmitter of the Bruker system using the 36-mm volume coil for transmission is linear only for flip angles above 30°; flip angles lower than 4° were not registered because the measured signal was too low to be in the linearity range of the RF probe used. An alteration of the pulse shape is observed for low flip angles. Because of the limited dynamic range of the modulator, the lobes of the sinc pulses use on the 4.7T system disappear for flip angles lower than 30°.

Experimental B1 maps and T1 correction

The figure 5 shows flip angles measured with the Low Angle Mapping method versus nominal flip angles. Figure 6 presents the flip angle map on the 1.5T Siemens system. The actual mean flip angle is 15% lower than the nominal flip angle, with a spatial variation of 0.2°. Figure 7 presents corrected and uncorrected T1 maps of the TO4 test object. Reference T1 value of 380ms was obtained with optimized SE measurements for the CuSO4 solution in the body of the test object.

Discussion

Nonlinearity of the radiofrequency transmitter

Since the flip angle depends upon other parameters such as the pulse shape, the magnitude of the RF pulse is not a direct measurement of the flip angle. Yet, it gives a general idea on how the actual flip angles scales up with the nominal flip angles. RF measurements shown figure 1 are consistent with the flip angle values shown in figure 5, calculated using the LAM method. Our study shows that the nonlinearity of the RF transmitter can be sometimes a dominant factor of bias in T1 measurement, even more important than inhomogeneities of the B1 field due to the transmitting coil geometry or B1 penetration effects. When using very low flip angles, the problem of linearity can no longer be ignored. According to the comparison between two different systems, the accuracy of the flip angles is strongly dependent on the configuration of the system. On the 4.7T system, the same measurements repeated with a 72-mm transmitting coil produced similar results but for lower nominal angle values.

We also observed a modification of the plot presented figure 1 when changing the pulse length. Thus, the calibration issue seems to be directly linked to the power delivered by the amplifier. As shown in Figs. 8 and9, even with a relatively homogeneous B1 field, the mean value of the flip angle can be extremely different from the nominal value, which can make T1 measurements impossible. The nonlinearity observed for the 1.5T system is not considerable, but as shown figure 7, the lack of accuracy for very low flip angles can lead to non negligible errors in T1 mapping.

Guidelines and limits

The LAM method is based on the approximation that B1 inhomogeneities have a negligible effect on the signal at the 90° flip angle, and the conditions of validity of this assumption must be satisfied to obtain reliable flip angle measurements. In view of the simulations shown in Fig. 2, the LAM method is optimal if the spatial variation of the 90° pulse does not exceed +/-10°. Under these conditions, the error due to the approximation is negligible (<3%) for flip angles of 1° to 70°. The method can still be used with spatial variations of the 90° pulse of +/-20°, with a 7% uncertainty on flip angles ranging from 1° to 30°. In our study, dieletric resonance effects were negligible. But at high field strength and in large objects, it may cause substantial B1 inhomogeneity in the 90° image.

The lowest flip angle that can be measured by the LAM method is determined solely by the SNR of the SPGR sequence at this flip angle. The measurement will be biased at very low SNR, if the noise noise distribution can no longer be considered as Gaussian. Since the number of excitations can be increased, and very short TR can be used for low flip angles there is no minimal angles value for our method.

The biased caused by incomplete relaxation can be problematic when dealing with long T1, typically in the brain or blood. For the 90° excitation, the complete relaxation is achieved when TR>5T1, and we recommend choosing at least TR>3T1. For lower flip angles, there is no analytical expression of the minimal TR to use. According to simulations TR=0.1*α (in °) is optimal for angles from 1° to 30°.

Comparison between the Low angle Mapping method and other B1 mapping approaches

Since the same sequence is used in the double-angle method, the comparison with Low Angle Mapping is straightforward. Both methods need the same acquisition time, so they differ only by their accuracy within the flip-angle range. The DAM is better for flip angles ranging from 50° to 90°, while both methods are equivalent in the range 35° to 50° and the LAM is better for flip angles ranging from 1° to 35°. Figure 4 shows a very good complementarity between the accuracy ranges of both methods.

Based on the studies of (Wade and Rutt, 2007) and [START_REF] Morrell | An analysis of the accuracy of magnetic resonance flip angle measurement methods[END_REF], we consider some other well known B1 mapping methods in our comparison. The phase sensitive method [START_REF] Morrell | A Phase-Sensitive Method of Flip Angle Mapping Magnetic Resonance in[END_REF] provides a good accuracy within a wider range than the DAM, yet the precision degrades for very low flip angles. However, by using a double angle, the method assumes the linearity of the flip angles. The Actual Flip Angle method (Yarnykh, 2007) has some advantages: linearity of the RF transmitter is not required and short TR are used, providing a better time-efficiency. But the method is not precise for low flip angles because the optimal ratio TR2/TR1 increases for lower flip angles, and the condition TR2 <<T1 must be satisfied. The LAM is not really competitive with recently developed methods, if compared over the usual range of flip angles higher than 20°, but it fills a gap in the range of flip angles covered by existing methods.

In vivo-applications

According to the amount of quantitative studies using SPGR sequence with low flip angles, the method can be widely used for in vivo imaging in protocols involving T1 mapping and/or DCE MRI [START_REF] Cheng | Rapid High-Resolution T1 Mapping by Variable Flip Angles: Accurate and Precise Measurements in the Presence of Radiofrequency Field Inhomogeneity Magnetic Resonance in[END_REF][START_REF] Deoni | High-Resolution T1 Mapping of the Brain at 3T with Driven Equilibrium Single Pulse Observation of T1 with High-Speed Incorporation of RF Field Inhomogeneities (DESPOT1-HIFI)[END_REF][START_REF] Treier | Optimized and combined T1 and B1 mapping technique for fast and accurate T1 quantification in contrast enhanced abdominal MRI Magnetic Resonance in[END_REF][START_REF] Vautier | 2D and 3D Radial Multi-Gradient-Echo DCE MRI in Murine Tumor Models With Dynamic R*2-Corrected R1 Mapping Magnetic Resonance in[END_REF][START_REF] Morrell | An analysis of the accuracy of magnetic resonance flip angle measurement methods[END_REF]. Most of publications on T1 mapping and DCE MRI that include a B1 correction finally never measure the actual value of the flip angles used in the sequence, and rely on a nominal B1 map. In contrast, our method is able to provide a direct measurement of the actual angle value for very low flip angles. For the 90° flip angles, the need to wait for the full recovery appears as a limitation for in vivo applications, especially for tissues with long T1. However, a small acquisition matrix (64x64) is generally sufficient to address both calibration and spatial variation issues. Moreover, parallel imaging method like GRAPPA or SENSE can be used, when available, to accelerate the acquisition of the 90° image. Since the problem of the linearity of the transmitter does not depend on the patient, it is possible to calibrate the flip angles once using test-objects, and to use another B1 mapping method to correct the inhomogeneity. However, the calibration depends on the configuration of the experiment i.e. the transmitting coil, the pulse width, the pulse shape. Therefore, the calibration would be required before each experiment but it would not provide information on the B1 map, which depends on the subject and the geometry of the acquisition. Thus, an in vivo flip angle mapping is preferable for quantitative studies. We applied our method for whole body perfusion imaging experiments on mice. The B1 correction enables the quantification of the contrast agent on a wider field of view and with a better precision. The method was also tested on T1 mapping of the human brain. In both applications B1 maps were acquired in less than 10 min. The method can be implemented directly on every MRI system, and can be used without any notion of sequence programming.

Opportunities for improvement

Currently, the minimal TR used with the 90° flip angle limits the methods on its own to 2D acquisition, but for rapid in vivo 3D B1 mapping, time efficiency needs to be improved. Fortunately, most of the major improvements of the DAM, especially concerning time-efficiency, can be applied to the Low Angle mapping method. [START_REF] Cunningham | Saturated Double-Angle Method for Rapid B1+ Mapping Magnetic Resonance in[END_REF] proposed a saturation pulse to eliminate all T1 weighting. (Wang et al, 2009) used a catalysed SPGR sequence to compensate the T1 weighting while using a short TR. Lately, Wade (2009) developed the Double Angle Look Locker for accelerated 3D flip angle mapping. Using such accelerated sequences offers two major advantages. Firstly, reduced acquisition time is an obvious asset for in vivo imaging. Moreover, the possibility of performing 3D acquisitions instead of multislice 2D acquisitions reduces the error in B1 mapping caused by slice selection. Optimizing the time efficiency of Low Angle Mapping will be the subject of future studies.

Conclusion

This study has shown the need for a B1 mapping protocol that is able to measure the lowest flip angles and take into consideration possible nonlinearity of the RF supply system. We propose a very simple and straightforward method able to map accurately the lowest flip angles, assuming reasonable spatial inhomogeneity in the 90° pulse. Our method fully supplements the range of methods based on the double-angle approach and makes it possible to obtain excellent flip-angle accuracy at all values of flip angle.

Figure 1

 1 Figure 1 Normalized magnitudes of the radiofrequency pulse for flip angles ranging from 1° to 90°. (a) Bruker Biospec 4.7T, with a 36-mm inner diameter volume transmitting coil (b) Siemens Avanto 1.5T, with body transmitting coil

Figure 2

 2 Figure 2 Bias on the estimated flip angle value due to inhomogeneity or systematic error of 5°, 10°, 15° and 20° in the 90°-flip angle Figure 3 shows the error on the flip angle estimation caused by a residual T1 weighting. The relative error does not depend on the flip angle measured. There is an overestimation of 15% of the actual flip angle value for TR=2T1 and 5% for TR=3T1, while the error is negligible for TR>5T1.

Figure 3

 3 Figure 3 Error in the flip angle estimation caused by residual T1 weightingFigure4shows the probability density functions of the LAM and the DAM. While the precision of the double-angle method degrades for flip angles lower than 20°, the precision of our method is almost constant for flip angles ranging from 1° to 70° and degrades for flip angles higher than 70°.

Figure 4

 4 Figure 4 Monte-Carlo simulations of the probability density functions of our method and the double-angle method. At top, with SNR=100 for the 90° image. At bottom, with SNR=30 for the 90° image.

Figure 5

 5 Figure 5 Flip angles measured with the low angle mapping method versus the nominal flip angles. a) on the Bruker 4.7T system and b) on the Siemens 1.5T system.

Figure 6 Figure 7

 67 Figure 6 Flip angle map and corresponding cross-section in the TO4 test-object at 1.5T for a nominal flip angle of 2°

Figure 9

 9 Figure8presents the flip-angle map at 4.7T, for a nominal flip angle of 4° with a 36-mm volume coil. The actual flip angle determined with the LAM method is 5 times lower than the nominal flip angles. For very low flip angles, the nonlinearity of the transmitter is a predominant source of error in relation to the B1 inhomogeneity. Figure9presents the uncorrected and corrected T1 maps calculated with VFA at TR=6 ms and flip angles of 4° and 30°. According to the SE reference measurements, T1 should be equal to 122 ms for the MnCl2 tube. In this case, the flip-angle error leads to the measurement of a negative T1 value of -25 ms without correction. With correction, we obtain a measured mean value of 130 ms.

Figure 8 Figure 9

 89 Figure 8 Flip-angle map and corresponding cross-section in the MnCl 2 tube at 4.7T for a nominal flip angle of 4°

  Wade T, Rutt B 2007 Comparison of Current B1-mapping Techniques Proceedings of 15th ISMRM meeting Wade T , Rutt B 2009 B1 correction using Double Angle Look-Locker (DALL) Proceedings of the International Society for Magnetic Resonance in Medicine 16th Scientific Meeting Wang D, Zuehlsdorff S, Larson AC 2009 Rapid 3D radiofrequency field mapping using catalyzed double-angle method NMR in Biomedicine 22 882-90 Weber H, Paul P, Elverfeldt DV, Hennig J, Zaitsev M 2010 Extended Multi-Flip-Angle B1 Mapping: A 3D Mapping Method for Inhomogeneous B1 Fields Concepts in Magnetic Resonance Part B (Magnetic Resonance Engineering) 37B 203-14 Yarnykh VL 2007 Actual Flip-Angle Imaging in the Pulsed Steady State: A Method for Rapid Three-Dimensional Mapping of the Transmitted Radiofrequency Field Magnetic resonance in Medicine 57 192-200

Acknowledgments

This work was supported by a financial grant from Region Bretagne. Dr M.S.N. Carpenter post-edited the English style.