Amélie Trotignon
email: amelie.trotignon@jku.at.

Discrete harmonic functions in the three-quarter plane

Keywords:

Discrete harmonic functions in the three-quarter plane

Introduction

In the two-dimensional continuous case, a continuous harmonic function is a function for which the standard Laplacian ∆ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 is zero. For example, in the cone {r exp(it) : 0 r < ∞, 0 t η} of opening angle η, then u(r, t) = r π/η sin π η t is the unique (up to multiplicative constants) positive harmonic function equal to zero on the boundary.

In the two-dimensional discrete case, consider the simplest Laplacian operator 1 4 [f (i + 1, j) + f (i, j + 1) + f (i -1, j) + f (i, j -1)] -f (i, j).

The function f (i, j) = i is positive harmonic in the right half-plane and equal to zero on y-axis.

In the positive quadrant, the function f (i, j) = ij is positive harmonic with Dirichlet boundary conditions. What about the three-quarter plane? Surprisingly, harmonic functions in the threequarter plan have more complex expressions, see equations [START_REF] Fayolle | Random walks in the quarter plane[END_REF] and [START_REF] Fayolle | Random walks in the quarter-plane with zero drift: an explicit criterion for the finiteness of the associated group. Markov Process[END_REF]. Unlike the two last examples, the three quadrants is not a convex cone and finding positive harmonic functions is more complicated. Although the difference between the intersection of two half-planes (the quarter plane) and union of two half-planes (the three-quarter plane) seems geometrically insignificant at first sight, Bousquet-Mélou has already noticed this complexification with a paths enumeration point of view: in [START_REF] Bousquet-Mélou | Square lattice walks avoiding a quadrant[END_REF] Bousquet-Mélou notes that studying the simple walks (with cardinal points jumps) is as difficult as studying the Gessel walks in the quadrant (a famous model with jumps {E, NE, W, SW}, solved after ten years of research). In this article, we develop a systematic approach to find positive discrete harmonic functions in the three-quarter plane with Dirichlet conditions.

Context. In the discrete case, planar lattice random walks in cones occupy a central position in probability and combinatorics. Harmonic functions play an important role in probability theory. Doob h-transform is a way to build conditioned random processes in cones from a random process and a positive harmonic function vanishing on the boundary of the cone. Finding positive harmonic functions for random processes is therefore a natural objective in the study of confined random walks. There are very few ways to compute discrete harmonic functions (see [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF] and references therein).

Most of walks study have been done on the quadrant, or more generally in convex cones. A natural generalization is to consider other domains of restriction and determine how the framework of (random) walks is different from the quarter plane to this region. Recently, non-convex cones, in particular the three-quarter plane

C = {(i, j) ∈ Z 2 : i 0 or j 0}, (1)
have been examined. Unlike the quarter plane, where generating functions involve only positive powers of the variables, in the three-quarter plane we face both positive and negative powers of the variables, making convergence issues arise. A natural strategy consists of cutting the three quadrants in some convex cones in which the generating functions are convergent. In [START_REF] Bousquet-Mélou | Square lattice walks avoiding a quadrant[END_REF], Bousquet-Mélou sees the three-quarter plane as the union of three quadrants and obtains some results for the simple and diagonals walks avoiding a quadrant. Integral expressions for the generating function of walks avoiding a quadrant with symmetric step sets for walks are derived in [START_REF] Raschel | On walks avoiding a quadrant[END_REF], where the three-quarter plane is seen as the union of two symmetric convex cones of opening angle 3π/4. Asymptotics of the number of excursions of walks with small steps in the three-quadrant is computed in [START_REF] Mustapha | Non-D-Finite Walks in a Three-Quadrant Cone[END_REF] by Mustapha. In this article, following [START_REF] Denisov | Random walks in cones[END_REF][START_REF] Bostan | Non-D-finite excursions in the quarter plane[END_REF], Mustapha expresses the critical exponent of harmonic functions in three quadrants as a function of the critical exponent of harmonic functions in a quadrant. When this exponent is not rational, then the generating function of walks is not D-finite and [START_REF] Mustapha | Non-D-Finite Walks in a Three-Quadrant Cone[END_REF]Thm 1.3] proves that the generating function of the walks of the 51 non-singular step sets with infinite group are not D-finite in the three-quarter plane (recall that neither are they in the quadrant).

In this article, we find an explicit expression for generating functions of discrete harmonic functions associated to random walks avoiding a quadrant with a mixed approach of [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF] and [START_REF] Raschel | On walks avoiding a quadrant[END_REF]. We focus on the analytic approach developed in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF], which consists in writing a functional equation for the generating function for a fixed harmonic function, transforming this functional equation into a boundary value problem and finally solving this problem, which results in an explicit expression for the generating function. We begin by making some assumptions on both the random walks and discrete harmonic functions for these random walks.

(H1) The walk is homogeneous inside the cone with transition probabilities {p i,j } -1 i,j 1 to the nearest neighbors;

(H2) The transition probabilities are symmetric (p i,j = p j,i ) and p 0,0 = p -1,1 = p 1,-1 = 0;

(H3) In the list p 1,1 , p 1,0 , p 0,-1 , p -1,-1 , p -1,0 , p 0,1 , there are no two consecutive zeros;

(H4) The drifts are zero:

-1 i,j 1 ip i,j = 0 and -1 i,j 1 jp i,j = 0.

We first suppose with (H1) the random walks to be homogeneous with small steps, which therefore provide us techniques and tools developed in [START_REF] Fayolle | Random walks in the quarter plane[END_REF]. Moreover, with assumption (H2), we suppose the walks to be symmetric with no anti-diagonal jumps. If we do not exclude non-zero probabilities p -1,1 and p 1,-1 , as noted in [START_REF] Raschel | On walks avoiding a quadrant[END_REF]Sec. 2], after the change of variable ϕ we would end up with random walks with non-zero probabilities p -2,1 and p 2,1 respectively for big jumps of vector (-2, -1) and (2, 1). Note that the analytic theory for walks with big steps is still incomplete [START_REF] Cohen | Boundary value problems in queueing system analysis[END_REF][START_REF] Cohen | Analysis of random walks, volume 2 of Studies in Probability, Optimization and Statistics[END_REF]. A related consequence would be the presence of more unknowns in the functional equation [START_REF] Raschel | On walks avoiding a quadrant[END_REF] (namely i 1 f (i -2, i)x i y i and i 1 f (i, i -2)x i y i ), making the problem not solvable with our techniques. The third hypothesis (H3) is not essential in the study, but automatically excludes degenerate cases which could have been studied with easier methods. Finally, we assume the walks to have zero drifts with (H4). Note that the zero and non-zero drifts are two very different frameworks [START_REF] Duraj | Random walks in cones: the case of nonzero drift[END_REF], and most results are given in the non-zero drift case. Furthermore, (H4) makes the random walks hit the negative axes almost surely. Observe that the hypotheses (H2) and (H4) are nonredundant: a model can satisfy (H2) and not (H4) 1 and vice versa2 . Figure 1 illustrates a possible transition probabilities set which satisfies (H1), (H2) and (H3). We will use the combinatorial step set terminology: the simple model is when

p 1,0 = p 1,0 = p -1,0 = p 0,-1 = 1/4, Gouyou-Beauchamps model has p 1,0 = p -1,1 = p -1,0 = p 1,-1 = 1/4 and Gessel model is p 1,0 = p 1,1 = p -1,0 = p -1,-1 = 1/4.
In addition, we ask the associated discrete harmonic functions f = (f (i, j)) (i,j)∈C to satisfy four properties:

(P1) For all i 1 or j 1, f (i, j) = -1 i 0 ,j 0 1 p i 0 ,j 0 f (i + i 0 , j + j 0 ); (P2) If i 0, f (i, 0) = 0 and if j 0, then f (0, j) = 0; (P3) If i > 0 or j > 0 then f (i, j) > 0.
We also make the hypothesis (P4) which, as we shall see (Remark 5), will be automatically satisfied for positive harmonic functions associated to symmetric step sets.

(P4) For all (i, j) ∈ C, f (i, j) = f (j, i).

The first property (P1) is the harmonicity condition, the second one (P2) is the zero condition on the boundary, the third one (P3) is the positivity condition within the cone and the last one (P4) is a symmetry condition (coming then from (H2)). In other words, we are interested in symmetric positive discrete harmonic functions for symmetric random walks with small steps constrained to the boundary of C. Thereafter, for such a fixed harmonic function f , we consider its generating function

H(x, y) = (i,j)∈C f (i, j)x i-1 y j-1 . (2)
We observe here one of the main difficulties of the three quadrants: the series H(x, y) is not convergent. Thereafter, we may see this object as a formal series. The goal is to find a formal explicit expression of H(x, y), and from this expression, it is not difficult to get the f (i, j) for all (i, j) ∈ C, with a formal power series expansion. Method. Working in non-convex cones, here in particular in the three-quarter plane, raises convergence problems. Indeed, it is not difficult to write a functional equation from the properties of harmonicity (P1) and vanishing conditions on the boundary (P2). However, even if this functional equation ( 13) seems close to the quarter plane case [START_REF] Mustapha | Non-D-Finite Walks in a Three-Quadrant Cone[END_REF] solved in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF], it is in fact fundamentally different: the three quadrants case involves negative powers of x and y making the series not convergent anymore. To remedy this difficulty, we follow the same strategy as in [START_REF] Raschel | On walks avoiding a quadrant[END_REF]: we divide the three quadrants into two symmetric convex cones and the diagonal (see Figure 5) and write a system of two functional equations (one for each cone). At first sight, this cut increases even more the level of difficulty: we have now two functional equations and more unknowns to deal with, but in the particular case of symmetric harmonic functions and symmetric transition probabilities, the system is composed of twice the same equation and the problem can be seen as a slightly different variation of the quadrant case (see Figure 7). In this symmetric case, we are able to use the tools and methods of [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]: transform the functional equation into a boundary value problem, solve it and write an explicit expression for the generating function. Finally, let us point out that this method of splitting the domain in two octants can also be applied in the quadrant in the symmetric case, and allow us to give alternative proofs of [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF].

p 1,0 p -1,0 p -1,0 = p 0,-1 p 1,0 = p 0,1 p 1,1 p -1,-1
Main results. Our main result is an expression of the diagonal section

D(x, y) = D ϕ (xy) = i 1 f (i, i)x i-1 y i-1 ,
see Lemma 16. Let w ϕ be a conformal mapping (see Section 3.1, w ϕ depends on the step set and satisfies an algebraic differential equation and can be simply expressed in terms of sine and arcsine functions) and G ϕ an explicit algebraic function defined from w ϕ . The generating function D(x, y) of discrete harmonic functions not necessarily positive (satisfying (P1) and (P2)) can be expressed as

D(x, y) = P ( w ϕ (xy)) G ϕ (xy) , P ∈ R[y]. (3)
In particular, taking P of degree 1, we get the unique positive discrete harmonic function (satisfying (P1), (P2) and (P3)), see Theorem 13. This expression of D ϕ (y) suffices to give a complete solution to the problem and Theorem 14 gives a formal explicit expression for H(x, y). This formal series is expressed with the kernel K(x, y), Figure 2: A random walk with associated angle θ is the three quadrants (resp. the quadrant), can be seen as an inhomogeneous random walk in the half-plane with associated angle π -θ 2 (resp. θ 2 ) in each quadrant.

a polynomial in x and y, and two conformal mappings W ϕ (x) and W ϕ (y). Each quantity depends on the transition probabilities of the random walk.

As mentioned earlier, for the simple step set, the positive discrete harmonic function in the half-plane is given by f (i, j) = i with critical exponent 1. In the positive quadrant, the positive discrete harmonic function is given by f (i, j) = ij with critical exponent 2. In this case, the value of the critical exponent in the quadrant (intersection of two half-planes) is twice the value of the critical exponent in the half-plane. Along our study, we find similar relations on harmonic functions in various quadrants (see Lemma 10 and Equation (66)). The growth of harmonic functions can be expressed from the angle θ defined by [START_REF] Fayolle | Random walks in the quarter-plane with zero drift: an explicit criterion for the finiteness of the associated group. Markov Process[END_REF]Sec. 1.2]

θ = arccos           - -1 i,j 1 ijp i,j   -1 i,j 1 i 2 p i,j   •   -1 i,j 1 j 2 p i,j             . (4)
In the three-quarter plane, the critical exponent of the harmonic function is π/(2π -θ) = π/(2θ ϕ ).

After the decomposition of the three quadrants and the changes of variables, the random walk in the three quadrants is equivalent to an inhomogeneous random walk in the half-plane (see Figure 11), with associated angle θ ϕ = π -θ/2 in each quadrant (see Figure 2a). We also recover the factor two between harmonic functions in the quadrant and harmonic functions in the half-plane: in the quarter plane, the critical exponent is π/θ = π/(2θ ψ ). With the same reasoning as the three quadrants case, a random walk associated with angle θ can be seen as an inhomogeneous random walk in the half-plane, with associated angle θ ψ = θ/2 in each quadrant (see Figure 2b).

Structure of the paper. In Section 2, we start with underlining an important tool used in this analytic study of harmonic functions of random walks with small steps. This object, denominated by K(x, y), is called the kernel of the walks and is a polynomial of degree 2 in x and y which encodes the transition probabilities of the walk. Further, we set up various functional equations for generating series of harmonic functions involving K(x, y).

In Section 3, we express explicitly generating functions of harmonic functions for some models of random walks. To come to these results, we transform a functional equation (built in the previous section) into a boundary value problem in Subsection 3.2, and the solution process of this problem can be found in Subsection 3.3. The resolution of the boundary value problem involves conformal gluing functions, introduced in Subsection 3.1. We end this part with the application of the results to the example of the simple random walks (Subsection 3.4).

In Appendix A, we suppose that we do not have the symmetry of the transition probabilities of the walks. We are still able to build kernel functional equations, but unfortunately there are too many unknowns and we are not able to solve this difficult problem yet.

In Appendix B, we apply this decomposition of the domain into two convex cones to symmetric harmonic functions of symmetric random walks in the quadrant and find partially the same results as [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]. We invite the reader to refer to this appendix along the consultation of the main body on the three quadrants.

Acknowledgments. I would like to thank Kilian Raschel for his strong support, advice and suggestions. I also would like to thank Samuel Simon for his help with the English. Finally, I thank the anonymous referee for his/her comments and suggestions. Most of the work presented in this article was done during my affiliation at Department of Mathematics, Simon Fraser University, Canada & Institut Denis Poisson, Université de Tours et Université d'Orléans, France.

Kernel functional equations

Kernel of the random walks

The kernel of the random walks, denoted by K(x, y), is an important object of the study. It appears in the various functional equations of this section, and we use it later to transform a functional equation into a boundary value problem. The kernel is characterized by the transition probabilities of the associated random walk by

K(x, y) = xy   -1 i,j 1 p i,j x -i y -j -1   . ( 5
)
The kernel is a polynomial of degree two in x and y (consequence of (H1) and (H3)) and can be written as

3 K(x, y) = α(x)y 2 + β(x)y + γ(x) = α(y)x 2 + β(y)x + γ(y), (6)
x 1 1

x 4 Figure 3: Cut plane C \ ([x 1 , x 4 ])
where

α(x) = p -1,-1 x 2 + p 0,-1 x; β(x) = p -1,0 x 2 -x + p 1,0 ; γ(x) = p 0,1 x + p 1,1 ; α(y) = p -1,-1 y 2 + p -1,0 y; β(y) = p 0,-1 y 2 -y + p 0,1 ; γ(y) = p 1,0 y + p 1,1 . (7)
We also define the discriminants in the x-plane and in the y-plane:

δ(y) = β(y) 2 -4 α(y) γ(y), δ(x) = β(x) 2 -4α(x)γ(x). (8)
The discriminant δ(x) (resp. δ(y)) is a polynomial of degree three or four (this is a consequence of (H2)). Hence there are four branch points x 1 , x 2 , x 3 , x 4 (resp. y 1 , y 2 , y 3 , y 4 ), with x 4 = ∞ (resp. y 4 = ∞) when δ(x) (resp. δ(y)) is of degree 3. Examples of kernel coefficients are given in Section 3.4.

Lemma 1 (Sec. 2.3 in [START_REF] Fayolle | Random walks in the quarter plane[END_REF] and Sec. 2.5 in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]). The branch points x i are real and

|x 1 | x 2 = 1 = x 3 |x 4 | ∞. (9)
More precisely, we have

x 1 ∈ [-1, 1) and x 4 ∈ (1, ∞) ∪ {∞} ∪ (-∞, -1]
. Furthermore, on the real line, δ(x) is negative if and only if x ∈ (x 1 , x 4 ) \ {1}. Symmetric results hold for the y i , roots of δ(y).

Let Y (x) (resp. X(y)) be the multivalued solution to K(x, Y (x)) = 0 (resp. K(X(y), y) = 0). These algebraic functions can be written as

Y (x) = -β(x) ± δ(x) 2α(x) and X(y) = -β(y) ± δ(y) 2 α(y) . (10)
Over the x-plane, the function Y has two branches Y 0 and Y 1 , both meromorphic on C \ [x 1 , x 4 ] (see Figure 3). We fix the notation by choosing

Y 0 = Y -and Y 1 = Y + . On the whole of C (see [13, Thm 5.3.3]), |Y 0 | |Y 1 |. ( 11
)
On the segment [x 1 , 1], Y 0 (x) and Y 1 (x) are complex conjugate and at the branch points x i , we have Y 0 (x i ) = Y 1 (x i ) (when finite), and this common value is denoted by Y (x i ). We introduce the curve

L by L = Y 0 ([x 1 , 1]) ∪ Y 1 ([x 1 , 1]) = {y ∈ C : K(x, y) = 0 and x ∈ [x 1 , 1]}. ( 12
)
We denote by G L the domain bounded by L which contains y 1 . Figure 4 presents examples of the curve L for the simple (p 1,0 = p 0,1 = p -1,0 = p 0,-1 = 1/4) and Gouyou-Beauchamps (p

1,0 = p -1,1 = p 0,-1 = p 1,-1 = 1/4) case.
Lemma 2 (Lem. 6.5.1 in [START_REF] Fayolle | Random walks in the quarter plane[END_REF]). The curve L in (12) is symmetric with respect to the horizontal axis, smooth except at Y (1) = 1 where it may have a corner point. At this point, the angle between the curve and the segment [y 1 , 1] is given by by (4).

Remark 3. The quotient

-1 i,j 1 ijp i,j ( -1 i,j 1 i 2 p i,j )•( -1 i,j 1 j 2 p i,j )
is the coefficient of correlation of the increments of the walk.

Similarly, we define the curve

M = X 0 ([y 1 , 1]) ∪ X 1 ([y 1 , 1]
), and the same symmetric results and notations hold. One has the following automorphism relations (see [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Sec. 5.3]):

X 0 : G L \ [y 1 , 1] → G M \ [x 1 , 1]
and

Y 0 : G M \ [x 1 , 1] → G L \ [y 1 , 1]
are conformal and inverse of one another. In particular, we have then X 0 (Y (x 1 )) = x 1 and Y 0 (X(y 1 )) = y 1 .

1 y 1 Y (x 1 ) θ θ 1 y 1 Y (x 1 )

A first functional equation

From the properties (P1) and (P2), we deduce a functional equation satisfied by the generating function H(x, y) defined in (2) for random walks under the hypothesis (H1)

K(x, y)H(x, y) = K(x, 0)H -0 (x -1 ) + K(0, y)H 0-(y -1 ) -K(0, 0)H 0,0 , (13)
where

H -0 (x -1 ) = i 0 f (i, 1)x i-1 , H 0-(y -1 ) = j 0 f (1, j)y j-1 and H 0,0 = f (1, 1). ( 14
)
This equation looks similar to the functional equation for discrete harmonic functions of random walks in a quadrant

Q = {(i, j) ∈ Z 2 : i 1 and j 1} (15)
written in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Sec. 2]. In this case, we assume that random walks satisfy the hypotheses (H1), (H2), (H3), (H4). We also ask the associated discrete harmonic functions to have the properties ( P1), ( P2), ( P3) where ( P1) For all i 1 and j 1, f (i, j) = -1 i 0 ,j 0 1 p i 0 ,j 0 f (i + i 0 , j + j 0 );

( P2) If i 0 or j 0, then f (i, j) = 0;

( P3) If i > 0 and j > 0 then f (i, j) > 0.

The generating function of such harmonic functions f

H(x, y) = i,j 1 f (i, j)x i-1 y j-1 (16)
satisfies the equation

K(x, y) H(x, y) = K(x, 0) H -0 (x) + K(0, y) H 0-(y) -K(0, 0) H 00 , (17)
where

H -0 (x) = i 1 f (i, 1)x i-1 , H 0-(y) = j 1 f (1, j)y j-1 and H 0,0 = f (1, 1). ( 18
)
As noted above, the structure of equations ( 13) and ( 17) are the same except that in the case of ( 13) there are infinitely many terms with positive and negative powers of x and y. As noticed in [START_REF] Raschel | On walks avoiding a quadrant[END_REF], this difference is not anecdotal, the series are not convergent anymore (for instance, it is not possible to evaluate them) and almost all the methodology of [START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF][START_REF] Raschel | Counting walks in a quadrant: a unified approach via boundary value problems[END_REF] can no longer be performed.

This difficulty appears as well in the study of counting walks avoiding a quadrant and in order to avoid this situation, in [START_REF] Bousquet-Mélou | Square lattice walks avoiding a quadrant[END_REF] Bousquet-Mélou views the three-quarter plane as the union of three quadrants and with a combinatorial approach, gets results on the simple and the diagonal walks. In [START_REF] Raschel | On walks avoiding a quadrant[END_REF], the three quadrants are split in two symmetric convex cones of opening angle 3π 4 . In this article, we follow the same strategy as in [START_REF] Raschel | On walks avoiding a quadrant[END_REF].

Functional equations in the 3π 4 -cones

The cone C is cut into three parts: the lower part {i 1, j i -1}, the diagonal {i = j} and the upper part {j 1, i j -1}. Let L(x, y) (resp. D(x, y) and U (x, y)) be the generating function of harmonic functions evaluated in the lower part (resp. diagonal and upper part), see Figure 5. By construction, we have

H(x, y) = L(x, y) + D(x, y) + U (x, y), (19)
where

L(x, y) = i 1 j i-1 f (i, j)x i-1 y j-1 , D(x, y) = i 1 f (i, i)x i-1 y i-1 and U (x, y) = j 1 i j-1 f (i, j)x i-1 y j-1 . L(x, y) U (x, y) D(x, y) D u (x, y) D ℓ (x, y)
K(x, y)L(x, y) = -p 0,1 x + p -1,0 x 2 y + 1 2 p 1,1 + p -1,-1 x 2 y 2 -xy D(x, y) + (p 1,0 y + p 1,1 ) L 0-(y -1 ) + 1 2 p 1,1 f (1, 1), ( 20
)
with L 0-(y -1 ) = j<0 f (1, j)y j-1 .
Remark 5. The symmetry of both the random walks (H2) and the harmonic function (P4) are crucial. In fact, we can directly observe the symmetry of the positive discrete harmonic function in the case of a symmetric random walk. Let S(n) be a symmetric random walk in the three quadrants and τ (i,j) the exit time of the random walk starting at (i, j) from C. Writing (p, q) = (r cos(t), r sin(t)), the harmonic function for the Brownian motion in the cone {R exp(is) : 0 R < ∞, 0 s 3π 2 } of opening angle 3π 2 (see Figure 6) is

u 3π 2 (r, t) = r 2/3 sin 2 3 t , t ∈ 0, 3π 2 .
After the change of variable (i, j) → (-j, i) = r cos t + π 2 , r sin t + π 2 (see again Figure 6), the harmonic function for the Brownian motion avoiding the negative quadrant is

u(r, t) = r 2/3 sin 2 3 t + π 2 , t ∈ - π 2 , π .
This harmonic function is symmetric in the left cone of Figure 6 with respect to the diagonal (u r, π 4 -t = u r, π 4 + t ). Moreover, there is a direct link between the positive discrete harmonic function V and the harmonic function u for the Brownian motion (see [START_REF] Denisov | Random walks in cones[END_REF]Lem. 12]):

V (i, j) = lim n→∞ E u((i, j) + S(n)); τ (i,j) > n , (i, j) ∈ C.
The exit time τ (i,j) and the harmonic function u are both symmetric in the symmetric cone C. Therefore the positive discrete harmonic function V is symmetric as well. (i, j) -→ (-j, i) The positivity of the harmonic function is crucial to deduce its symmetry from the symmetry of the random walk. Let us give the following counter-example, again in the three quadrants: the non-symmetric function f (i, j) = i 2 -j 2 is harmonic for the symmetric simple walks. Note that this function is not positive everywhere in the three quadrants.

Proof of Lemma 4. Thanks to (P1) and (P2), we can easily write a functional equation for L(x, y) and D(x, y). Let D u (x, y) (resp. D ℓ (x, y)) be the generating function of a harmonic function evaluated in the upper (resp. lower) diagonal:

D u (x, y) = i 1 f (i -1, i)x i-2 y i-1 and D ℓ (x, y) = i 1 f (i, i -1)x i-1 y i-2 .
We have

L(x, y) =   -1 i,j 1 p i,j x -i y -j   L(x, y)+ p 0,1 y -1 + p -1,0 x D(x, y)-p 0,-1 y + p 1,0 x -1 D ℓ (x, y) -p 1,0 x -1 + p 1,1 x -1 y -1 L 0-(y -1 ) + p 1,0 x -1 f (1, 0)y -1 , (21)
D(x, y) = p 1,1 x -1 y -1 + p -1,-1 xy D(x, y) -p 1,1 x -1 y -1 f (1, 1) + p 1,0 x -1 + p 0,-1 y D ℓ (x, y) -p 1,0 x -1 f (1, 0)y -1 + p 0,1 y -1 + p -1,0 x D u (x, y) -p 0,1 y -1 f (0, 1)x -1 . (22)
Due to the symmetry of the cut and the random walks (p i,j = p j,i , hypothesis (H2)), we can simplify the last equation and get

D(x, y) = p 1,1 x -1 y -1 + p -1,-1 xy D(x, y) -p 1,1 x -1 y -1 f (1, 1) + 2 p 1,0 x -1 + p 0,-1 y D ℓ (x, y) -2p 1,0 x -1 f (1, 0)y -1 . (23)
Plugging ( 23) into [START_REF] Xu | Quarter-plane lattice paths with interacting boundaries: Kreweras and friends[END_REF] and multiplying by xy, we get [START_REF] Raschel | On walks avoiding a quadrant[END_REF].

In Equation [START_REF] Raschel | On walks avoiding a quadrant[END_REF], the bivariate generating function L(x, y) is related to the bivariate generating function D(x, y) and the univariate generating function L 0-(y -1 ). In order to simplify the functional equation [START_REF] Raschel | On walks avoiding a quadrant[END_REF], we perform the following change of variable ϕ(x, y) = (xy, x -1 ).

(24)

L ϕ (x, y) D ϕ (y) D ℓ ϕ (y)
L ϕ (x, 0) Equation ( 20) is transformed into

K ϕ (x, y)L ϕ (x, y) = -x α ϕ (y) + 1 2 β ϕ (y) D ϕ (y) + K ϕ (x, 0)L ϕ (x, 0) + 1 2 p 1,1 f (1, 1), (25) with 
  K (ϕ(x, y)) = 1 x K ϕ (x, y), L (ϕ(x, y)) = xL ϕ (x, y) = x i,j 1 f (j, j -i)x i-1 y j-1 , D (ϕ(x, y)) = D ϕ (y) = i 1 f (i, i)y i-1 , (26)
and

K ϕ (x, y) = α ϕ (x)y 2 + β ϕ (x)y + γ ϕ (x) = α ϕ (y)x 2 + β ϕ (y)x + γ ϕ (y), δ ϕ (y) = β ϕ (y) 2 -4 α ϕ (y) γ ϕ (y), δ ϕ (x) = β ϕ (x) 2 -4α ϕ (x)γ ϕ (x). ( 27
)
The functional equation (25) now has a closer structure to the functional equation in the quadrant [START_REF] Mustapha | Non-D-Finite Walks in a Three-Quadrant Cone[END_REF] (see Figure 7), and to solve this problem, we can use some tools stated in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]. The significant difference is the mixed factor in x and y in front of D ϕ (y), whereas in the quarter plane case, see Equation [START_REF] Mustapha | Non-D-Finite Walks in a Three-Quadrant Cone[END_REF], the generating function H(x, y) is decomposed into a sum of two univariate functions: one section on the x-axis, namely K(x, 0) H(x, 0), and the other on the y-axis, namely K(0, y) H(0, y).

After the change of variable ϕ, we consider positive harmonic functions associated to random walks in the quadrant (see Figure 7). In [19, Sec. 2], it is proved that for positive discrete harmonic function f in the quarter plane, the horizontal section H -0 (x) = i 1 f (i, 1)x i-1 , the vertical section H 0-(y) = j 1 f (1, j)y j-1 and further K(x, y) H(x, y) = K(x, y) (i,j)∈Q f (i, j)x i-1 y j-1 are bounded at least on {(x, y) ∈ C 2 : |x| < 1, |y| < 1}. The horizontal section L ϕ (x, 0) and the vertical section D ϕ (y) are thus bounded at least in the unit disc. Furthermore, with the functional equation (25), we can deduce that K ϕ (x, y)L ϕ (x, y) is bounded at least for |x| < 1 and |y| < 1 as well.

Expression for the generating functions

We start this section with the introduction of an important conformal gluing function W (x) that maps the domain G M to the complex plane cut along a segment. Then we transform the functional equation (25) into a boundary value problem, which is a problem involving both regularity and boundary conditions. Finally we solve this problem and end up with an explicit expression for the generating function H(x, y). The boundary value problem satisfied by the generating function D ϕ (y) and ultimately the generating function H(x, y) will be both expressed in terms of W . Further, all quantities with a ϕ-tag are objects defined from the changed random walks with kernel K ϕ (x, y), as in Section 2.3.

Conformal gluing function

Definition 6 (Conformal gluing functions). A function W is said to be a conformal gluing function for the set G M if:

• W is injective in G M ; • W is analytic on G M ; • For all x on M \ {1}, W (x) = W (x). • W is continuous on G M \ {1} and W (1) = ∞.
Remark 7. In fact, to be a conformal gluing function, W only needs to be injective in G M , analytic on G M and to satisfy W (x) = W (x) for x ∈ M, as soon as W (x) is defined and finite. Moreover, W has an infinite value at least on one point of M (or W would be constant, see [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Lem. 4]). In this article, we choose W to be infinite at 1 (if W m is a conformal gluing function taking the value ∞ anywhere else on M, then we can consider W = 1/(W m -W m (1))).

The expression of such a function is computed in [14, Sec. 2.2]. Let

T (x) = 1 1 3 -2f (x) δ ′′ (1) , f (x) = δ ′′ (x 4 ) 6 + δ ′ (x 4 ) x-x 4 if x 4 = ∞, δ ′′ (0) 6 + δ ′′′ (0)x 6 if x 4 = ∞. ( 28
) Then W (x) = sin π θ arcsin(T (x)) - π 2 2 , ( 29
)
with θ defined in (4). Moreover, there exists c = 0 such that for x in the neighborhood of 1,

W (x) = c + o(1) (1 -x) π/θ . ( 30
)
As in [14, Sec. 2.2] and [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Sec. 3.4], the expression of W (x) in ( 29) is valid only for |T (x)| 1.

When |T (x)| 1, writing arcsin(T ) = π/2 ± i ln(T + √ T 2 -1), we can write W (x) = - 1 4 T (x) + T (x) 2 -1 2π/θ -2 + T (x) -T (x) 2 -1 2π/θ . ( 31
)
Lemma 8. The function W defined in (29) has a zero at x 1 and is equal to 1 at X(y 1 ). Moreover, for x ∈ G M , it satisfies the following differential equation

δ(x)W ′ (x) 2 = π θ 2 -δ ′′ (1) 2 W (x)(1 -W (x)). (32)
Proof. We start from (29). After differentiation and squaring, we can write

W ′ (x) 2 = π θ 2 4T ′ (x) 2 1 -T (x) 2 W (x)(1 -W (x)). ( 33
)
The strategy now is to write δ(x) under its factorized form (remember that δ has four roots x 1 , 1, 1, and x 4 with possibly x 4 = ∞ when δ is of degree 3). If

x 4 = ∞, then T (x) 2 = 1-x 1 1-x and if x 4 = ∞, then T (x) 2 = (1-x 1 )(x 4 -x) (x 4 -x 1 )(1-x) . In both cases, 4T ′ (x) 2 1-T (x) 2 = -δ ′′ (1)
2δ(x) (and δ ′′ (1) < 0). We also note that T (x 1 ) = 1 which implies that W (x 1 ) = 0 (and because W is injective, it is the only zero). The only zero of W ′ is at X(y 1 ) (see [START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method[END_REF]Sec. 5.3]), and from (33), we deduce that W (X(y 1 )) = 1.

Similarly, we define W a conformal gluing for G L in the sense of Definition 6. We show that W and W are strongly related as

W (Y 0 (x)) = -W (x) + 1.
Indeed, see [18, Thm 6 and Rem 6]4 , W (Y (x)) is a conformal gluing for G M but may not be the one defined by (29). However, we know for sure that W (Y (x)) = aW (x) + b, with a, b ∈ C. By plugging in x = x 1 and x = X(y 1 ), we deduce a = -1 and b = 1.

Thereafter, we are interested in a conformal gluing function which vanishes at 0. We consider w defined by

w(x) = W (x) -W (0). ( 34
)
The function w is a conformal gluing function in the sense of Definition 6, vanishes at 0, and satisfies the differential equation (which can be easily derived from (32))

δ(x)w ′ (x) 2 = π θ 2 - δ ′′ (1) 2 (w(x) + W (0)) (w(X(y 1 )) -w(x)) . ( 35
)
Remark 9. Let w (resp. w ϕ and w ϕ ) be a conformal gluing function for G L (resp. G Mϕ and G Lϕ ) which vanishes at 0. Similar results and properties hold for those functions.

The conformal gluing functions w ϕ and w ϕ are defined from θ ϕ which can be computed with (4) and transition probabilities ϕ(p i,j ) = p i-j,i . The angles θ ϕ and θ are simply related (see Figure 2a), as stated in the following lemma. Relations between θ and θ ϕ will be ultimately interpreted as relations between the growths of harmonic functions.

Lemma 10. The angle between the curve L ϕ and the segment [y ϕ,1 , 1] is given by

θ ϕ = π - θ 2 , ( 36
)
with θ defined in (4).

Proof. Writing c = -1 i,j 1 ijp i,j

( -1 i,j 1 i 2 p i,j )•( -1 i,j 1 j 2 p i,j ) , we have θ = arccos(-c). Similarly, we write θ ϕ = arccos(-c ϕ ). We have

c ϕ = 1 -c 2 ⇒ θ ϕ = arccos - 1 -c 2 = π - 1 2 arccos(-c) = π - θ 2 .

Boundary value problem

Lemma 11. The generating function D ϕ (y) can be analytically continued from the unit disc D to the domain G Lϕ ∪ D and is continuous on G Lϕ \ {1}. Moreover, for all y ∈ L ϕ \ {1}, D ϕ (y) satisfies the following boundary condition

g ϕ (y) g ′ ϕ (y) D ϕ (y) - g ϕ (ȳ) g ′ ϕ (ȳ) D ϕ (ȳ) = 0, (37)
where g ϕ is defined below in (40) and w ϕ is a conformal gluing function defined in Section 3.1.

Remark 12. The boundary value problem in Lemma 11 is very close to the boundary value problem stated in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Lem. 3].

Proof. We first assume the analyticity and continuity of D ϕ (y) and we begin to prove the boundary (37). We start to evaluate the functional equation (25

) at Y 0,ϕ (x) for x close to [x ϕ,1 , 1): 0
= -x α ϕ (Y ϕ,0 (x)) + 1 2 β ϕ (Y ϕ,0 (x)) D ϕ (Y ϕ,0 (x)) + K ϕ (x, 0)L ϕ (x, 0) + 1 2 p 1,1 f (1, 1). (38)
By letting x go to any point of [x ϕ,1 , 1] with a positive (resp. negative) imaginary part, we obtain two new equations. We subtract these two equations and get:

X 0 (y) α(y) + 1 2 β(y) D ϕ (y) -X 0 (ȳ) α(ȳ) + 1 2 β(ȳ) D ϕ (ȳ), y ∈ L ϕ \ {1}.
With [START_REF] Cohen | Boundary value problems in queueing system analysis[END_REF] this last equation can be simplified as

δ ϕ (y)D ϕ (y) -δ ϕ (ȳ)D ϕ (ȳ) = 0, y ∈ L ϕ \ {1}. (39)
This boundary condition is the homogeneous equation of the boundary condition in [20, Lem. 5],

with the same difficulty to deal with a non-meromorphic prefactor in G Lϕ , namely δ ϕ (y) in (39). The idea is to introduce a new function F ϕ , analytic in G Lϕ with finite limits on L ϕ , which satisfies the decoupling condition

δ ϕ (y) δ ϕ (ȳ) = F ϕ (y) F ϕ (ȳ) , ∀y ∈ L ϕ \ {1},
which allow us to write the boundary condition (39) as

F ϕ (y)D ϕ (y) -F ϕ (ȳ)D ϕ (ȳ) = 0.
Such a function F ϕ is defined in [START_REF] Raschel | On walks avoiding a quadrant[END_REF]Sec 3.5]. It can be constructed from a conformal mapping g ϕ from G Lϕ onto the unit disc, with the property that g ϕ (ȳ) = g ϕ (y). We have F ϕ (y) = gϕ(y) g ′ ϕ (y) and one can choose g ϕ defined in G Lϕ by

g ϕ (y) = h 2 w ϕ (Y ϕ (x ϕ,1 )) w ϕ (y) -1 , (40)
with h(y) = -y + y 2 -1 and w ϕ given in Section 3.1. Finally, the boundary condition (39) can be rewritten as (37) With the same reasoning as [START_REF] Raschel | On walks avoiding a quadrant[END_REF]Sec. 3.3], we can prove that D ϕ (y) is analytic in G Lϕ , has finite limits on L ϕ \ {1} and then is continuous on G Lϕ \ {1}. We start with proving the analytic continuation of D ϕ (y) from the unit disc D to G Lϕ . From the functional equation ( 25), for

y ∈ {y ∈ C : |X ϕ,0 (y)| 1} ∩ D}, 2K ϕ (X ϕ,0 (y), 0) L ϕ (X ϕ,0 , 0) + δ(y)D ϕ (y) + p 1,1 f (1, 1) = 0.
The latter equation defined on G Lϕ ∩ D can be continued in G Lϕ ∪ D. The generating function D ϕ (y) is analytic on D, and on G Lϕ ∪ D \ D, D ϕ (y) has the same singularities as δ ϕ (y) and X ϕ,0 (y), namely the branch cut [y 1 , y 4 ]. However, this segment does not belong to G Lϕ ∪ D\D , then D ϕ (y) can be analytically continued to G Lϕ . Then we prove that D ϕ (y) has finite limits on L ϕ \ {1}. From equation (38), we only need to study the zeros of [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Cor. 5.3.5]), it recurs to study the zeros of δ ϕ for y in G Lϕ ∪ D \ D. The discriminant δ ϕ vanishes at y ϕ,1 , 1, y ϕ,4 and only 1 belongs to the last set. Then D ϕ (y) has finite limits on L ϕ \ {1}.

x α ϕ (Y ϕ,0 (x)) + 1 2 β ϕ (Y ϕ,0 ) for x in [x ϕ,1 , x ϕ,2 ]. Thanks to the relation X ϕ,0 (Y ϕ,0 (x)) = x valid in G Mϕ (see

Solution of the boundary value problem

In this section we solve the boundary value problem stated in Lemma 11 and obtain in Theorem 13 an explicit expression for the generating function D ϕ (y) = i 1 f (i, i)y i-1 . Finally, Theorem 14 gives a formal expression of the generating function H(x, y) = (i,j)∈C f (i, j)x i-1 y j-1 . Along the proof Theorem 13, we also state Lemma 16 which gives an expression of D ϕ (y) for a family of harmonic function non necessarily positive.

Theorem 13. The generating function D ϕ (y) can be written as

D ϕ (y) = - f (1, 1) w ′ ϕ (0) π θ ϕ - δ ′′ ϕ (1) 2 δ ϕ (y) 1 -W ϕ (0) W ϕ (y), (41)
with θ ϕ , w ϕ (y) and W ϕ defined in Subsection 3.1 and δ ϕ in (27).

Theorem 14. Let f be a harmonic function associated to a random walk in the three-quarter plane with hypotheses (H1), (H2), (H3) and (H4). The generating function H(x, y) of f can be formally written as the finite sum of convergent generating functions (see [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF])

H(x, y) = - f (1, 1) w ′ ϕ (0) π θ ϕ 1 -W ϕ (0) - δ ′′ ϕ (1) 2 1 K(x, y) 1 -W ϕ (y -1 ) + 1 -W ϕ (x -1 ) 2 - W ϕ (xy) δ ϕ (xy) x -1 + y -1 α ϕ (xy) + β ϕ (xy)   + W ϕ (xy) δ(xy)   . ( 42
)
Remark 15. In equation ( 42), up to the multiplicative constantf (1,1)

w ′ ϕ (0) π θϕ 1 -W ϕ (0) - δ ′′ ϕ (1)
2 , the terms

1 K(x, y)   -y -1 α ϕ (xy) + 1 2 β ϕ (xy) δ ϕ (xy) W ϕ (xy) + 1 -W ϕ (y -1 ) 2   , W ϕ (xy) δ ϕ (xy)
,

and 1 K(x, y)   -x -1 α ϕ (xy) + 1 2 β ϕ (xy) δ ϕ (xy) W ϕ (xy) + 1 -W ϕ (x -1 ) 2   ,
contribute respectively for the generating functions L(x, y) in the lower cone {i 1, j i -1}, the diagonal D(x, y) and the generating function U (x, y) in the upper cone {j 1, i j -1}.

Lemma 16. Let g ϕ and D ϕ be as in Lemma 11. There exists a polynomial P ∈ R[y] such that

g ϕ (y) g ′ ϕ (y) D ϕ (y) = P ( w ϕ (y)). (43)
Remark 17. More generally, for any polynomial P of degree n, we get a harmonic functions satisfying (P1) and (P2) (but not necessarily (P3)). Remark 19 illustrates this statement.

Proof of Lemma 16. The conformal gluing function w ϕ (y) defined in Subsection 3.1 transforms the domain G Lϕ into the complex plane cut by the segment I ϕ = [ w ϕ (Y ϕ (x ϕ,1 )), 1). Let w -1 ϕ be the inverse function of w ϕ . We denote by w -1 ϕ (z) + and w -1 ϕ (z) -the left and right limits of w -1 ϕ on I ϕ . The latter are complex conjugate on I. Let

∆ ϕ (y) = 1 gϕ(y) g ′ ϕ (y) D ϕ (y) and Z ϕ = z ∈ I ϕ : g ϕ g ′ ϕ D ϕ • w -1 (z) = 0 .
The set Z ϕ is finite, otherwise by the principle of isolated zeros, gϕ(y) g ′ ϕ (y) D ϕ (y) would be equal to zero on the whole of G Lϕ . Let N ϕ be the cardinal of Z ϕ . With Lemma 11, we have

∆ ϕ • w -1 (z) + -∆ ϕ • w -1 (z) -= 0 ∀z ∈ I ϕ , ∆ ϕ • w -1 is analytic on C \ I ϕ , which implies that ∆ ϕ • w -1 is holomorphic in C. Furthermore, as ∆ ϕ • w -1 ϕ (z) - z 0 ∈Zϕ 1 (z -z 0 )
is bounded in C, then by Liouville's theorem, the latter is constant in the whole complex plane. In addition, there exists a polynomial P of degree N ϕ such that

gϕ g ′ ϕ D ϕ • w -1 ϕ (z) = P (z), hence (43).
Proof of Theorem 13. In particular take P of degree one in (43), the generating function D ϕ (y) can be written as

D ϕ (y) = µ w ϕ (y) + ν g ϕ (y)/g ′ ϕ (y) . ( 44
)
The expression of . Furthermore, as the function w is decreasing on (X(y 1 ), 1), taking the square root of (35) gives

δ(x)w ′ (x) = - π θ - δ ′′ (1) 2 w(Y (x 1 )) -w(y) w(y) + W (0).
Then, thanks to equation ( 30), there exists a c g = 0 such that for y in the neighborhood of 1,

g ϕ (y) g ′ ϕ (y) = - w ϕ (y) w ϕ (Y ϕ (x ϕ,1 )) θ ϕ π d ϕ (y) - d ′′ ϕ (1) 2 w ϕ (y) + W ϕ (0) = c g + o(1) (1 -y) π/(2θϕ)-1 .
and finally, there exists a c ′ D = 0 such that

D ϕ (y) = c ′ D + o(1) (1 -y) π/θϕ-π/(2θϕ)+1 = c ′ D + o(1) (1 -y) π/(2θϕ)+1 = c ′ D + o(1) (1 -y) π/(2π-θ)+1 . (45)
The asymptotics of D ϕ (y) in the last equation ( 45) is exactly the asymptotics found in [17, Eq. (1.4)], see Remark 18. By unicity of the positive harmonic function, P is a polynomial of degree one. Going back to (44) and noticing that g ϕ (0)/g ′ ϕ (0) = 0, we deduce ν = -w ϕ (0) = 0. Then, computing the first term in the expansion of (44) at y = 0, we find µ = f (1,1) w ′ ϕ (0) . Remark 18. In [START_REF] Mustapha | Non-D-Finite Walks in a Three-Quadrant Cone[END_REF], Mustapha computes the asymptotic of the number of small steps walks excursions in the three-quarter plane. In particular, his equation (1.4) shows that the polynomial exponent α C in the three quadrants can be easily expressed as a function of the critical exponent α Q in the quadrant. From this equation we have for y close to 1, a constant c H = 0 such that H(x, 0) = c H +o (1) (1-x) π/(2π-θ) , and we deduce that there exists c D = 0 such that D ϕ (y) =

c D +o (1)
(1-y) π/(2π-θ)+1 . Let us point out that the positivity of the harmonic function (property (P3)) is crucial in the proof of asymptotic results in [START_REF] Mustapha | Non-D-Finite Walks in a Three-Quadrant Cone[END_REF].

Proof of Theorem 14. The expression of D ϕ (y) in Theorem 13 suffices to calculate H(x, y). Indeed, to have an expression of H(x, y), we only need to have one of D(x, y) and L(x, y) (thanks to the symmetry of the problem we have U (x, y) = L(y, x)). We begin by plugging in Y ϕ,0 (x) to the functional equation (25) and get

K ϕ (x, 0)L ϕ (x, 0) = - 1 2 δ ϕ (Y ϕ,0 (x))D ϕ (Y ϕ,0 (x)) + p 1,1 f (1, 1) = - f (1, 1) 2   1 w ′ ϕ (0) π θ ϕ - δ ′′ ϕ (1) 2 1 -W ϕ (0) 1 -W ϕ (x) + p 1,1   . ( 46
) L(x, y) U (x, y) D(x, y) D u (x, y) D ℓ (x, y) ϕ U ϕ L L ϕL (x, y) U ϕU (x, y) D ϕL (y) = D ϕU (y) D u ϕU (y) D ℓ ϕL (y)
Figure 9: Decomposition of the three-quarter plane, associated generating functions and changes of variable Remark 21. Notice the symmetry between x and y in (58). In the case of a symmetric random walk (i.e. if p i,j = p j,i ) the two functional equations are the same.

Proof. The strategy of the proof is similar to the proof of Lemma 4. In addition to the functional equations for L(x, y) and D(x, y), see ( 21) and ( 22), we can write a functional equation for U (x, y):

U (x, y) =   -1 i,j 1 p i,j x -i y -j   U (x, y)+ p 1,0 x -1 + p 0,-1 y D(x, y)-p -1,0 x + p 0,1 y -1 D u (x, y) -p 0,1 y -1 + p 1,1 x -1 y -1 U -0 (x -1 ) + p 0,1 y -1 f (0, 1)x -1 . (59)
Mixing equations ( 21), (59) and ( 22) and multiplying by xy, we get (58).

In this system of two functional equations, the bivariate generating function U (x, y) (resp. L(x, y)) is related to the bivariate generating functions D(x, y), D ℓ (x, y) (resp. D u (x, y)) and the univariate generating function U -0 (x -1 ) (resp. L 0-(y -1 )). In order to simplify this system of functional equations, we perform two changes of variables ϕ L for the lower part and ϕ U for the upper part with ϕ U (x, y) = (x, x -1 y) and ϕ L (x, y) = (xy, x -1 ).

This change of variables transforms the upper part {j 1, i j -1} into the left quadrant {i -1, j 0} and the lower part {i 1, j i -1} into the right quadrant {i 1, j 0} (see Figure 9). Note that the diagonal is changed by both ϕ L and ϕ U into the positive y-axis (see Figure 9). The step set is changed as well, and walks in the three-quadrant can be seen as inhomogeneous walks in the half-plane with two different step sets in each quadrant and a mixed step set on the positive y-axis (see Figure 10). Other type of inhomogeneous walks are studied in [START_REF] Bouaziz | Discrete harmonic functions on an orthant in Z d[END_REF][START_REF] Buchacher | Inhomogeneous restricted lattice walks[END_REF].

In the symmetric case (symmetry of the random walks and hence the symmetry of the unique positive harmonic function), the study of harmonic functions of random walks in the three quadrant is then equivalent to the study of harmonic functions of random walks reflected on the y-axis and constrained by the x-axis in the positive quadrant (see Figure 11). Articles [START_REF] Beaton | Exact solution of some quarter plane walks with interacting boundaries[END_REF][START_REF] Xu | Quarter-plane lattice paths with interacting boundaries: Kreweras and friends[END_REF] work on problem in the same vein: their authors study walks in the quadrant with different weights on the boundary.

p 1,0 p 1,1 p 0,1 p -1,0 p -1,-1 p 0,-1 p 1,0 p 1,2 p 0,1 p 0,-1 p -1,0 p 0,1 p 1,1 p 1,0 p 1,0 p 0,1 p 1,1 p -1,0 p 0,-1 p -1,-1 p 1,0 p 0,1 p 1,1 p -1,0 p 0,-1 p 1,1 p 1,0 p 0,1 p -1,-1 p -1,0 p 0,-1 p 1,1 p 1,0 p 0,1 p 0,-1 p 1,1 p 1,0 p 0,1 p -1,-1 p -1,0 p 0,-1 p 1,1 p 1,0 p 0,1 p -1,0
Figure 10: Random walks avoiding a quadrant (on the left) can be seen as walks in the half-plane with probability transitions ϕ U ((p i,j ) -1 i,j 1 ) in the left quadrant and ϕ L ((p i,j ) -1 i,j 1 ) on the right quadrant (on the right). Unfortunately, due to number of unknown functions, we are not able to solve this system of functional equations yet. Let us end this section by pointing out that the split cone along the diagonal can also be related to the Join-the-Shortest-Queue model (JSQ). Supposed that there are two lines (see Figure 12, left), each of them with a service time exponentially distributed of rate r 1 and r 2 and that the customers arrive according to a Poisson process. The clients choose the shorter queue and if both lines happened to have the same length, the costumers pick one or the other with probability p 1 or p 2 . A common question in queuing theory is to obtain closed-form expression for the stationary distribution. This JSQ problem can be modeled by random walks in the quarter plane split into two octants, each axis representing the length of each line (see Figure 12, right). The symmetric case (when r 1 = r 2 and p 1 = p 2 = 1/2) is solvable, see [START_REF] Adan | Analysis of the asymmetric shortest queue problem[END_REF][START_REF] Foley | Join the shortest queue: stability and exact asymptotics[END_REF][START_REF] Kurkova | Malyshev's theory and JS-queues. Asymptotics of stationary probabilities[END_REF] and [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Chap. 10] for reference. On the other side, the non-symmetric case is still an open problem.

B Discrete harmonic functions in the quadrant

Even if discrete harmonic functions of random walks in the quarter plane are already studied in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF], we want to point out that the strategy of splitting the domain into three parts (the upper part, the diagonal and the lower part) can also be performed in the quarter plane Q defined in [START_REF] Foley | Join the shortest queue: stability and exact asymptotics[END_REF] allow us to find explicit expressions of harmonic functions. We suppose that random walks satisfy the hypotheses (H1), (H2), (H3), (H4), and associated discrete harmonic functions f the properties ( P1), ( P2), ( P3). The generating function of such harmonic functions is defined in [START_REF] Kurkova | Malyshev's theory and JS-queues. Asymptotics of stationary probabilities[END_REF]. We split the quadrant Q in three parts: the lower part {i 2, 1 j i -1}, the diagonal and the upper part {j 2, 1 i j -1}. As in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF], by construction we have H(x, y) = L(x, y) + D(x, y) + U (x, y),

where L(x, y) = i 2 1 j i-1 f (i, j)x i-1 y j-1 denotes the generating function of harmonic functions associated to random walks ending in the lower part, D(x, y) = i 1 f (i, i)x i-1 y i-1 ending on the diagonal and U (x, y) = j 2 1 i j-1 f (i, j)x i-1 y j-1 in the upper part (see Figure 13).

We can write a functional equation for each section and finally get a functional equation in terms of L(x, y), L -0 and D(x, y). Finally, noticing that K(x, 0) = x 4 and that H(x, 0) = f (1, 1) + L(x, 0), from (69) we get

H(x, 0) = f (1, 1) (1 -x) 2 , (71)
and this result matches the computation in [19, Eq. 2.6].

Figure 1 :

 1 Figure 1: Models satisfying (H1), (H2) and (H3).

Figure 4 :

 4 Figure 4: The curve L for Gouyou-Beauchamps model (on the left) and for the simple model (on the right). In the case of Gouyou-Beauchamps model θ = π/4, and in the case of the simple model, the curve is simply the unit circle and θ = π/2.

Figure 5 :

 5 Figure 5: Decomposition of the three-quarter plane and associated generating functions

3π 2 Figure 6 :

 26 Figure 6: Various cones of opening angle 3π 2 , see Remark 5. On the left the cone {R exp(is) : 0 R < ∞, 0 s 3π 2 } and the right the cone {R exp(is) : 0 R < ∞, -π 2 s π}

Figure 7 :

 7 Figure 7: Lower convex cone after the change of variable

ϕ

  (y) where the function g is defined in (40) can be simplified into

Figure 11 :

 11 Figure 11: Random walks with symmetric probability transitions avoiding a quadrant can be seen as random walks reflected on the y-axis and constrained by the x-axis.

Figure 12 :Figure 13 :

 1213 Figure 12: Left: the JSQ model can be represented as a system of two queues, in which the customers choose the shortest one (the green one, on the picture). Right: representation of the JSQ model as an inhomogeneous random walk in the quadrant

Lemma 22 .

 22 For any random walks with property (H1) and (H2), the generating function L(x, y)On the other side, we have             δ ψ (x) = (x 2 -6x+1)(x-1)we have then W ψ (x) = 1 -64x 2 4(x -1) 4 .

For example the random walk with transition probabilities p-1,0 = p0,-1 = 1/4 and p1,1 = 1/2.

For instance the random walk with transition probabilities p1,0 = p1,1 = p0,-1 = 1/5 and p-1,0 = 2/5.

We follow the notations of[START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF] which are slightly different from the usual ones in[START_REF] Fayolle | Random walks in the quarter plane[END_REF]. Indeed,[START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF] points out that the kernel in (5) is the reciprocal one of the kernel Q(x, y) in[START_REF] Fayolle | Random walks in the quarter plane[END_REF]: K(x, y) = x 2 y 2 Q(1/x, 1/y).

In fact, in[START_REF] Raschel | Counting walks in a quadrant: a unified approach via boundary value problems[END_REF] Rem. 6], it is said that if W is a suitable mapping, then any aW (x)+b cW (x)+d with a, b, c, d ∈ C such that adbc = 0 is also a suitable mapping. In our case, we are interested in mapping with a pole at 1. As Y (1) = 1, by letting x goes to 1, we get c = 0. Hence we only need to find a and b such that W (Y (x)) = aW (x) + b.

A. Trotignon was supported by the Austrian Science Fund (FWF) grant FWF05004.

Together with (25), we can write an expression of L ϕ (x, y) and thanks to the symmetry, we get for free an expression of U ϕ (x, y) = L ϕ (x -1 , y), and with a series expansion of L ϕ (x, y), U ϕ (x, y) and D ϕ (y), the value of f (i, j) for all (i, j) ∈ C. Nevertheless, if we still want to write a formal expression of the generating function H(x, y) defined in [START_REF] Beaton | Exact solution of some quarter plane walks with interacting boundaries[END_REF], we can use the following relations K ϕ y -1 , 0 L ϕ (y -1 , 0) = K(0, y)L 0,-(y -1 ) and D ϕ (xy) = D(x, y),

and from them we can write

We have now all the ingredients to write the following theorem which gives an expression of H(x, y).

Example of the simple random walk

In this subsection, we apply the results of Subsection 3.3 to the simple random walk. The change of variable ϕ defined in (24) transforms the simple random walk into the Gessel random walk (see Figure 8). We first need to compute the angle θ ϕ , the conformal gluing functions W ϕ , W ϕ and w ϕ .

Then we get an expression of D ϕ (y) and L ϕ (x, 0) and end up with a series expansion of H(x, y).

The simple walk is defined by p 1,0 = p -1,0 = p 0,-1 = p 0,1 = 1 4 . We easily have θ = π 2 and together with (36), we deduce θ ϕ = 3π 4 . From Equations ( 5) and ( 26), the kernel after the change of variable is defined by

Conformal gluing functions. On one side, we have

, and finally

On the other side,

, and finally

Series expansion of D ϕ (y) and L ϕ (x, 0). Noticing that w ϕ (0) = -16/9, W ϕ (0) = 0, thanks to Theorem 13, we have

W ϕ (y)

Thanks to (46) with K ϕ (x, 0) = 1/4, we have

Series expansion of L ϕ (x, y). To begin with, we write an expression of K ϕ (x, y)L ϕ (x, y). With (25), we get

Remark 19. The harmonic function f (i, j) = ij is a positive harmonic function of the simple walk in the quarter plane. In the three-quarter plane, this function is still harmonic, but does not satisfy (P3). However, there exists a polynomial P as in (43

A quick study of exponent in the same idea of Section 3.3 shows that P should be of degree 2. Then there exists (a, b, c) ∈ R such that

We have

, and evaluating this last equation with y = 0 gives c = 0. Then, dividing by w ϕ (y) and letting y go to 0 gives b = - 9 16 . Finally, examining the first and second term of the expansion of (56) at y = 0 gives a = (57)

A Non-symmetric case

In this section (except clearly stated) we suppose that the probability transitions of the random walks we consider are not symmetric and a priori neither is the harmonic function. In other words, the random walks satisfy hypotheses (H1), ( Ĥ2), (H3) and (H4) with ( Ĥ2) We assume that the transition probabilities p 0,0 = p -1,1 = p 1,-1 = 0.

and discrete harmonic functions associated to these random walks satisfy the properties (P1), (P2), (P3) but not necessarily (P4). Like in Subsection 2.3, we split the three-quadrant into two symmetric convex cones of opening angle 3π 4 , and split the generating function H(x, y) into three generating functions: L(x, y) for harmonic function in the lower part, D(x, y) for the ones on the diagonal and U (x, y) for the ones in the upper part (see [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]). Since we do not have the symmetry conditions anymore, instead of one functional equation as in Lemma 4, we end up with a system of two functional equations.

Lemma 20. For any random walks with property (H1), the generating functions L(x, y) and U (x, y) satisfy the following system of functional equations

satisfies the following functional equation

with

In order to simplify the last functional equation (61), we apply the following change of variables

The equation ( 61) is changed into

and

The angle θ ψ is also simply related to θ (see Figure 2b) as

The scheme of the proof is the same as the proof of Lemma 10. Writing θ ψ = arccos(-c ψ ), we have

2 , hence (66). The functional equation ( 63) is very close to the functional equation in the three-quarter plane (25), the only difference is the additional term p 0,1 x. We can then write a similar boundary value problem as Lemma 11.

Lemma 23. The generating function D ψ (y) is analytic in G L ψ and continuous on G L ψ \ {1}. Moreover, for all y ∈ L ψ \ {1}, D ψ (y) satisfies the following boundary condition

where

and w ψ is a conformal gluing function defined in Subsection 3.1. Moreover, the expression of D ψ (y) is the same as in the three-quarter plane and we have the following theorem, which is a quarter plane equivalent to Theorem 13.

Theorem 24. The generating function D ψ (y) can be written as

with θ ψ , w ψ (y) and W ψ defined in Subsection 3.1 and δ ψ in (65).

From the expression of D ψ in (68) and the functional equation ( 63), we get an expression for L ψ (x, 0):

1 -W (0) 1 -W ψ (x) -(p 0,1 x + 1 2 p 1,1 ) f (1, 1), (69) and from (30) there exists k = 0 such that, for x in the neighborhood of 1,

This asymptotic results matches that in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]. We end this section with the example of the simple random walk. The application ψ changes the simple random walk into the Gouyou-Beauchamps random walk (see Figure 14). In order to compute L(x, 0), we need to calculate θ ψ , δ ψ , w ψ and W ψ . We easily have θ ψ = π


K Berent

M Calvo-Dahlborg

Jakub Cieslak

Ulf Dahlborg

Mössbauer investigations of the FeCoNiPdx and CrxFeCoNiPd high entropy alloys

whether they are published or not. The documents may come   L'archive ouverte pluridisciplinaire


Xixi Dong
email: xixi.dong@brunel.ac.uk

Peijie Li

Sajjad Amirkhanlou

Shouxun Ji
email: shouxun.ji@brunel.ac.uk

Pjotr S Popel

Ulf Dahlborg

Monique Calvo-Dahlborg

evidence of disruption of Si-rich microstructure in engineering-lightweight Al-12.2at.%Si alloy melt above liquidus temperature

The exploration of microstructures in high temperature alloy melts is important for manufacturing of metallic components but extremely challenging. Here, we report experimental evidence of the disruption of Si-rich microstructure in engineering-lightweight Al-12.2at.%Si alloy melt at 1100 °C, via melt-spinning (MS) of Al 1-x Si x (x = 0.03,0.07,0.122,0.2) alloy melts from different initial melt temperatures, 800 °C and 1100 °C, under the super-high cooling rate of ~ 10 6 °C/s, in cooperation with the small angle neutron scattering (SANS) measurement. Si particles in 1100 °C MS alloys are abnormally smaller and increased in number at Al-12.2at.%Si, compared with 800 °C MS alloys, which demonstrates the disruption of Si-rich microstructure in Al-12.2at.%Si alloy melt at 1100 °C. SANS experiment verifies that large quantities of small (0-10 nm) Si-rich microstructures and small quantities of large (10-240 nm) Si-rich microstructures exist in Al-12.2at.%Si alloy melt, and the large Si-rich microstructures disrupt into small Si-rich microstructures with increasing of melt temperature from 800 to 1100 °C. Microstructure analysis of the MS alloys indicates that the large Si-rich microstructures in Al-12.2at.%Si alloy melt are probably aggregates comprising multiple small Si-rich microstructures. This work also provides a pathway for the exploration of microstructures in other high temperature alloy melts.

 . However, the exploration of microstructure in alloy melts is always extraordinary challenging, due to the high temperature of alloy melts that usually ranges from several hundreds to ~ 3500 °C.

The Al-Si based alloys are important lightweight engineering alloys in automotive, aerospace and other industries, which constitute ~ 90 % of all aluminium shape castings 8-14 . As a special system, Al-Si alloy comprises two elements with a large melting point discrepancy at 754 °C (Al: 660 °C, Si: 1414 °C), which provides theoretical possibility for the existence of Si-rich microstructure in the molten state 15 . In addition, the measurement of irreversible changes in physical properties such as density during heating and cooling cycles supported the microstructure evolution in Al-Si alloy melts with changing of melt temperature 16 , however, we can hardly get information of the detail microstructure evolution in the Al-Si alloy melts from the measurement of physical properties.

open

X-ray diffraction (XRD) was applied to study the microstructure in Al-Si alloy melts [START_REF] Bian | Liquid structure of Al-12.5% Si alloy modified by antimony[END_REF][START_REF] Srirangam | Effect of strontium on liquid structure of Al-Si hypoeutectic alloys using high-energy X-ray diffraction[END_REF][START_REF] Dahlborg | Structure of molten Al and eutectic Al-Si alloy studied by neutron diffraction[END_REF] , and diffraction can reflect the atomic scale local short-range order structure information in Al-Si alloy melts such as the statistical mean first or second nearest neighbor atomic distance, as the measured intensity of diffraction is proportional to the scattering length of individual atomic nuclei. Unfortunately, it is hard to reveal the microstructure that is larger than the atomic scale in Al-Si alloy melts under diffraction. Molecular dynamics (MD) simulations were also performed to study the microstructure in Al-Si alloy melts [START_REF] Ji | Ab initio molecular dynamics simulation on temperature-dependent properties of Al-Si liquid alloy[END_REF][START_REF] Wang | Structure and dynamics of liquid Al 1-x Si x alloys by ab initio molecular dynamics simulations[END_REF][START_REF] Khoo | Ab initio molecular dynamics simulations of molten Al 1-x Si x alloys[END_REF] , and scarce information on the nanoscale microstructure was obtained by MD simulations, due to the limited number of atoms available to be calculated. In recent years, the technical advances in X-ray synchrotron radiation and imaging [START_REF] Cai | 4D synchrotron X-ray tomographic quantification of the transition from cellular to dendrite growth during directional solidification[END_REF][START_REF] Guo | Dendritic evolution during coarsening of Mg-Zn alloys via 4D synchrotron tomography[END_REF][START_REF] Xu | Heterogeneous nucleation and grain growth of inoculated aluminium alloys: an integrated study by in-situ X-radiography and numerical modelling[END_REF][START_REF] Aucott | Revealing internal flow behaviour in arc welding and additive manufacturing of metals[END_REF] have made the direct imaging of micron-scale structure such as the growing dendrite in alloy melts available during the solidification process, but the direct imaging of nanoscale microstructure in alloy melts is still fiction technically. However, this does not mean that no way for the unveiling of the mystery of the detail microstructure evolution in Al-Si alloy melts that has been long-standing blind spot of science.

In the present work, we report experimental evidence of the disruption of Si-rich microstructure in the Al-12.2at.%Si alloy melt above liquidus temperature, via the melt-spinning of a serials of Al 1-x Si x (x = 0.03,0.07,0.122,0.2) alloy melts from two different initial melt temperatures, 800 °C and 1100 °C, under the super-high cooling rate of ~ 10 6 °C/s, in cooperation with the small angle neutron scattering measurement of the Al-12.2at.%Si alloy melt.

Results

Abnormal decrease of Si particle size in 1100 °C MS Al-12.2at.%Si alloy. XRD analysis of MS

Al-Si alloys. Figure 1a-d show the XRD patterns of the 800 °C and 1100 °C MS Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloys, respectively. The XRD patterns of all the investigated MS Al-Si alloys are present in the peaks of Al phase and Si phase, indicating that the microstructure of the MS Al-Si alloys is composed of the Al matrix phase and Si phase. The XRD intensity of the Al(111) main peak of both 800 °C and 1100 °C MS Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloys decreases monotonically with the increase of Si content, except the 1100 °C MS Al-12.2at.%Si alloy whose Al(111) peak intensity shows an abnormal back- ing increase, when compared with the Al(111) peak intensity of the 1100 °C MS Al-7at.%Si alloy. The volume fraction of Al matrix phase that takes the majority in the investigated MS Al-Si alloys decreases with increasing Si content, and there is a big decrease in the volume fraction of the Al matrix phase from 12.2at.%Si to 20at.%Si, which might lead to the general decrease of the XRD intensity of Al with increasing Si content and the small XRD intensity of Al at the MS Al-20at.%Si alloys. 2a-d present the medium magnification (×15 k) SEM micrographs taken from the center of the cross-section of the 800 °C MS Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloy ribbons, sequentially, and Fig. 2e-h show the corresponding statistical size distribution of Si particles in Fig. 2a-d. Si particles in the 800 °C MS Al-Si alloys show a near lognormal size distribution, with particle size ranges of (0, 220 nm), (0, 260 nm), (0, 320 nm) and (0, 440 nm) for the Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si MS alloys, respectively. Figure 2i-l present the medium magnification (×15 k) SEM micrographs taken from the center of the cross-section of the 1100 °C MS Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloy ribbons, sequentially, and Fig. 2m-p show the corresponding statistical size distribution of Si particles in Fig. 2i-l. Si particles in 1100 °C MS Al-Si alloys also show a near lognormal size distribution, with particle size ranges of (0, 340 nm), (0, 400 nm), (0, 200 nm) and (0, 440 nm) for the Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si MS alloys, separately. The size of Si particles in the 800 °C MS Al-Si alloys increases monotonically with the increase of Si content. Different from the 800 °C MS condition, with increasing Si content, the size of Si particles in the 1100 °C MS Al-Si alloys increases in a non-monotonic way with an abnormal decrease occurs at Al-12.2at.%Si. Simultaneously, the abnormal increase of the number of Si particles was observed at Al-12.2at.%Si for the 1100 °C MS Al-Si alloys.

SEM analysis of MS Al-Si alloys. Figure

Confirmation of abnormal decrease of Si particle size. During the MS process, the cooling rate decreases gradually from the wheel side to the free surface of the MS alloy ribbons, which results in the gradual increase of the size of phases across the cross-section of the MS alloy ribbons from the while side to the free surface [START_REF] Dong | Gradient microstructure and multiple mechanical properties of AlSi9Cu alloy ribbon produced by melt spinning[END_REF][START_REF] Dong | Two directional microstructure and effects of nanoscale dispersed Si particles on microhardness and tensile properties of AlSi7Mg melt-spun alloy[END_REF] . Therefore, the size of Si particles across the whole cross-section (near the wheel side, in the center and near the free surface) of the 1100 °C MS Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloy ribbons was studied, to confirm the abnormal decrease of Si particle size in the 1100 °C MS Al-12.2at.%Si alloy. Figure 3a-d,e-h,i-l,mp show the detail cross-section SEM morphology of the 1100 °C MS Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloy ribbons, respectively. From the low magnification (×1 k) cross-section morphology shown in Fig. 3a,e,i,m, the thickness of the 1100 °C MS Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloy ribbons is ~ 55 μm. Figure 3b,f,j,n, Fig. 3c,g,k,o and Fig. 3d,h,l,p show the high magnification (×30 k) morphology of Si particles near the wheel side, in the center and near the free surface of the MS alloy ribbons, respectively. The size of Si particles in the 1100 °C MS Al-Si alloys increases gradually across the cross-section from the wheel side to the free surface due to the decreasing cooling rate. With the increase of Si content, the size of Si particles across the whole cross-section of the 1100 °C MS Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloy ribbons first increases, then decreases abnormally at Al-12.2at.%Si, after increases again. Thus the abnormal decrease of Si particle size at the 1100 °C MS Al-12.2at.%Si alloy is the fact across the whole cross-section of the 1100 °C MS Al-12.2at.%Si alloy, and it is not a fluke by loosely taking the size of finer Si particles near the wheel side of the 1100 °C MS Al-12.2at.%Si alloy while taking the size of larger Si particles far from the wheel side of the other 1100 °C MS alloys.

For consistency, the medium magnification (×15 k) SEM micrographs taken from the center of the crosssection were used for the statistics of the size of Si particles in the investigated MS Al-Si alloy ribbons. Figure 4a presents the evolution of the statistical mean size of Si particles in the 800 °C and 1100 °C MS Al-Si alloys versus Si content. All of the investigated MS Al-Si alloys show the increase in Si particle size with the increase of both Si content and initial MS melt temperature, except for the 1100 °C MS Al-12.2at.%Si alloy that shows an abnormal and(d, h, l, p) near the free surface of the melt-spinning alloy ribbons. decrease in Si particle size. Figure 4b shows the XRD intensity of the Al(111) main peak of all the investigated MS Al-Si alloys, basing on the XRD patterns shown in Fig. 1. For 800 °C MS Al-Si alloys, with increasing Si content, the Al(111) XRD intensity decreases monotonically, which is consistent with the monotonic increase of Si phase in the alloys observing by SEM. For 1100 °C MS Al-Si alloys, with increasing Si content, the Al(111) XRD intensity decreases in a non-monotonic way, and the abnormal increase of the Al(111) XRD intensity at Al-12.2at.%Si is self-consistent with the abnormal decrease of Si particle size under this condition observing by SEM. The self-consistent SEM and XRD analysis of the 800 °C and 1100 °C MS Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloys confirms the abnormal decrease of Si particle size and the abnormal increase of Si particle number in the 1100 °C MS Al-12.2at.%Si alloy, which demonstrates the disruption of Si-rich microstructure in the Al-12.2at.%Si alloy melt with the increase of melt temperature from 800 to 1100 °C.

Small angle neutron scattering of Al-12.2at.%Si alloy melt. Small angle X-ray scattering (SAXS) [START_REF] Feld | Nanocomposites of highly monodisperse encapsulated superparamagnetic iron oxide nanocrystals homogeneously dispersed in a poly(ethylene oxide) melt[END_REF][START_REF] Rieker | Particle size distribution inferred from small-angle X-ray scattering and transmission electron microscopy[END_REF] and small angle neutron scattering (SANS) [START_REF] Popel | Metastable microheterogeneity of melts in eutectic and monotectic systems and its influence on the properties of the solidified alloy[END_REF][START_REF] Potton | A new method for the determination of particle-size distributions from small-angle neutron-scattering measurements[END_REF][START_REF] Borodin | Liquid structure with nano-heterogeneity promotes cationic transport in concentrated electrolytes[END_REF] have been applied to study the micro-heterogeneous structure in solid materials and solutions especially the colloidal suspension system, as the measured intensity of small angle scattering (SAS) is proportional to the contrast that is given by the difference in the scattering length density between the micro-heterogeneous structure and the matrix, and there is SAS signal for the microheterogeneous system, while there is no SAS signal for the homogeneous system without micro-heterogeneous structure embedded in the matrix [START_REF] Paradies | Particle size distribution and determination of characteristic properties of colloidal bismuth-silica compounds by small-angle X-ray scattering and inelastic light scattering[END_REF][START_REF] Shim | Tailored colloidal stability and rheological properties of graphene oxide liquid crystals with polymer-induced depletion attractions[END_REF] . Considering the abnormal decrease of Si particle size in the 1100 °C MS Al-12.2at.%Si alloy, SANS was applied to further explore the microstructure evolution in the Al-12.2at.%Si alloy melt.

Figure 5 shows the size distribution of the micro-heterogeneous structure in the Al-12.2at.%Si alloy melt at 800 °C and 1100 °C measuring by SANS, and the nanoscale micro-heterogeneous structure in the Al-12.2at.%Si alloy melt comes from the aggregation of Si atoms, as it has the contrast difference with the Al-rich melt matrix under SANS. The Si-rich micro-heterogeneous structure in the Al-12.2at.%Si alloy melt exists in two size families, i.e., large quantities of small Si-rich micro-heterogeneous structure ranging between 0 and 10 nm and small quantities of large Si-rich micro-heterogeneous structure ranging between 10 and 240 nm. Large quantities of small (0-6 nm) micro-heterogeneous structure was measured in the Sn-26.1at.%Pb alloy melt by SANS [START_REF] Popel | Metastable microheterogeneity of melts in eutectic and monotectic systems and its influence on the properties of the solidified alloy[END_REF] , therefore the size of the micro-heterogeneous structure in alloy melts depends on the alloy system. With the increase of the Al-12.2at.%Si melt temperature from 800 to 1100 °C, the proportion of the small (0-10 nm) Si-rich microheterogeneous structure in the alloy melt increases, while the proportion of the large (10-240 nm) Si-rich microheterogeneous structure in the alloy melt decreases significantly, which further confirms the disruption of large Si-rich micro-heterogeneous structure in the Al-12.2at.%Si alloy melt into small Si-rich micro-heterogeneous structure under the high temperature of 1100 °C.

TEM analysis of MS Al-Si alloys.

The above mentioned MS and SANS study demonstrates the Si-rich micro-heterogeneous structure in Al-12.2at.%Si alloy melt at 800 °C, so the 800 °C MS Al-12.2at.%Si alloy was further studied by TEM. Among the eight MS conditions, Si particles in the 800 °C MS Al-3at.%Si alloy are the smallest and closest to the Si-rich micro-heterogeneous structure in the molten state, therefore the 800 °C MS Al-3at.%Si alloy was also analyzed by TEM. Si-rich particle aggregates were observed in the Al grains of the 800 °C MS Al-3at.%Si and Al-12.2at.%Si alloy under TEM, as shown in Fig. 6. Figure 6a presents the scanning TEM (STEM) image of a particle aggregate (160 nm) in the Al grain of the 800 °C MS Al-3at.%Si alloy, and the magnification (Fig. 6b) shows that the particle aggregate in Fig. 6a comprises multiple small particles with the sizes smaller than 30 nm. Figure 6c presents the STEM composition mapping of the particle aggregate shown in Fig. 6a,b and the Al matrix surrounding the particle aggregate, the enrichment of Si and the infertility of Al can be found in the area of the particle aggregate when compared to the surrounding Al matrix, while the concentration of oxygen (O) in the particle aggregate is uniformly the same as that in the surrounding Al matrix, and the concentration of O is much lower than that of Al and Si, which demonstrates that the particle aggregate is not the Al-based or Si-based oxide and the particle aggregate is rich in Si. The uniform appearance of trace O in the STEM mapping was due to the inevitable oxidation of the surface of the TEM sample. Figure 6d shows the bright-field TEM image of a Si-rich particle aggregate in the Al grain of the 800 °C MS Al-12.2at.%Si alloy, and the particle aggregate also comprises multiple small nanoscale Si-rich particles below 50 nm, while the size of small Si-rich particles in the particle aggregate of the 800 °C MS Al-12.2at.%Si alloy is slightly larger than that of the 800 °C MS Al-3at.%Si alloy.

Discussion

It is understandable that the size of Si particles in the 800 °C and 1100 °C MS Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloys increases with the increase of both Si content and initial MS melt temperature, due to the increasing growth tendency of Si phase under higher Si content and the decreasing cooling rate under higher initial MS melt temperature. However, Si particles in the 1100 °C MS Al-12.2at.%Si alloy show an abnormal decrease in Si particle size together with an abnormal increase of Si particle number, and this clearly demonstrates the disruption of Si-rich microstructure in the Al-12.2at.%Si alloy melt under the high temperature of 1100 °C. The decrease of the ratio of large Si-rich micro-heterogeneous structure and the increase of the ratio of small Si-rich micro-heterogeneous structure under SANS measurement further confirm that the large (10-240 nm) Si-rich micro-heterogeneous structures in the Al-12.2at.%Si alloy melt disrupt into the small (0-10 nm) Si-rich micro-heterogeneous structures with the increase of melt temperature from 800 to 1100 °C. The microstructure evolution with temperature was sporadically evidenced in other systems such as the In-Sn80 and the La 50 Al 35 Ni 15 alloy melts [START_REF] Zu | Observation of an anomalous discontinuous liquid-structure change with temperature[END_REF][START_REF] Xu | Evidence of liquid-liquid transition in glass-forming La 50 Al 35 Ni 15 melt above liquidus temperature[END_REF] .

The TEM analysis in Fig. 6 shows the entrapment of Si-rich particle aggregates in the Al grains of the 800 °C MS Al-12.2at.%Si alloy. According to the thermal conduction model [START_REF] Xu | Cooling rate and microstructure of rapidly solidified Al-20 wt.% Si alloy[END_REF] of the MS process, the mean cooling rate of the 800 °C MS here was calculated as ~ 2.79 × 10 6 °C/s, and the Si particle aggregates entrapped in the Al grains had limited time to grow during MS. In addition, the size of the Si-rich particle aggregates entrapped in the Al grains of the 800 °C MS Al-12.2at.%Si alloy agrees with the large (10-240 nm) Si-rich micro-heterogeneous structure in the Al-12.2at.%Si alloy melt measured by SANS. From Fig. 6, the size of small Si-rich particles in the particle aggregate of the 800 °C MS Al-12.2at.%Si alloy is slightly larger than that of the 800 °C MS Al-3at.%Si alloy, which indicates the inevitable growth of the small Si-rich particles in the particle aggregate under the super high cooling rate of the MS process, and the size of the small Si-rich particles in the particle aggregate of the MS alloy fits with the small (0-10 nm) Si-rich micro-heterogeneous structure in the Al-12.2at.%Si alloy melt measured by SANS, when considering the growth effect during MS. Furthermore, the Si-rich particle aggregate can be determined not precipitating out during the MS process, as the precipitation phase is normally precipitated out in the form of single particle rather than aggregate, and the Si-rich particle aggregate entrapped in the Al grain of the 800 °C MS Al-12.2at.%Si alloy is close to its original state in the alloy melt. Moreover, hardly were Si-rich particle aggregates observed in the Al grains of the 1100 °C MS Al-12.2at.%Si alloy, which supports the disruption of large Si-rich aggregates in the Al-12.2at.%Si alloy melt at 1100 °C. Therefore, the large (10-240 nm) Si-rich micro-heterogeneous structures in the Al-12.2at.%Si alloy melt are probably Si-rich aggregates comprising multiple small (0-10 nm) Si-rich micro-heterogeneous structures, due to the agglomeration effects [START_REF] Mackay | General strategies for nanoparticle dispersion[END_REF] of nanostructures, and the large Si-rich aggregates in the Al-12.2at.%Si alloy melt might be significantly disrupted into multiple small Si-rich microstructures at 1100 °C. The degrees of superheat of the Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloys are 459 °C, 485 °C, 523 °C and 406 °C, respectively, at 1100 °C. The superheat of the eutectic Al-12.2at.%Si alloy is the largest among the four investigated Al-Si alloys at the same MS temperature of 1100 °C, and the superheat of 523 °C is sufficient for the significant disruption of Sirich microstructure in the Al-12.2at.%Si alloy melt, which results in the abnormal decrease of Si particle size in the 1100 °C MS Al-12.2at.%Si alloy. It can be estimated that the Si-rich microstructure in the other three Al-Si alloys can also be significantly disrupted if the superheat is sufficient, and the sufficient superheat for the significant disruption of Si-rich microstructure in the other three Al-Si alloys might be 523 °C or even higher, due to the higher liquidus temperatures and atomic binding in the other three Al-Si alloys than that of the eutectic Al-12.2at.%Si alloy. Nanoscale Si particles were observed absorbing on the surface of the AlB 2 nucleation particles for the grain refinement of Al-Si alloys [START_REF] Chen | Grain refinement of hypoeutectic Al-Si alloys with B[END_REF] , and these nanoscale Si particles might be originated from the small (0-10 nm) Si-rich micro-heterogeneous structures in the Al-Si alloy melts.

Considering that the melting points of nanoparticles have size effects [START_REF] Couchman | Thermodynamic theory of size dependence of melting temperature in metals[END_REF][START_REF] Pan | Melting the ice: on the relation between melting temperature and size for nanoscale ice crystals[END_REF] , the melting points of spherical pure Si nanocrystals in the atmosphere of Al-12.2at.%Si alloy melt were determined, to reveal the existing state of the Si-rich micro-heterogeneous structures in the Al-12.2at.%Si alloy melt. Wautelet 40, Nanda et al. [START_REF] Nanda | Liquid-drop model for the size-dependent melting of low-dimensional systems[END_REF] and Lu et al. [START_REF] Lu | Size-, shape-, and dimensionality-dependent melting temperatures of nanocrystals[END_REF] proposed different theoretical models estimating the melting point dependence of spherical nanocrystals versus size. For spherical pure Si nanocrystals, the theoretical melting points given by Wautelet's model were relatively close to experimental values given by Goldstein [START_REF] Goldstein | The melting of silicon nanocrystals: submicron thin-film structures derived from nanocrystal precursors[END_REF] (Fig. 7a). However, a correction factor k (0.785) needs to be added into Wautelet's theoretical model, to fit the experimental melting points of pure Si nanocrystals in the atmosphere of vacuum: where T 0 (1687 K) is the melting point of bulk Si, β (1.88 nm) is a constant, d is the diameter of the Si nanocrystal, and d 0 (0.235 nm) is the interatomic distance in crystalline Si.

According to the Clapeyron equation, melting point changes with pressure. Based on Eq. ( 1), the real melting point of spherical pure Si nanocrystals in the atmosphere of air is: where P 0 is the standard atmospheric pressure (1.01325 × 10 5 Pa), β α H m (50.21 kJ•mol -1 ) is the molar melting heat of Si, and β α V m (-1.131 cm 3 mol -1 ) is the change of molar volume before and after the melting of Si. In Al-12.2at.%Si alloy melt, the spherical pure Si nanocrystals will suffer from an extra surface tension. Based on the Clapeyron equation and Eq. ( 2), the real melting points of spherical pure Si nanocrystals in the atmosphere of Al-12.2at.%Si alloy melt are then:

where σ (0.35241 J m -2 ) [START_REF] Gunduz | The measurement of solid-liquid surface energies in the Al-Cu, Al-Si and Pb-Sn Systems[END_REF] is the surface tension.

According to Eq. ( 3), the melting points of spherical pure Si nanocrystalline particles in the atmosphere of Al-12.2at.%Si alloy melt were calculated, as shown by the red curve in Fig. 7a, which are ~ 4-6 °C lower than that in the atmosphere of air. The melting points of the pure Si nanocrystalline particles with diameters of 3, 4, 5, 6, 7, 8, 9 and 10 nm were derived as 216, 423, 548, 631, 691, 736, 771 and 799 °C, respectively, in the atmosphere of Al-12.2at.%Si alloy melt. The melting point of Si-rich micro-heterogeneous structure is lower than that of the pure Si nanocrystalline particle with the same size, in the atmosphere of Al-12.2at.%Si alloy melt. Therefore, the small (0-10 nm) Si-rich micro-heterogeneous structures in the 800 °C Al-12.2at.%Si alloy melt exist in the state of droplet with non-crystalline structure (Fig. 7c), rather than the state of solid crystals with crystalline structure (Fig. 7b), as their melting points are lower than the melt temperature. Correspondingly, the large (10-240 nm) Sirich aggregates comprising multiple small (0-10 nm) Si-rich micro-heterogeneous structures in the Al-12.2at.%Si alloy melt exist in the state of droplet aggregates. The state of droplet in turn supports the spherical shape of the Si-rich micro-heterogeneous structures in Al-12.2at.%Si alloy melt, due to the surface tension.

Earlier a thermodynamic metastable colloidal model [START_REF] Popel | Metastable colloidal states of liquid metallic solutions[END_REF] of liquid metallic solutions was proposed to explain the coexistence of micro-heterogeneous droplets in alloy melts, and the nanoscale micro-heterogeneous droplets in Sn-Pb alloy melt 5 could be thermodynamically metastable by a thin transition layer with a thickness of ~ 0.3 nm [START_REF] Popel | Metastable colloidal states of liquid metallic solutions[END_REF] , to reach the local minimum of the free energy of the system. Here the droplet state Si-rich micro-heterogeneous (1) structure is possibly metastable in the Al-12.2at.%Si alloy melt via a thin transition layer, which provides the thermodynamic base for the disruption of the Si-rich micro-heterogeneous structure in the Al-12.2at.%Si alloy melt with the increase of melt temperature from 800 to 1100 °C.

T mv = kT 0 1 - β d (d > 10d 0 ) (2)
T ma = kT 0 1 - β d 1 + P 0 � β α V m � β α H m (d > 10d 0 ) (3) T mm = kT 0 1 - β d 1 + P 0 � β α V m � β α H m 1 + 4σ � β α V m d� β α H m (d > 10d 0 )

conclusion

In summary, we provide experimental evidence for the disruption of Si-rich micro-heterogeneous structure in the engineering-lightweight Al-12.2at.%Si alloy melt above liquidus temperature. The size of Si particles in 800 °C and 1100 °C MS Al 1-x Si x (x = 0.03, 0.07, 0.122, 0.2) alloys increases with the increase of both Si content and initial MS melt temperature, except for the 1100 °C MS Al-12.2at.%Si alloy that shows an abnormal decrease in Si particle size together with an abnormal increase in Si particle number, which demonstrates the disruption of Si-rich microstructure in Al-12.2at.%Si alloy melt at 1100 °C. Small angle neutron scattering experiment further verifies that large quantities of small (0-10 nm) Si-rich micro-heterogeneous structures and small quantities of large (10-240 nm) Si-rich micro-heterogeneous structures exist in Al-12.2at.%Si alloy melt, and the large Si-rich micro-heterogeneous structures in the Al-12.2at.%Si alloy melt into small Si-rich micro-heterogeneous structures with increasing of melt temperature from 800 °C to 1100 °C. Si-rich particle aggregates comprise multiple small Si-rich particles were observed entrapping in the Al grains of the 800 °C MS Al-12.2at.%Si alloy, which agree in size with the large Si-rich micro-heterogeneous structures in the Al-12.2at.%Si alloy melt, and this indicates that the large (10-240 nm) Si-rich micro-heterogeneous structures in the Al-12.2at.%Si alloy melt are probably Si-rich aggregates comprising multiple small (0-10 nm) Si-rich micro-heterogeneous structures. The method of MS from different initial temperatures and compositions in cooperation with the small angle neutron scattering measurement presented in this work also provides a pathway for the exploration of microstructures in other high temperature alloy melts.

Materials and methods

A series of high purity hypoeutectic Al-3at.%Si and Al-7at.%Si, eutectic Al-12.2at.%Si and hypereutectic Al-20at.%Si binary Al-Si alloys were prepared by the arc-melting of the high purity Al (5N) and Si (5N). The alloys were remelted three times under the inert Ar atmosphere, to ensure the composition homogenization of the prepared high purity Al-Si alloys. The prepared high-purity Al-3at.%Si, Al-7at.%Si, Al-12.2at.%Si and Al-20at.%Si alloy melts were rapidly solidified with super-high cooling rate (~ 10 6 °C/s) into thin ribbons by the MS [START_REF] Dong | Gradient microstructure and multiple mechanical properties of AlSi9Cu alloy ribbon produced by melt spinning[END_REF][START_REF] Salehi | Structure and properties of nanostructured aluminum A413.1 produced by melt spinning compared with ingot microstructure[END_REF][START_REF] Li | Giant magnetocaloric effect in melt-spun Ni-Mn-Ga ribbons with magneto-multistructural transformation[END_REF] method under high-vacuum (3.3E-5 Pa) condition. In the melt-spinner, alloy charges of ~ 3 g were melted in a Ø 12 mm crucible with 5 mm × 0.5 mm orifice and ejected by argon onto a rotating copper wheel with the linear velocity of 32.8 m•s -1 . All alloy melts were MS from two different initial melt temperatures, 800 °C and 1100 °C. The melt temperatures during MS were measured and controlled with an accuracy of ±6 °C. The resultant MS Al-Si alloy ribbons were ~ 5 mm wide and ~ 55 μm thick.

The prepared high purity Al-12.2at.%Si alloy melt was measured by SANS at 800 °C and 1100 °C. The sample for SANS was contained in a single crystalline sapphire container with a diameter of 10 mm and a wall thickness of 50 μm. The incident neutron wavelength was chosen as 8 Å, to avoid the multiple scattering in the container walls. After rising to the melt temperatures of 800 °C and 1100 °C, the alloy melt was held for 1 h before SANS measurement, and the SANS measurement time was 1 h afterwards.

SEM, XRD and TEM were used to characterize the microstructure of the prepared MS Al-Si alloy ribbons. The MS alloy ribbons for SEM investigation were polished and etched by 0.5 vol.% HF. SEM observation was performed on JSM-7001F. The XRD measurements were carried out in a Rigaku H2500 diffractometer using Cu Ka1 radiation at 40 kV and 20 mA in the 2θ range from 25° to 85°. The MS alloy ribbons for TEM investigation were ion-beam milled under a constant preparation temperature of ~ -10 °C. The Tecnai G 2 F20 and JEOL-2100 TEM were used at 200 kV for bright-field imaging, scanning TEM imaging and composition mapping.

For consistency and accuracy, the medium magnification (×15k) SEM micrographs taken from the center of the cross-section of the MS Al-Si alloy ribbons were used for the statistics of the size and distribution of Si particles in each of the MS alloys. The statistics was conducted in the software Nano Measurer 1.2.5 by measuring all the sizes of Si particles in the micrographs. The particle size distribution under SANS measurement was determined using the Generalized Indirect Fourier Transformation method as developed in the GIFT computer code [START_REF] Brunner-Popela | Small-angle scattering of interacting particles I Basic principles of a global evaluation technique[END_REF] .

Figure 1 .

 1 Figure 1. X-ray diffraction patterns of the Al-Si melt-spinning alloys rapid solidified from melts at 800 °C and 1100 °C. (a) Al-3at.%Si melt-spinning alloy; (b) Al-7at.%Si melt-spinning alloy; (c) Al-12.2at.%Si meltspinning alloy; (d) Al-20at.%Si melt-spinning alloy.

Figure 2 .

 2 Figure 2. Medium magnification (×15 k) SEM micrographs taken from the cross-section center of the Al-Si melt-spinning alloy ribbons and corresponding statistical size distribution of Si particles in the Al-Si meltspinning alloy ribbons rapid solidified from melts at (a-h) 800 °C and (i-p) 1100 °C: (a, e, i, m) Al-3at.%Si; (b, f, j, n) Al-7at.%Si; (c, g, k, o) Al-12.2at.%Si; (d, h, l, p) Al-20at.%Si.

Figure 3 .

 3 Figure 3. Cross-section SEM morphology of (a-d) Al-3at.%Si, (e-h) Al-7at.%Si, (i-l) Al-12.2at.%Si and (mp) Al-20at.%Si melt-spinning alloy ribbons rapid solidified from melts at 1100 °C: (a, e, i, m) low magnification (×1 k) cross-section morphology, and high magnification (×30 k) morphology of Si particles (b, f, j, n) near the wheel side, (c, g, k, o) in the center, and (d, h, l, p) near the free surface of the melt-spinning alloy ribbons.

Figure 4 .

 4 Figure 4. Abnormal decrease of Si particle size in 1100 °C melt-spinning Al-12.2at.%Si alloy. (a) Evolution of statistical mean size of Si particles in the investigated 800 °C and 1100 °C melt-spinning Al-Si alloys versus Si content; (b) Evolution of the XRD intensity of the Al(111) main peak of the investigated 800 °C and 1100 °C melt-spinning Al-Si alloys versus Si content.

Figure 5 .

 5 Figure 5. Small angle neutron scattering results showing the distribution of the Si-rich microstructure in the Al-12.2at.%Si alloy melt at 800 °C (blue curve) and 1100 °C (red curve). (a) Distribution of small Si-rich microstructure bellow 10 nm, (b) Distribution of large Si-rich microstructure above 10 nm.

Figure 6 .

 6 Figure 6. TEM images showing the Si-rich particle aggregates in the Al grains of the melt-spinning Al-Si alloys. (a) Scanning TEM image of the Si-rich particle aggregate and (b) its magnification in the 800 °C meltspinning Al-3at.%Si alloy; (c) Scanning TEM composition mapping of the Si-rich particle aggregate in (a) and (b); (d) Bright-field TEM image of the Si-rich particle aggregate and (e) its magnification in the 800 °C meltspinning Al-12.2at.%Si alloy.

Figure 7 .

 7 Figure 7. Existing state of the Si-rich microstructure in the Al-12.2at.%Si alloy melt. (a) Melting points of solid state spherical Si nanocrystalline particle versus size under various atmospheres; (b) Schematic diagram showing the solid state spherical Si nanocrystalline particle with crystalline structure; (c) Schematic diagram showing the droplet state Si-rich microstructure in Al-12.2at.%Si alloy melt with non-crystalline structure.

Scientific RepoRtS | (2020) 10:12979 | https://doi.org/10.1038/s41598-020-69972-2

Acknowledgements

The work was supported by the Innovate UK project and the National Basic Research Program of China.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Author contributions

X.D. conceptualized the method, conducted the experiments, analyzed the results and prepared the article. P.L. discussed the results and revised the manuscript. S.A. assisted the TEM characterization and discussed the results. S.J. discussed the results and revised the manuscript. P.P., U.D. and M.D. assisted the experiments, discussed the results and revised the manuscript.

competing interests

The authors declare no competing interests.