Serge Cohen

Xavier Bressaud

Transition of the simple random walk on the graph of the ice-model

Keywords: Random walk, Markov Chain AMS classification (2000): 05C81, 60F05

The 6-vertex model is a seminal model for many domains in Mathematics and Physics. The sets of configurations of the 6-vertex model can be described as the sets of paths in multigraphs. In this article the transition probability of the simple random walk on the multigraphs is computed. The unexpected point of the results is the use of continuous fractions to compute the transition probability.

Introduction

In this article we are interested in a simple random walk Y K on particular (multi)graphs G K indexed by an integer K. The set of vertices of G K is V K def = {-1, 1} K and the set E K of edges is related to the square ice model [START_REF] Lieb | Residual entropy of square ice[END_REF], which is a particular case of the 6-vertex model.

As a motivation for the study of the transition probability of Y K let us recall briefly some facts concerning the square ice model. The configuration set of the ice model can be viewed as the set of functions h on a n × K rectangular lattice called height functions. If a, b = {z ∈ Z, a ≤ z ≤ b}, and if h : 0, n × 0, K → Z, we denote the gradient of a function h by ∇h a function on the edges of the lattice such that ∇h((j ± 1, k), (j, k)) = h(j ± 1, k) -h(j, k) and ∇h((j, k ± 1), (j, k)) = h(j, k ± 1) -h(j, k) for j, k ∈ Z. A function h is called an height function if h(0, 0) = 0, and ∇h(e) = ±1 for every edge e of the lattice. Let us denote by H 0 n,K = {h : 0, n × 0, K , h(x, y) ∈ Z, h(0, 0) = 0, ∇h(e) = ±1}

the set of height functions. If h ∈ H 0 1,1 , the only possible ∇h are depicted on the figures below, where the values of the height function on the corner are written in green and ±1 is abbreviated in ±,

1 + + 2 + 0 + 1 1 + - 0 + 0 --1 1 + -0 - 0 + 1 -1 - --2 - 0 --1 -1 - + 0 - 0 + 1 -1 - + 0 + 0 --1
The six possible choices of signs on the edges are in contrast to the 16 possibilities if we don't require the existence of an height function. It explains why this model is also called a 6-vertex model.

We may also wonder, what are the condition on a, b ∈ V K , so that there exists h ∈ H 0 1,K with ∇h((0, k + 1), (0, k)) = a k , ∇h((1, k + 1), (1, k)) = b k , ∀k = 0 to K -1.

Let us take a = (1, 1, -1, -1) and b = (1, -1, 1, -1). In this case there is only one height function that can be defined on the rectangle, where we put the list of signs of the digits of a and b starting from the bottom of the rectangle as 0

- + 1 - 1 + - 2 + 2 - + 1 - 1 + + 2 + 0 + 1
One can also check that there exists no height function if you take a = (1, 1, -1, -1) and b = (-1, -1, 1, -1). The set of edges E K of G K defined rigorously later in Definition 2.2 is such that (a, b) ∈ E K if there exists at least an height function on the 1 × K rectangle with a on the left side and b on the right. It is an elementary fact that there exists at most one possibility if a = b and two if a = b that is proved in section 2.

For a n×K rectangle the existence of an height function h n,K corresponds to a unique path of length n in G K . For instance

1 + - 2 + -1 - + 2 + + 3 + ❴ ❴ ❴ • + -h n,K (n, 4) - 2 - + 1 - + 2 - -1 - + 2 - ❴ ❴ ❴ • - + h n,K (n, 3) - 1 + + 2 + + 3 + -2 - + 3 + ❴ ❴ ❴ • + + h n,K (n, 2) + 0 + 1 + 2 + 3 -2 ❴ ❴ ❴ • + h n,K (n, 1)
it is the path

a 1 = (1, 1, -1), a 2 = (1, -1, 1), a 3 = (1, -1, -1), a 4 = (-1, -1, 1), a 5 = (1, -1, 1), . . . , a n-1 = (1, -1, 1), a n = (1, -1, -1) ∈ G 3 .
A more detailed introduction to ice and 6-vertex model can be found in [START_REF] Baxter | Exactly solved models in statistical mechanics[END_REF].

Lieb in [START_REF] Lieb | Residual entropy of square ice[END_REF] counts the number of height functions with periodic boundary conditions. With our notations it means K = n and Lieb considers an height function h n on the torus T n . In statistical mechanic the uniform distribution is taken on all height functions on T n and Duminil-Copin et al. 2019 [START_REF] Duminil-Copin | Logaritmic variance for the height fonction of square-ice[END_REF] show what is called a delocalization behavior.

Theorem 1.1. There exist c, C ∈ (0, ∞) such that for every n ≥ 1 and every

u, v ∈ T n , such that u -v 1 ≥ 2, c log u -v 1 ≤ var [(h n (u) -h n (v))] ≤ C log u -v 1 , where • 1 is the L 1 distance in T n .
In Boissard et al. [START_REF] Boissard | Diffusivity of a random walk on random walks[END_REF] we consider a stationary simple random walk Y K on G K . It endows the set of paths with length n with a non uniform distribution if K > 1. In this case it has been shown in [START_REF] Boissard | Diffusivity of a random walk on random walks[END_REF] that ∀i, j ∈ {0, . . . , K},

lim n→+∞ var [h n,K (n, j) -h n,K (0, i)] n = 2 K + 2 .
Still for finite K the variance is computed in [START_REF] Espinasse | A combinatorial approach to a model of constrained random walkers[END_REF] for a periodic boundary condition in the variable between 1 and K, and in [START_REF] Lammers | Diffusivity of a walk on fractures of a hypertorus[END_REF] for other constraints.

The easiest way to describe the simple random walk on G K in the stationary regime is to state that the distribution of the pair (Y K (0), Y K (1)) is the uniform distribution on the edges E K . One aim of this article is to provide a formula for the transition probability of the simple random walk that starts from a given vertex. Surprisingly enough the formula in the Theorem 3.1 uses the continuous fraction associated to the length of the constancy blocks of the digits of the given vertex. The authors see two useful consequences of this result. The first one is the fact that the transition probability of the simple random walk will considerably make easier and faster simulations of the walk. See [START_REF] Montegut | Limite d'échelle de marche aléatoires contraintes[END_REF] for a survey of previous simulation methods, which are both memories and computationally intensive due to the difficulty to describe easily the neighbors of a given vertex in G K for large K. In [START_REF] Montegut | Limite d'échelle de marche aléatoires contraintes[END_REF] it was pretty long to obtain simulations for K > 30, whereas the simulations using the method of this article allow us to attain K = 1000 on the same computer. It opens perspectives since simulations may be useful when n = K and n large enough to have a law of large numbers. Moreover the same techniques used for Theorem 3.1 yields a formula for the degree of a given vertex in G K , which is related to a Fibonacci type sequence associated to the length of the constancy blocks of the digits of the vertex in Proposition 3.1. Another interesting consequence of Theorem 3.1 is that it can be extended to the case K = ∞. This in turns gives a sense to the decay of the variance of the height function in G ∞ . The definitions and the models are given in section 2. We think that the model for K = ∞ yields an alternative to classical theormodynamic limits that are usually used for infinite lattices. In section 3 the main theorems are written and they are proved in the following sections.

The model

Let us first construct the (multi)graph G K such that there is one to one correspondence between edges of G K and height functions with a on the left and b on the right.

Definition 2.1. For a, b ∈ V K = {-1, 1} K let
us denote by a ∼ b such that there exists an height function

h ∈ H 0 1,K with ∇h((0, k + 1), (0, k)) = a k , ∇h((1, k + 1), (1, k)) = b k .
One can first remark that if b = a there are two height functions such that a ∼ a. Namely one can take h + such that ∇h

+ ((0, k), (1, k)) = +1, ∀k = 0 to K and h -such that ∇h -((0, k), (1, k)) = -1, ∀k = 0 to K. If b = a, there is a first digit k 0 = inf{0 < k|b k = a k },
and one can check that there exists an height function h such that ∇h((0, k +1), (0, k)) = a k , ∇h((1, k+1), (1, k)) = b k , ∀k = 0 to k 0 -1. ∇h((0, k-1), (1, k-1)) = a k 0 , ∀k ≤ k 0 and ∇h((0, k 0 ), (1, k 0 )) = -a k 0 . Then, by induction, one can define

k i = inf{k i-1 < k|b k = a k }, for i ≥ 1, if k i-1 is not the last index such that b k = a k . One can also check that ∇h((0, k -1), (1, k -1)) = a k i , ∀k i-1 < k ≤ k i . This implies that for the subsequence (k i ) of indices such that a k = b k a k i = (-1) i a k 0 and b k i = (-1) i+1 a k 0 . Hence if a = b and a ∼ b there is one height function such that h ∈ H 0 1,K with ∇h((0, k + 1), (0, k)) = a k , ∇h((1, k + 1), (1, k)) = b k , ∀k = 0 to K -1.
Let us also remark that if a ∼ b, the non vanishing coordinates of the vector (b-a) ∈ {-2, 0, 2} K have alternate signs. We summarize this characterization by writing that b -a follows the alternate signs rule. Please note that by convention b -a will follow the alternate signs rule if there are 0 or 1 non-vanishing coordinates in b -a. With this convention we just proved the following Proposition. We can define the (multi) graph G K as follows.

Definition 2.2. Let V K def = {-1, 1} K , and let E K be the subset of V K × V K defined by : A pair (a, b) ∈ V K × V K such that a = b belongs to E K if non vanishing coordinates of the vector (b -a) ∈ {-2, 0, 2} K have alternate signs.
For every vertex a ∈ V K there are two edges from a to a E K denoted by (a, a) + and (a, a) -. For every

K ∈ N the (multi)graph G K def = (V K , E K ).
The motivation for the two loops at any vertex a is the existence of the two height functions h + and h -. The definition of G K is written to have a one to one correspondence between E K and height functions with a on the left and b on the right.

In this article we consider Y K the simple random walk on G K starting from the invariant probability on V K . As usual for stationary simple random walks on (multi)graphs, the uniform distribution on the set of edges E K is the same as the distribution of (Y K (0), Y K (1)). Please note that when

Y K (0) = Y K (1) we have to toss a coin independently of what happens before to choose between (Y K (0), Y K (0)) + and (Y K (0), Y K (0)) -.
Obviously the invariant probability on V K can be expressed as the ratio of the degree of any vertex divided by the cardinal of E K . But it turns out that a sequential depiction of the uniform probability on E K allows us to extend our model to K = ∞. Lemma 2.1. Let ǫ be a random variable such that P(ǫ = -1) = P(ǫ = 1) = 1 2 , let (α k ) k≥1 be an i.i.d. sequence of Bernoulli random variables with parameter 1 3 and let (β k ) k≥1 be an i.i.d. sequence of random variables such that P(β k = -1) = P(β k = 1) = 1 2 . The previous random variables are mutually independent. Let us define ∀k ≥ 2 :

γ k def = ǫ(-1) k-1 i=1 α i , with the convention that γ 1 = ǫ.
Let us also define ∀k ≥ 1 :

A k = (1 -α k )β k + α k γ k (1)
B k = (1 -α k )β k -α k γ k . ( 2
)
The distribution of the pair

((A k ) 1≤k≤K , (B k ) 1≤k≤K
) is the uniform distribution on E K , if we denote by (A, B) K the edge defined by :

(A, B) K def = ((A k ) 1≤k≤K , (B k ) 1≤k≤K ) if (A k ) 1≤k≤K = (B k ) 1≤k≤K ((A k ) 1≤k≤K , (B k ) 1≤k≤K ) ǫ if (A k ) 1≤k≤K = (B k ) 1≤k≤K . Then (A, B) K L = (Y K (0), Y K (1)), (A k ) 1≤k≤K L = Y K (0) and (B k ) 1≤k≤K L = Y K (0). Proof. Let us first prove that (A, B) K ∈ E K . We remark that B k -A k = -2α k γ k , then it is vanishing if α k = 0,
and the alternate signs rule is fulfilled because every time α k = 1, γ k has a different sign from γ k-1 . Let us denote by D K the cardinal of E K . A simple computation yields D 1 = 6 and by induction D K = 2 × 3 K . It is also obvious to check P((A, B) 1 = e) = 1 6 for every edge in E 1 . Let us assume that P((A, B) K = e) = 1 D K is true for every edge in E K . Let us consider u ′ , v ′ ∈ V K and denote by u ′ ± 1 the vertex in V K+1 obtained by concatenating ±1 on the right of u ′ . Then for

P((A, B) K+1 = (u ′ 1, v ′ 1)) = P((A, B) K = (u ′ , v ′ ) ∩ α K+1 = 0 ∩ β K+1 = 1) = 1 D K 2 3 1 2 = 1 D K+1 .
The same holds for (u ′ -1, v ′ -1). If the concatenated digit to u ′ is different from the one concatenated to v ′ and u ′ = v ′ there is only one possible choice which leads to a non vanishing probability depending on the last digit that differs between u ′ and v ′ . Then for this choice

P((A, B) K+1 = (u ′ ± 1, v ′ ∓ 1)) = P((A, B) K = (u ′ , v ′ ) ∩ α K+1 = 1) = 1 D K 1 3 = 1 D K+1 .
The proof is complete when we consider the case u ′ = v ′ and in this case we get also P((A, B) K+1 = (u ′ ± 1, v ′ ∓ 1)) = 1 D K+1 thanks to the distribution of ǫ. The other claims follow easily.

Results

Transition probability for finite K.

The first aim of this article is to compute the transition probability of the stationary Markov chain associated with the simple random walks on the (multi)graphs G K for K ∈ N. In the following we are conditioning the distribution of (B k ) k≤K of Lemma 2.1 with respect of events of the form (A k ) k≤K = (a k ) k≤K where (a k ) k≤K is a deterministic sequence.

Let us fix some notations for a deterministic sequence (a k ) k≤K ∈ {-1, 1} K that starts with a 1 = 1. Then, on the m-th block of constancy of a, the digits are equal to i m = (-1) m+1 (if a 1 = -1, then i m = (-1) m ). By convention we set S 0 = 0, and for m ≥ 1, we assume that the m-th block of constancy of a starts with S m-1 + 1 and stops with S m .

Let us denote for m ≥ 1, and a given (a k ) k≤K ∈ {-1, 1} K the set

A (m) = {A ∈ {-1, 1} K such that A S m-1 +1 = i m , . . . , A Sm = i m } (3)
of sequences A which are equal to a on the m-th block of constancy of a.

Hence {A = a} = ∩ N m=1 A (m)
where N is the number of blocks of constancy of a.

Remark 3.1. In the following we are conditioning the distribution of B with respect of events of the form {A = a}, where a is a deterministic sequence. Once a is given, so is the sequence S and the conditioning with respect of A (m) actually means with respect of the event {A S m-1 +1 = i m , . . . , A Sm = i m }. We will use the abuse of notation P(.|a), P(.|A (m) ) in the sequel.

The length of the m-th block of constancy of a is denoted by

M m = S m -S m-1 . (4)
Let for 1

≤ m ≤ n ≤ N x n m def = 1 M m + 1 M m+1 + 1 ...+ 1 Mn+1 = [M m , M m+1 , . . . , M n , 1], (5)
if n < m we set x n m) . Let (E m ) 1≤m≤N be a sequence of random variables on {1, . . . , M m } which encode the digit that is changed in

A (m) if ǫ m = 1.
One further constraint due to the alternate signs rule is that when ǫ m = 1, ǫ m+2k = 0 on the event that ǫ m+1 = 0, . . . , ǫ m+(2k-1) = 0. In other words there cannot be change of digits in two successive blocks of constancy of a that have an even difference of indexes, since the a k 's are the same on those blocks.

Let us define the subset of C n of {0, 1} n for which this constraint is fulfilled and introduce some notations.

Let e ∈ {0, 1} n and 1 ≤ m ≤ n, such that e m = 1, then either ∀1 ≤ m ′ < m, e m ′ = 0, or we can define the last index before m such that e m ′ = 1 m(e) = max{1 ≤ m ′ < m, e m ′ = 1}.

By abuse of notation we will often denote m(e) as m. The constraint of the alternate signs rule for e and m is m -m = 2k -1, for k ≥ 1. Then The computations can be split in three cases. First the initializing phase, where the distribution of ǫ 1 is computed. Second if (ǫ 1 , . . . , ǫ m-1 ) = (0, . . . , 0), i.e. when b = a up to the m -1-th block of constancy of a, we call this case the subsequent phase when previously there is no change. Third (ǫ 1 , . . . , ǫ m-1 ) = (e 1 , . . . , e m-1 ) ∈ C * m-1 , which means that there is at least one digit b m ′ = a m ′ for m ′ < m and b m = a m will respect the alternate signs rule, this last case will be called the subsequent phase when previously there is at least one change.

C n def = {e ∈ {0, 1} n , ∀m ≤ n, e m = 1 | either ∀1 ≤ m ′ < m, e m ′ = 0, or m -m = 2k -1, k ≥ 1},
The conditional probability P(Y K (1) = b|Y K (0) = a) is then described by the following Theorem that yields the distribution of the (ǫ m ) 1≤m≤N . Theorem 3.1. Let N be the number of blocks of constancy of a. The distribution of (ǫ m ) 1≤m≤N is given by :

• Initializing phase

P(ǫ 1 = 1|a) = M 1 M 1 + 1 + x N 2 . (6)
• Subsequent phase when previously there is no change

P(ǫ m = 1|ǫ m-1 = 0, . . . , ǫ 1 = 0, a) = M m M m + 1 + x N m+1 . (7)
• Subsequent phase when previously there is at least one change. Let us now investigate as an example the probability of loops :

P((Y K (0), Y K (1)) = (a, a) + |Y K (0) = a) = P((Y K (0), Y K (1)) = (a, a) -|Y K (0) = a) Corollary 3.1 (Loop). If ǫ m = 0 for 1 ≤ m ≤ N which is equivalent to b = a, P(((ǫ 1 , . . . , ǫ N ) = (0, . . . , 0)) = 2P((Y K (0), Y K (1)) = (a, a) + |Y K (0) = a), and
P((Y K (0), Y K (1)) = (a, a) + |Y K (0) = a) = 1 2(M 1 + 1 + x N 2 ) N k=2 x N k . ( 9
)
If you give yourself a ∈ V K , the list of the neighbors can be generated by induction. But, when K is large, it is time consuming. Actually the same type of arguments used for the previous theorem, yield a way to obtain this cardinal, which will be denoted by deg K a.

p m+2 = M N -m-1 p m+1 + p m . ( 10
)
Then deg K a = p N + p N -1 .
For instance, when a = ((-1) k ) ∈ V K , N=K, and M m = 1, ∀m ≤ N. In this case (p m ) and deg K (a) are Fibonacci sequences.

Transition probability for K = ∞

One important consequence of the previous results is the fact that the graph andE ∞ is defined with the same alternate signs rule as for finite K. The Lemma 2.1 is still true when K infinite.

G K is defined for K = ∞. Let us be more precise. When K = ∞, V ∞ def = {-1, 1} N ,
Please note that the law of large number implies that lim

K→∞ 1 K K k=1 (1 -α k )β k = 0. Moreover K k=1 α k γ k = ±1. Hence lim K→∞ 1 K K k=1 A k = 0 and the se- quence (A k ) k≤K is not constant for k big enough almost surely. It means that if K = ∞, N = ∞ almost surely.
When K = ∞ the continuous fraction in ( 5) is converging when n → ∞ toward an irrational number that will be denoted by x ∞ m . When K = ∞, Lemma 2.1 still yields the distribution (A, B) of the "uniform "measure on E ∞ . "Uniform "is between quotes, since a has an infinite number of neighbors. In this case P(A = a) = 0, ∀a ∈ V ∞ , but the distribution of B conditioned on A is properly defined. Moreover the previous Theorem still has a generalization when we consider that the number of constancy blocks of a is almost surely infinite and use the definition of the continuous fraction as a limit. The next theorem, which is mutatis mutandis the theorem when K finite, yields for a ∈ V ∞ a regular version of the conditional distribution of B conditioned on A = a. Theorem 3.2. The distribution of (ǫ m ) 1≤m is given by :

• Initializing phase

P(ǫ 1 = 1|a) = M 1 M 1 + 1 + x ∞ 2 .
(11)

• Subsequent phase when previously there is no change

P(ǫ m = 1|ǫ m-1 = 0, . . . , ǫ 1 = 0, a) = M m M m + 1 + x ∞ m+1 . (12)
• Subsequent phase when previously there is at least one change. For all e ∈ C * m-1 such that ē = (e 1 , . . . , e m-1 , 1)

∈ C m P(ǫ m = 1|(ǫ 1 , . . . , ǫ m-1 ) = e, a) = M m x ∞ m . ( 13
)
∀m ≥ 1 the distribution of the E m 's is uniform on {1, . . . , M m } conditionally to the event ǫ m = 1.

Remark 3.3. Please note that the probability of a loop in G ∞ is vanishing.

Let us illustrate the Theorem with an example. If a = ((-1) k ) ∈ V ∞ , we can better understand what is P(|a). In this case ∀m ≥ 1, M m = 1 and

x ∞ m = 1 1 + x ∞ m+1 , ∀m ≥ 1. Hence ∀m ≥ 1, x ∞ m = -1 + √ 5 2
.

The depiction of P(|a) is easier in this case because x ∞ m does not depend on m and we will write p = -1+ √ 5 2

. The initializing phase yields

P(ǫ m = 1|ǫ m-1 = 0, . . . , ǫ 1 = 0, a) = 1 -p.
Let T be the index of the first digit of B which is different from the digit in A, the distribution of T conditioned on A = a has a geometric distribution with parameter p.

The distribution of the digits of B conditioned on A = a for the indices after T are given by the subsequent phase when previously there is at least one change. The easy part is to get P(B t+1 = A t+1 |a, T = t) = p. Then we will derive from an independent sequence (η m ) m≥1 of Bernoulli variables with parameter p a sequence (ǫ m ) m≥1 of random variables which almost surely satisfy the alternate signs rule. The heuristic is to substitute to every η = 0 two 0's in ǫ. More precisely let (η m ) m≥1 be a sequence of independent Bernoulli random variables with parameter p, define the following sequence (ǫ m ) m≥1 of random variables by induction as follows : if η 1 = 1, ǫ1 = 1. If η 1 = 0, ǫ1 = 0, ǫ2 = 0. Then suppose (ǫ m ) 1≤m≤M has been defined using n η's. If η n+1 = 1, ǫM+1 = 1. If η n+1 = 0, ǫM+1 = 0, ǫM+2 = 0. Please remark that almost surely (ǫ m ) 1≤m≤M ∈ C M . Then equation (13) shows that the distribution of (B k+T ) k≥1 conditionally to a and T is the distribution of (-1) k+ǫ k .

Proofs of the result 4.1 Conditional independence

Let us now introduce the conditional independence with respect of γ k , which is an important tool for our computations. Let us denote by

σ i l def = σ(A k , B k , l ≤ k ≤ i), for 2 ≤ l ≤ i. By convention σ i 1 def = σ(A k , B k , 1 ≤ k ≤ i, ǫ). Lemma 4.1. For every 1 ≤ l ≤ i < j σ i l is independent of σ j i+1 conditionally to γ i+1 .
The proof of this Lemma comes from the definitions of Lemma 2.1. For every m ≤ n, let us define

u n m def = P(γ S m-1 +1 = i m ∩ A (m) ∩ . . . ∩ A (n) ) and v n m def = P(γ S m-1 +1 = -i m ∩ A (m) ∩ . . . ∩ A (n) ).
With the help of Lemma 4.1 we can compute u n m , v n m , by induction starting from u n n , v n n , and we get the following result. for 1 ≤ m ≤ n ≤ N, where x n m is the continued fraction defined in [START_REF] Lammers | Diffusivity of a walk on fractures of a hypertorus[END_REF]. Proof. By conditioning and Lemma 4.1 we may write

u n m = P(γ S m-1 +1 = i m , A (m) |γ Sm+1 = i m ) × P(γ Sm+1 = -i m+1 , A (m+1) ∩ . . . ∩ A (n) ) + P(γ S m-1 +1 = i m , A (m) |γ Sm+1 = -i m ) × P(γ Sm+1 = i m+1 , A (m+1) ∩ . . . ∩ A (n) ).
This equation yields

u n m = P(γ S m-1 +1 = i m ∩ A (m) |γ Sm+1 = i m )v n m+1 ( 16
) + P(γ S m-1 +1 = i m ∩ A (m) |γ Sm+1 = -i m )u n m+1 .
Observe that

{γ S m-1 +1 = i m ∩ A (m) ∩ γ Sm+1 = i m } = {α l = 0, β l = i m for S m-1 + 1 ≤ l ≤ S m } ∩ {γ S m-1 +1 = i m }. ( 17
)
The two events on the right hand side are independent and the probability of the first one is 1 3 Mm . Therefore

P(γ S m-1 +1 = i m ∩ A (m) |γ Sm+1 = i m ) = P(γ S m-1 +1 = i m ∩ A (m) ∩ γ Sm+1 = i m ) P(γ Sm+1 = i m ) . Hence P(γ S m-1 +1 = i m ∩ A (m) |γ Sm+1 = i m ) = 1 3 Mm . ( 18
)
With a similar argument we get

P(γ S m-1 +1 = i m ∩ A (m) |γ Sm+1 = -i m ) = M m 3 Mm . (19)
Moreover

v n m = P(γ S m-1 +1 = -i m ∩ A (m) |γ Sm+1 = -i m ) × P(γ Sm+1 = i m+1 ∩ A (m+1) ∩ . . . ∩ A (n) ) + P(γ S m-1 +1 = -i m ∩ A (m) |γ Sm+1 = i m ) × P(γ Sm+1 = -i m+1 ∩ A (m+1) ∩ . . . ∩ A (n) ).
Note that P(γ S m-1 +1 = -i m ∩ A (m) ∩ γ Sm+1 = i m ) = 0, since, on this event, the value -i m of γ S m-1 +1 does not fit the value of a l 's on A (m) . Hence all the α l = 0 and γ Sm+1 = γ S m-1 +1 . It follows that

v n m = P(γ S m-1 +1 = -i m ∩ A (m) |γ Sm+1 = -i m )u n m+1 . (20)
The equation ( 14) is the consequence of equations ( 16), (20), ( 18), (19).

(ǫ m = 1|ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ N n=1 A (n) ) for N > m ≥ 2.
The numerator of the conditional probability is

P(ǫ m = 1, ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ N n=1 A (n) ) = P(ǫ m = 1, ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ N n=1 A (n) , γ Sm+1 = i m+1 ) = 2P(ǫ m = 1, ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ m n=1 A (n) ) × P(γ Sm+1 = i m+1 , ∩ N n=m+1 A (n) ) = 4P(ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ m-1 n=1 A (n) , γ S m-1 +1 = i m ) × P(ǫ m = 1, A (m) )u N m+1 ,
where the last two equalities come from conditional independence with respect to γ taken at the convenient index. For the denominator of the conditional probability of the title of the section, the same kind of manipulations yield

P(ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ N n=1 A (n) ) = 2P(ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ m-1 n=1 A (n) )P(γ S m-1 +1 = i m , ∩ N n=m A (n) ) = 2P(ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ m-1 n=1 A (n) )u N m .
Then

P(ǫ m = 1|ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ N n=1 A (n) ) = 2P(ǫ m = 1, A (m) ) u N m+1 u N m and P(ǫ m = 1, A (m) ) = 1 2 M m 3 Mm yield P(ǫ m = 1|ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ N n=1 A (n) ) = M m 3 Mm u N m+1 1 3 Mm (M m u N m+1 + v N m+1 )
, If we consider the numerator of the conditional probability of the second line we get :

P(ǫ m = 1, ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∀1 ≤ i ≤ m -2k, ǫ i = e i , a)
is also equal to

P(ǫ m = 1, ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, γ S m-2k +1 = i m-(2k-1) , ∀1 ≤ i ≤ m -2k, ǫ i = e i , a).
Then by conditioning by γ S m-2k +1 = i m-(2k-1) it is also equal to

P(ǫ m = 1, ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∩ i≥m-(2k-1) A (i) |γ S m-2k +1 = i m-(2k-1) ) × P(∀1 ≤ i ≤ m -2k, ǫ i = e i , ∩ m-2k i=1 A (i) |γ S m-2k +1 = i m-(2k-1) ) 2 .
But the same argument holds also for the denominator and the factor

P(∀1 ≤ i ≤ m -2k, ǫ i = e i , ∩ m-2k i=1 A (i) |γ S m-2k +1 = i m-(2k-1)
) is simplified in the conditional probability P(ǫ m = 1|ǫ m-1 = 0, . . . , ǫ m-(2k-1) = 1, ∀1 ≤ i ≤ m -2k, ǫ i = e i , a), which yields equation (31). The previous argument works for m < N, but it can be generalized to m = N and also to the case K = ∞. Then P(γ 1 = ν, ǫ m-1 = 0, . . . , ǫ 1 = 0, ∩ N n=1 A (n) ) = 2P(γ 1 = ν, ǫ 1 = 0, A (1) )P(ǫ m-1 = 0, . . . , ǫ 2 = 0, γ S 1 +1 = ν, ∩ N n=2 A (n) ) = P(ǫ 1 = 0, A (1) )P(ǫ m-1 = 0, . . . , ǫ 2 = 0, γ S

M m 3 Mm 1 u m m + v m m (36) = M m M m + 2 . ( 37
)
Equations ( 33) and (37) yield [START_REF] Montegut | Limite d'échelle de marche aléatoires contraintes[END_REF] in view of the convention x n n+1 = 1.

Distribution of the E m 's

To conclude the proof of Theorem 3.1 and 3.2, let us prove that ∀m ≥ 1 the distribution of the E m 's is uniform on {1, . . . , M m } conditionally to the event ǫ m = 1. Let us fix m 0 such that 2 ≤ m 0 ≤ N. We have to show that ∀j 0 ∈ {1, . . . , M m 0 } P(E m 0 = j 0 |a, ǫ m 0 = 1)

does not depend on j 0 . We have P(E m 0 = j 0 , ∩ N m=1 A (m) , ǫ m 0 = 1) = P(E m 0 = j 0 , ǫ m 0 = 1, γ S m 0 -1 +1 = i m 0 , γ Sm 0 +1 = i m 0 +1 , ∩ N m=1 A (m) )

which is also equal to 2P(∩ m 0 -1 m=1 A (m) , γ S m 0 -1 +1 = i m 0 ) × P(E m 0 = j 0 , ǫ m 0 = 1, γ S m 0 -1 +1 = i m 0 , γ Sm 0 +1 = i m 0 +1 , ∩ N m=m 0 +1 A (m) ) and = 4P(∩ m 0 -1 m=1 A (m) , γ S m 0 -1 +1 = i m 0 ) × P(E m 0 = j 0 , ǫ m 0 = 1, A (m 0 ) )P(γ S m 0 -1 +1 = i m 0 +1 , ∩ N m=m 0 +1 A (m) ) thanks to the conditional independence when γ is given. Furthermore P(E m 0 = j 0 , A (m 0 ) , ǫ m 0 = 1) = P(α j 0 = 1, ∀i = j 0 , α i = 0, A (m 0 ) )

does not depend on j 0 and consequently P(E m 0 = j 0 , A (m 0 ) , ǫ m 0 = 1) does not depend on j 0 . It is also the case for P(E m 0 = j 0 , ∩ N m=1 A (m) , ǫ m 0 = 1|a, ǫ m 0 = 1) since P(∩ N m=1 A (m) , ǫ m 0 = 1) does not depend on j 0 . The proof is easier when m 0 = 1.

Proposition 2 . 1 (

 21 Alternate signs rule). If a, b ∈ V K a ∼ b ⇔ b -a follows the alternate signs rule.

  convention. Let us now describe when Y K (0) = a and Y K (1) = b the sequence of the digits of b by comparison with those of a. Please remark that at most one digit b k of b is different of a k in any block of constancy A (m) of a, because of the alternate signs rule. Let (ǫ m ) 1≤m≤N be {0, 1}-valued random variables such that ǫ m = 1 if and only if there is one change of digit between a and b in the m-th block of constancy A (

  and C * n def = C n \ {e ∈ {0, 1} n |∀1 ≤ m ≤ n, e m = 0}. Let us remark that if e ∈ C n and n ′ ≤ n, (e 1 , . . . , e n ′ ) ∈ C n ′ . The definition of C n yields that if e / ∈ C n , P((ǫ 1 , . . . , ǫ n ) = e) = 0. Moreover if e ∈ C m-1 and ē = (e 1 , . . . , e m-1 , 1) / ∈ C m , it means that P((ǫ 1 , . . . , ǫ m ) = ē) = 0. Therefore we only have to compute P(ǫ m = 1|(ǫ 1 , . . . , ǫ m-1 ) ∈ C m-1 ) when (ǫ 1 , . . . , ǫ m ) ∈ C m .

1 . 3 . 2 .

 132 For all e ∈ C * m-1 such that ē = (e 1 , . . . , e m-1 , 1) ∈ C m P(ǫ m = 1|(ǫ 1 , . . . , ǫ m-1 ) = e, a) = M m x N m . (8) ∀1 ≤ m ≤ N the distribution of the E m 's is uniform on {1, . . . , M m } conditionally to the event ǫ m = Remark Equations (8) implies roughly that the law of ǫ m , conditioned to a and to the fact that there has been a change in digits of a before the index m, is independent of the last time a change has been made and what happens before, if it is of positive probability. It is a consequence of the fact that the right hand side of (8) does depend neither on the last time a change has been made nor on e ∈ C * m-1 . Theorem 3.1 allows us to compute for any b ∼ a, the P(Y K (1) = b|Y K (0) = a). Any pair (a, b) ∈ E K yields a sequence e ∈ {0, 1} N , where e m = 1 if and only if there is one digit b k in A (m) such that b k = a k . Because of the alternate signs rule e ∈ C N and the probability P((ǫ 1 , . . . , ǫ N ) = e) is a product of the probabilities given in Theorem 3.1 by applying successive conditioning.

Proposition 3 . 1 .

 31 If a ∈ V K , corresponds to N blocks of constancy of lengths M 1 , . . . , M N , we can define by induction a sequence (p m ) 0≤m≤N such that p 0 = 1, p 1 = M N + 1, and ∀ 0 ≤ m ≤ N -2,

Lemma 4 . 2 .

 42 For every m < n,

. 4 . 3 . 1 P

 431 The equation (14) yields for m + 1 ≤ n, (ǫ m = 1|(ǫ 1 , . . . , ǫ m-1 ) = e, a) For all e ∈ C * m-1 such that ē = (e 1 , . . . , e m-1 , 1) ∈ C m m(e) = m-(2k-1) for a k depending on e. Let us first compute P

4. 3 . 2 P+1 ν=- 1 P(γ 1 =

 3211 (ǫ m = 1|ǫ m-1 = 0, . . . , ǫ 1 = 0, ∩ N n=1 A (n) ) Let us start with the denominator and assume until further notice that N > m ≥ 2,P(ǫ m-1 = 0, . . . , ǫ 1 = 0, ∩ N n=1 A (n) ) = ν, ǫ m-1 = 0, . . . , ǫ 1 = 0, ∩ N n=1 A (n) ).

1

 1 +1 = ν, ∩ N n=2 A (n) n = 0, A (n) )P(γ S m-1 +1 = ν, ∩ N n=m A (n) ). P(ǫ m = 1|ǫ m-1 = 0, . . . , ǫ 1 = 0, ∩ m n=1 A (n) ) = P(ǫ m = 1, A (m) )

Acknowledgment

The authors would like to thank James Norris and Fabien Montegut for fruitful discussions.

Moreover

u n n = P(γ S n-1 +1 = i n , a j = i n , S n-1 + 1 ≤ j ≤ S n )

= P(γ S n-1 +1 = i n , β j = i n , α j = 0, S n-1 + 1 ≤ j ≤ S n ) + Sn j=S n-1 +1

P(γ S n-1 +1 = i n , ∀j = j 0 β j = i n , α j = 0, and α j 0 = 1) = 1 2

which yields (15).

Remark 4.1. Please note that we can use the induction of Lemma 4.2 even if K = ∞. We recall that, when n → ∞, x n m is converging to the irrational number that we will denote by x ∞ m .

Initializing phase

In this part we will compute the conditional probability given a that there is a digit b j = a j in the first block of constancy of a namely A (1) . Hence when K is finite if a has N blocks of constancy we are aiming for P(ǫ 1 = 1|∩ N m=1 A (m) ). When K is infinite we want to compute the limit of the previous probability when N → ∞. For N ≥ n ≥ 2, we start by computing

.

Since P(γ S 1 +1 = ±1) = 1 2 and P(ǫ 1 = 0, A (1) , γ S 1 +1 = ±1) = P(ǫ 1 = 0, A (1) , ǫ = ±1);

this yields

which can be written

Similarly

Since P(γ

) = 0, by summing the previous probabilities, we obtain

and

Equation ( 25) yields equation ( 6) and becomes

when K is infinite by letting n → ∞. When N = 1, x N 2 is not defined but we may compute P(ǫ 1 = 1|A (1) ) as follows. If N = 1 it means that the vertex a has a single block of constancy. Hence A (1) = {a}, and the vertex a has M 1 neighbors in the graph which are different from a and there are two edges that starts from a and ends at a. Hence, when N = 1, equation (25) becomes

which is coherent with the convention x 1 2 = 1.

When K is infinite, we can let N → ∞ to obtain

When m = N, we have to change the computation of the numerator. Actually in this case

Since

we get

Because of ( 23)

Hence equation ( 28) is always verified. Let us recall that for all e ∈ C * m-1 such that ē = (e 1 , . . . , e m-1 , 1) ∈ C m m(e) = m-(2k -1) for a k depending on e. Let us now add the conditioning by what happens before m -(2k -1) for all k such that 2k -1 < m, and all sequences (e i ) 1≤i≤m-2k ∈ C m-2k we will prove that

The previous formula is also true for m = N. Moreover

for N > m. Hence

We have used again

M m 3 Mm and Lemma 4.2 in the previous computations. When K = ∞ we let N → ∞ in (33) and we get

When m = N,

Then

.

If we recall x N N = 1 M N +1 and x N N +1 = 1 it yields equation (9).

Proof of the proposition 3.1

Let us first recall that there are two loops in E K with base point a ∈ V K : (a, a) + , (a, a) -. So a is counted twice in the cardinal of the set of the neighbors of a, which is the degree of a. Then by definition x

which is a consequence of (6,8). Then

Hence deg K (a) = p N + p N -1 .

The proof of Proposition 3.1 can be made directly without using Theorem 3.1. It is for the sake of brevity that we view it as a Corollary of Theorem 3.1.


O Chdil

M Balli

N Brahiti

R Essehli

Patricia De Rango

P Fournier

S Naamane

K El Maalam

O Mounkachi

K El Maalam

Depicting the roles of CuO secondary phase and heat treatment in driving the magnetic and magnetocaloric features of Pr 2/3 Sr 1/3 MnO 3 manganite

Keywords: Magnetocaloric effect, Manganites composites, Secondary phase, Heat treatment, Grains size

come

Depicting the roles of CuO secondary phase and heat treatment in driving the magnetic and magnetocaloric features of Pr2∕ 3Sr1∕ 3MnO3 manganite

Introduction

Near-room temperature solid-state magnetic refrigeration is a new and rapidly growing area of materials science and engineering [START_REF] Gschneidner | Thirty years of near room temperature magnetic cooling: Where we are today and future prospects[END_REF][START_REF] Gutfleisch | Magnetic materials and devices for the 21st century: stronger, lighter, and more energy efficient[END_REF][START_REF] Liu | Giant magnetocaloric effect driven by structural transitions[END_REF][START_REF] Balli | A pre-industrial magnetic cooling system for room temperature application[END_REF][START_REF] Engelbrecht | Experimental results for a novel rotary active magnetic regenerator[END_REF][START_REF] Moya | Giant and reversible extrinsic magnetocaloric effects in La 0.7 Ca 0.3 MnO 3 films due to strain[END_REF], due to its high thermal efficiency and the absence of air pollutants. This makes magnetic refrigeration a serious alternative to conventional techniques [START_REF] Tishin | The magnetocaloric effect and its applications[END_REF][START_REF] Balli | Advanced materials for magnetic cooling: Fundamentals and practical aspects[END_REF]. Currently, there are several prototype devices worldwide that use the magnetocaloric effect to generate higher temperature spans and cooling powers [START_REF] Yu | A review of magnetic refrigerator and heat pump prototypes built before the year 2010[END_REF][START_REF] Scarpa | A classification methodology applied to existing room temperature magnetic refrigerators up to the year 2014[END_REF]. However, there is currently no mass-market product available for purchase [START_REF] Kitanovski | Present and future caloric refrigeration and heat-pump technologies[END_REF]. This is because various hurdles must be overcome before magnetic refrigeration can be commercialized, such as corrosion and mechanical brittleness issues, as well as the discovery of a magnetocaloric material with large MCE over a wide temperature range combined with low hysteresis, low cost, and high electrical resistance. Magnetic refrigeration is based on the magnetocaloric effect (MCE) of magnetic materials, which are used to create a regenerator matrix representing the heart of magnetocaloric systems [START_REF] Smith | Materials challenges for high performance magnetocaloric refrigeration devices[END_REF][START_REF] Gottschall | Making a cool choice: the materials library of magnetic refrigeration[END_REF][START_REF] Kitanovski | Energy applications of magnetocaloric materials[END_REF]. The MCE is defined as the change in temperature (magnetic entropy) of a magnetic material when it is adiabatically exposed to an external magnetic field (isothermally) [START_REF] Franco | The magnetocaloric effect and magnetic refrigeration near room temperature: materials and models[END_REF]. The atomic and spin latices are the two major contributors to the overall change in entropy. Under the influence of an external magnetic field, the coupling between atomic and magnetic lattices, in typical ferromagnets and paramagnets, results in the material cooling (heating) when the magnetic moments are disordered (ordered) [START_REF] Oliveira | Theoretical aspects of the magnetocaloric effect[END_REF]. An effective refrigerant material must have a higher ΔS m and ΔT ad values, high thermal conductivity, and low specific heat capacity [START_REF] Tegus | Transition-metalbased magnetic refrigerants for room-temperature applications[END_REF][START_REF] Terada | High-efficiency magnetic refrigeration using holmium[END_REF]. Various families of materials were implemented in magnetic cooling, including gadolinium, rare earth alloys, and intermetallics [START_REF] Franco | Magnetocaloric effect: From materials research to refrigeration devices[END_REF][START_REF] Gimaev | Review on magnetic refrigeration devices based on htsc materials[END_REF][START_REF] Kamran | Review on the developments of active magnetic regenerator refrigerators-evaluated by performance[END_REF][START_REF] Krenke | Inverse magnetocaloric effect in ferromagnetic Ni-Mn-Sn alloys[END_REF][START_REF] Barman | Caloric effects in perovskite oxides[END_REF][START_REF] Zarkevich | Viable materials with a giant magnetocaloric effect[END_REF]. Unfortunately, the majority of these materials exhibit chemical instabilities (corrosion and oxidation) in aqueous environments, being harmful to active magnetic refrigerators (AMR) [START_REF] Smith | Materials challenges for high performance magnetocaloric refrigeration devices[END_REF][START_REF] Gottschall | Making a cool choice: the materials library of magnetic refrigeration[END_REF][START_REF] Franco | The magnetocaloric effect and magnetic refrigeration near room temperature: materials and models[END_REF]. Additionally, most of the reported MCE materials include costly components or require complicated and expensive production processes [START_REF] Kitanovski | Energy applications of magnetocaloric materials[END_REF][START_REF] Franco | Magnetocaloric effect: From materials research to refrigeration devices[END_REF][START_REF] Zhang | Solutions to obstacles in the commercialization of room-temperature magnetic refrigeration[END_REF].

In this context, the mixed-valence manganese perovskites, with the chemical formula R 1-x A x MnO 3 , in which R is a rare-earth cation (Pr 3+ , La 3+ , Nd 3+ , Y 3+ ) and A an alkali or alkaline earth cation (Sr 2+ , Ca 2+ , Ba 2+ , K + , Na + ), have recently attracted the interest of researchers due to their significant magnetocaloric effects near ambient temperature and the possibility of tailoring their magnetocaloric properties because of the strong coupling between their spin, charge, orbital, and lattice degrees of freedom [START_REF] Bahl | High performance magnetocaloric perovskites for magnetic refrigeration[END_REF][START_REF] Balli | Review of the magnetocaloric effect in RMnO 3 and RMn 2 O 5 multiferroic crystals[END_REF][START_REF] Chaudhary | Iron and manganese based magnetocaloric materials for near room temperature thermal management[END_REF]. Furthermore, they are also suitable for magnetic refrigeration applications owing to their relatively low cost, easy preparation, increased chemical stability, reduced magnetic hysteresis, and low eddy current heating [START_REF] Phan | Review of the magnetocaloric effect in manganite materials[END_REF]. Manganites with mixed-valence states may be considered as a solid solution combining end members like La 3+ Mn 3+ O 2- 3 and Ca 2+ Mn 4+ O 2- 3 , resulting in mixed-valence La 3+ 1-x Ca 2+

x Mn 3+ 1-x Mn 4+

x O 2- 3 compounds [START_REF] Morelli | Magnetocaloric properties of doped lanthanum manganite films[END_REF][START_REF] Guo | Large magnetic entropy change in perovskite-type manganese oxides[END_REF]. Typically, end members are antiferromagnetic and insulating, but solid solutions with x = 1/3 are ferromagnets and conductors [START_REF] Coey | Mixed-valence manganites[END_REF]. Therefore, the specific composition, R 2/3 A 1/3 MnO 3 , is attractive for magnetic refrigeration, since a ratio of Mn 4+ /Mn 3+ equal to 0.5 allows combining large magnetization and high Curie temperature [START_REF] Guillou | Development of a new magnetocaloric material used in a magnetic refrigeration device[END_REF]. Moreover, the MnO 6 octahedral environment in manganites may be altered chemically by substituting A-or Mn-sites, which primarily affects the charge density and produce lattice distortions. The doping of A 2+ site with divalent ions leads to mixed-valence manganite systems in which ferromagnetic (FM) double exchange (DE) Mn 3+ -O 2--Mn 4+ competes with antiferromagnetic (AFM) super-exchange Mn 3+ -O 2--Mn 3+ [START_REF] Zener | Interaction between the d-shells in the transition metals. ii. ferromagnetic compounds of manganese with perovskite structure[END_REF][START_REF] Anderson | Considerations on double exchange[END_REF][START_REF] Kaplan | Physics of manganites[END_REF][START_REF] Rocco | Dependence of the magnetocaloric effect on the A-site ionic radius in isoelectronic manganites[END_REF]. The magnetic and magnetocaloric properties of these materials are highly dependent on the strength of DE interactions between Mn 3+ /Mn 4+ ions, which are caused by the e g electrons migration between two partly filled d-orbitals [START_REF] Ahn | Magnetic properties and colossal magnetoresistance of La(Ca)MnO 3 materials doped with Fe[END_REF]. Moreover, the discrepancy in the average ionic radii of A-site cations results in variations in the Mn-O-Mn bond angles and Mn-O bond lengths, leading to the octahedral tilting of MnO 6 [START_REF] Thaljaoui | Magnetocaloric effect of monovalent K doped manganites Pr 0.6 Sr 0.4-x K x MnO 3 (x= 0 to 0.2)[END_REF][START_REF] Boudjada | Impact of sintering temperature on the magnetic and magnetocaloric properties in Pr 0.5 Eu 0.1 Sr 0.4 MnO 3 manganites[END_REF][START_REF] Daivajna | Magnetocaloric effect in pristine and Bi-doped Pr 0.6 Sr 0.4 MnO 3 manganite[END_REF][START_REF] Szewczyk | Specific heat and magnetocaloric effect in Pr 0.6 Sr 0.4-x Ag x MnO 3 manganites[END_REF][START_REF] Gamzatov | Correlation of the magnetocaloric effect and magnetostriction near the first-order phase transition in Pr 0.7 Sr 0.2 Ca 0.1 MnO 3 manganite[END_REF][START_REF] Batdalov | Magnetocaloric properties in the Pr 0.7 Sr 0.3-x Ca x MnO 3 : Direct and indirect estimations from thermal diffusivity data[END_REF]. In contrast to the A-site substitution, the replacement of a transition metal ion at the Mn site may result in altered magnetic interactions between Mn and the replaced ion, transforming then the magnetic order from FM to an AFM state or conversely [START_REF] Nasri | Study of magnetic transition and magnetic entropy changes of Pr 0.6 Sr 0.4 MnO 3 and Pr 0.6 Sr 0.4 Mn 0.9 Fe 0.1 O 3 compounds[END_REF][START_REF] Gamzatov | Specific heat, thermal diffusion, thermal conductivity and magnetocaloric effect in Pr 0.6 Sr 0.4 Mn 1-x Fe x O 3 manganites[END_REF][START_REF] Gamzatov | Heat capacity and the magnetocaloric effect in Pr 0.6 Sr 0.4 Mn 1-x Fe x O 3 manganite[END_REF][START_REF] Snini | Structural, magnetic and magnetocaloric investigations in Pr 0.67 Ba 0.22 Sr 0.11 Mn 1-x Fe x O 3 (0 ≤ x ≤ 0.15) manganite oxide[END_REF].

Numerous techniques have been proposed with the aim of tailoring the magnetocaloric effect of R 1-x A x MnO 3 manganite oxides, including hydrostatic pressure [START_REF] Markovich | Pressure effects on the magnetic and transport properties of Pr 1-x Sr x MnO 3 crystals near the percolation threshold[END_REF][START_REF] Markovich | Pressure effect on magnetism in phaseseparated Cr-doped Pr 0.5 Ca 0.5 Mn 1-x Cr x O 3 manganites[END_REF][START_REF] Thiyagarajan | Effect of hydrostatic pressure on magnetic entropy change and critical behavior of the perovskite manganite La 0.4 Bi 0.3 Sr 0.3 MnO 3[END_REF], doping with other rare earth elements [START_REF] Khlifa | Structural, magnetic and magnetocaloric properties of K-doped Pr 0.8 Na 0.2-x K x MnO 3 manganites[END_REF][START_REF] Arun | Mixed rare earth oxides derived from monazite sand as an inexpensive precursor material for room temperature magnetic refrigeration applications[END_REF], and the insertion of secondary phases [START_REF] Vandrangi | Enhanced magnetocaloric effect driven by interfacial magnetic coupling in self-assembled Mn 3 O 4 -La 0.7 Sr 0.3 MnO 3 nanocomposites[END_REF][START_REF] Marzouki-Ajmi | 11 Magnetic and magnetocaloric study of polycrystalline (1-x)La 0.65 Ca 0.35 MnO 3 /xFe 2 O 3 composites[END_REF][START_REF] Vadnala | Investigation of near room temperature magnetocaloric, magnetoresistance and bolometric properties of Nd 0.5 La 0.2 Sr 0.3 MnO 3 : Ag 2 O manganites[END_REF][START_REF] Nasri | Impact of CuO phase on magnetocaloric and magnetotransport properties of La 0.6 Ca 0.4 MnO 3 ceramic composites[END_REF][START_REF] Mohamed | Annealing temperature effect on magnetic and magnetocaloric properties of manganites[END_REF][START_REF] Mohamed | Tuning magnetoresistive and magnetocaloric properties via grain boundaries engineering in granular manganites[END_REF][START_REF] Gavrilova | Magnetic and magnetocaloric properties of (1-x)La 0.7 Sr 0.3 MnO 3 /xNaF composites[END_REF][START_REF] Amri | Synthesis and characterization of (1 x)(La 0.6 Ca 0.4 MnO 3 )/x(Sb 2 O 3 ) ceramic composites[END_REF]. In fact, both the structural temperature transition and the Curie temperature (T c ) increase linearly upon increasing the applied hydrostatic pressure [START_REF] Markovich | Pressure effects on the magnetic and transport properties of Pr 1-x Sr x MnO 3 crystals near the percolation threshold[END_REF]. Moreover, as revealed by Sethupathi et al. [START_REF] Mahato | Large magnetic entropy change in nanocrystalline Pr 0.7 Sr 0.3 MnO 3[END_REF], the increase of praseodymium (Pr) content in Pr 1-x Sr x MnO 3 decreases only the material Curie temperature, whereas the magnetocaloric properties remain almost unchanged. Apart from chemical replacement and hydrostatic pressure, nanoscale particle size reduction has been utilized to modify the physical characteristics of manganites [START_REF] Ng | Grain size effect on the electrical and magneto-transport properties of nanosized Pr 0.67 Sr 0.33 MnO 3[END_REF][START_REF] Souza | Effect of nanoscale size reduction on the magnetic properties of Pr 0.6 Sr 0.4 MnO 3[END_REF]. Additionally, the strain effect has also been shown to be an efficient approach for enhancing the magnetocaloric characteristics of oxides [START_REF] Bouhani | Engineering the magnetocaloric properties of PrVO 3 epitaxial oxide thin films by strain effects[END_REF]. On the other hand, inserting a secondary phase into the manganite-based materials has been extensively investigated, initially for the purpose of improving low-field magnetoresistance (LFMR) characteristics [START_REF] Fan | Microstructure, magnetic, and magnetoresistance properties of La 0.7 Sr 0.3 MnO 3 : CuO nanocomposite thin films[END_REF][START_REF] Chen | Silver addition in polycrystalline La 0.7 Ca 0.3 MnO 3 : Large magnetoresistance and anisotropic magnetoresistance for manganite sensors[END_REF][START_REF] Shah | Electrical, magnetotransport and magnetization studies of (1-x)La 0.7 Sr 0.3 Mn 0.94 Co 0.06 O 3 /xZnO (x= 0, 0.04) composite[END_REF]. The reported studies demonstrate that incorporation of an insulating secondary phase into the manganite materials is an efficient method for creating phase boundaries and accordingly generating a colossal low-field magnetoresistance [START_REF] Huang | Lowfield MR behaviour in La 0.67 Ca 0.33 MnO 3 /ZrO 2 composite system[END_REF][START_REF] Karmakar | Study of grain boundary contribution and enhancement of magnetoresistance in La 0.67 Ca 0.33 MnO 3 /V 2 O 5 composites[END_REF][START_REF] Yang | Fabrication of 0-3 type manganite/insulator composites and manipulation of their magnetotransport properties[END_REF][START_REF] Gao | Tunable low-field magnetoresistance properties in (La 0.7 Ca 0.3 MnO 3 ) 1-x : (CeO 2 ) x vertically aligned nanocomposite thin films[END_REF]. On the other hand, these findings have motivated researchers to investigate the impact of adding secondary phases to the manganite matrix aiming to tailor and/or enhance the magnetocaloric properties. In the reported work by El Maalam et al. [START_REF] Maalam | Composite (La 0.45 Nd 0.25 )Sr 0.3 MnO 3 /5CuO materials for magnetic refrigeration applications[END_REF], it was shown that the mixture of La 0.45 Nd 0.25 Sr 0.3 MnO 3 with a small amount of CuO (typically 5% weight ratio) induces a slight decrease in the Curie temperature and enhances the magnetocaloric effect, which is, according to the authors, attributed to the larger grain size. Bhatt et al. [START_REF] Bhatt | Near room temperature magneto-transport (TCR & MR) and magnetocaloric effect in Pr 2/3 Sr 1/3 MnO 3 : Ag 2 O composite[END_REF] have also investigated the effect of adding Ag 2 O to the pristine Pr 2/3 Sr 1/3 MnO 3 oxide, where it has been demonstrated that the magnetic entropy change exhibits surprisingly a rising trend as Ag 2 O concentration increases without affecting T c . Recently, Neupane et al. [START_REF] Neupane | Magnetocaloric study of La 0.45 Nd 0.25 Sr 0.3 MnO 3 /MO (MO= CuO, CoO, and NiO) nanocomposites[END_REF] reported a study in which they discussed the influence of NiO, CuO, and CoO transition metal oxides on the nanomaterial La 0.45 Nd 0.25 Sr 0.3 MnO 3 magnetocaloric effect. Their research work demonstrates the possibility of a potential nanocomposite with increased magnetocaloric characteristics. Especially, it was demonstrated that the La 0.45 Nd 0.25 Sr 0.3 MnO 3 -2.5 Wt. % CuO nanocomposite displays ΔS m -3.95 J kg -1 K -1 under a magnetic field of 5 T, which is 110% higher than its equivalent of pristine material. This is associated with the increase of T c from 240 K (pristine) to 284 K (composite). However, the driving mechanisms behind such marked changes regarding T c and the magnetic entropy changes are not clearly explained. In addition, Neupane et al. [START_REF] Neupane | Magnetocaloric study of La 0.45 Nd 0.25 Sr 0.3 MnO 3 /MO (MO= CuO, CoO, and NiO) nanocomposites[END_REF] findings are in full contradiction with El Maalam et al. [START_REF] Maalam | Composite (La 0.45 Nd 0.25 )Sr 0.3 MnO 3 /5CuO materials for magnetic refrigeration applications[END_REF] results.

From a practical point of view, the Pr 1-x Sr x MnO 3 series are one of the most interesting perovskites for magnetic refrigeration due to their large magnetization near to the ambient temperature and their strong chemical resistance that compensates for their relatively moderate magnetocaloric effect [START_REF] Phan | Large magnetocaloric effect in single crystal Pr 0.63 Sr 0.37 MnO 3[END_REF][START_REF] Markovich | Magnetic properties of perovskite manganites and their modifications[END_REF]. Additionally, the substitution approach can be used to modify the Curie temperature, with a change of about 4 K in T c per percent (%) of Sr [START_REF] Shaikh | Structural and electrical properties of Pr 1-x Sr x MnO 3 (x= 0.25, 0.3, 0.35 and 0.4) manganites[END_REF][START_REF] Saw | Study of transport, magnetic and magnetocaloric properties in Sr 2+ substituted praseodymium manganite[END_REF]. In the performed study by Repaka et al. [START_REF] Maheswar Repaka | Magnetocaloric effect and magnetothermopower in the room temperature ferromagnet Pr 0.6 Sr 0.4 MnO 3[END_REF], the magnetization and the magnetocaloric effect of the polycrystalline Pr 0.6 Sr 0.4 MnO 3 were particularly investigated. It was observed that the studied compound shows a second-order FM to paramagnetic (PM) transition at T c = 305 K, with a magnetic entropy change of 2.185 J kg -1 K -1 and 3.416 J kg -1 K -1 for a field change of 1.5 T and 3 T, respectively. Furthermore, Chanda et al. [START_REF] Chanda | Effects of external magnetic field and hydrostatic pressure on magnetic and structural phase transitions in Pr 0.6 Sr 0.4 MnO 3[END_REF] have demonstrated that for the same composition (Pr 0.6 Sr 0.4 MnO 3 ), a structural transition from the orthorhombic to the monoclinic phase occurs at T s = 86 K. The latter significantly increases under the application of hydrostatic pressure to reach 113 K, while the Curie temperature remains almost unchanged. On the other hand, Guillou et al. [START_REF] Guillou | Development of a new magnetocaloric material used in a magnetic refrigeration device[END_REF] have reported an experimental study on Pr 0.66 Sr 0.34 MnO 3 synthesized by the solid-state reaction method followed by a pelletization process. Particularly, it was shown that the Pr 0.66 Sr 0.34 MnO 3 compound exhibits an FM-PM transition at T c = 295 K. Moreover, under a magnetic field change of 1 T, a magnetic entropy variation and an adiabatic temperature change of 2.3 J kg -1 K -1 and 1.1 K can be reached respectively. Furthermore, the implementation of Pr 0.66 Sr 0.34 MnO 3 in a reciprocating AMR-magnetic cooling device enables to generate a temperature span of roughly 5 K under a magnetic field of 1 T [START_REF] Guillou | Development of a new magnetocaloric material used in a magnetic refrigeration device[END_REF]. On the other hand, we have recently reported a detailed numerical study regarding the real potential of Pr 0.6 Sr 0.4 MnO 3 in magnetic cooling. The obtained results have unveiled the high performance of this material particularly when fully taking advantage of its MCE. Furthermore, even though the Pr 0.6 Sr 0.4 MnO 3 exhibits a relatively low magnetocaloric effect at practical fields of 1 T to 2 T, the coefficient of performance can be increased by a factor of two provided that magnetic forces are balanced in magnetocaloric devices [START_REF] Chdil | Assessment of near Pr 2/3 Sr 1/3 MnO 3 oxide in magnetic cooling[END_REF].

Herein, the role of CuO and additional heat treatments (HT) in tailoring the structural, magnetic, and magnetocaloric properties of the near Pr 2/3 Sr 1/3 MnO 3 is deeply investigated. Keeping in mind the fact that previous studies did not provide a satisfactory explanation on how the secondary phases impact the magnetism and MCEs of manganites [START_REF] Nasri | Impact of CuO phase on magnetocaloric and magnetotransport properties of La 0.6 Ca 0.4 MnO 3 ceramic composites[END_REF][START_REF] Maalam | Composite (La 0.45 Nd 0.25 )Sr 0.3 MnO 3 /5CuO materials for magnetic refrigeration applications[END_REF][START_REF] Bhatt | Near room temperature magneto-transport (TCR & MR) and magnetocaloric effect in Pr 2/3 Sr 1/3 MnO 3 : Ag 2 O composite[END_REF][START_REF] Neupane | Magnetocaloric study of La 0.45 Nd 0.25 Sr 0.3 MnO 3 /MO (MO= CuO, CoO, and NiO) nanocomposites[END_REF]. In this way it is worthy to note that CuO oxide is a semiconductor with a narrow bandgap (E g ) of 1.2 eV [START_REF] Zhou | Effect of second introduced phase on magnetotransport properties of La 2/3 Sr 1/3 MnO 3 /0[END_REF] and an antiferromagnetic order taking place at the Neel temperature T N = 230 K [START_REF] Díaz-Guerra | Exchange bias in single-crystalline CuO nanowires[END_REF]. It crystallizes in the monoclinic structure with the lattice parameters of a = 4.6837 Å, b = 3.4226 Å, and c = 5.1288 Å, and β = 99.54 [START_REF] Siemons | Tetragonal CuO: End member of the 3d transition metal monoxides[END_REF]. As a first step in our work, the PSMO (95%)-CuO (5%) (PSMO-CuO) composites were prepared via solid-state reaction. The lattice parameters, the interatomic distances, as well as the bond angles were determined by refining the X-ray diffraction (XRD) data. The energy-dispersive X-ray spectroscopy (EDX) technique was used to verify the chemical compositions. The isothermal and isofield magnetic measurements were carried out by using the Quantum Design Physical Property Measurement System (PPMS).

Experimental details

The polycrystalline PSMO samples were prepared by using the solid-state reaction method. The use of the latter for synthesizing manganites and composite materials seems to be the most convenient since it has good grain growth distribution, excellent process control, and appropriate process time. Stoichiometric proportions of Pr 6 Sr 11 , SrO, and MnO 2 powders with high purity were mixed and grounded in an agate mortar for 1 hour. The mixture was annealed at 1000 °C for 12h and at 1200 °C for 24h in air, with intermediate grinding. Then, the calcined powder was divided into four equal quantities. PSMO (2h HT) and PSMO (24h HT) samples were heat-treated for two hours and 24 hours, respectively, at 1200 °C. The two remaining quantities were used to prepare the PSMO-CuO composites, where a small quantity of copper oxide (5% weight ratio) was added and well mixed with PSMO to have a homogenous powder. The resulting mixtures denoted as PSMO-CuO (2h HT) and PSMO-CuO (24h HT) were calcined at 1200 °C for 2h and 24h, respectively. To check the crystalline quality of prepared samples, XRD measurements were performed at room temperature using a Bruker D8 ADVANCE diffractometer with Cu-Kα radiation over the 2θ-angular range of 10°< 2θ < 80°with a step size of 0.01°. The particle sizes and the surface morphology were analyzed by employing an Analytical Scanning Electron Microscope (ASEM) model X. Magnetization data were collected over the temperature range of 10-350 K under magnetic fields going from 0 to 7 T by using a superconducting quantum interference device magnetometer from Quantum Design, model MPMS XL.

Results and discussion

Structural characterization

The X-ray diffraction patterns of PSMO (2h HT -24h HT) and PSMO-CuO (2h HT -24h HT) samples at room temperature are shown in Fig. 1. HighScore Plus software was used to analyze the existing phases, and the Rietveld method was adopted to refine the XRD data using the FullProf refinement software [START_REF] Degen | The highscore suite[END_REF][START_REF] Roisnel | Computer program fullprof[END_REF]. The observed and calculated patterns are successfully superposed with a reliability factor (R WP ) less than 7%, unveiling the excellent quality of refinement [START_REF] Toby | R factors in rietveld analysis: How good is good enough?[END_REF]. The analysis of XRD data as reported in Fig. 1 shows that the compounds are single phased without any impurities. This means that no additional phases (e.g., Mn 3 O 4 , Mn 5 O 8 , Pr 6 O 11 ) and/or no additional impurities at the Mn sites (e.g., Cr, Co, Fe, Ni) can be found other than PSMO in the pristine samples and PSMO-CuO in the composite samples. At room temperature, PSMO samples are well crystallized in the orthorhombic structure with the Pbnm (n°62) space group. For the PSMO-CuO samples, the coexistence of copper oxide and PSMO can be clearly observed with a strong orientation along the (200) direction. The refinement results for the studied samples are summarized in Table 1.

The XRD data examination revealed that adding a small amount of CuO to the pristine PSMO led to a minor modification in the lattice constants and the cell volume. Moreover, it is worthy to mention that for all studied samples, c/ √ 2 < b < a. However, the addition of CuO resulted in a significant change in the Mn-O II -Mn bond angle that decreases from about 162°f or PSMO to about 156°for PSMO in the composite samples. The average Mn-O-Mn bond angle distortion of MnO 6 octahedra in the studied samples is defined as <ω> = 180°-<Mn-O-Mn> [START_REF] Shaikh | Structural and electrical properties of Pr 1-x Sr x MnO 3 (x= 0.25, 0.3, 0.35 and 0.4) manganites[END_REF][START_REF] Alonso | Evolution of the jahn-teller distortion of mno6 octahedra in RMnO 3 perovskites (R= Pr[END_REF]. From Table 1, it is noticed that the value of < ω > increases when adding the CuO secondary phase, indicating that the latter enhances the octahedral MnO 6 distortion. Moreover, the variation in bond angles and lengths leads to a change in the effective one-electron bandwidth W Mn-O-Mn (electron hopping interaction for Mn 3+ and Mn 4+ ions), described as [START_REF] Radaelli | Structural effects on the magnetic and transport properties of perovskite A 1-x A x MnO 3 (x= 0.25, 0.30[END_REF],

W Mn-O-Mn ∝ cos π -θ Mn-O-Mn 2 d 3.5 Mn-O , (1)
where θ Mn-O-Mn is the Mn-O-Mn bond angle, whereas d Mn-O represents the Mn-O bond length. Table 1 includes the estimated values of W, which decreases for the composite samples as compared to pristine ones. The narrowing of the one-electron bandwidth indicates that the double exchange (Mn 3+ -O 2--Mn 4+ ) interaction is less dominant than the superexchange (Mn 3+ -O 2--Mn 3+ ) coupling or other electronic instabilities [START_REF] Tokura | Critical features of colossal magnetoresistive manganites[END_REF]. Additionally, the lower W Mn-O-Mn implies a possible weakening of the octahedron electron transition probability.

On the other hand, scanning electron micrographs of the studied samples are reported in Fig. 2. The surface of obtained pure PSMO perovskite powders after the three heat treatment modes, shown in Fig. 2(a, b, andc), is composed of almost accumulated spherical grains with an average grain size of 0.6 μm and a homogeneous microstructure. It is worth mentioning that the addition of two or twenty-four hours of heat treatment has no effect on the grain size. However when the CuO phase is mixed with the pristine compound under the same heat treatment conditions, the PSMO grains began to merge together, while their boundaries became challenging to detect due to their large size as can be seen in Fig. 2(d ande). The CuO effect is readily apparent in the PSMO mean grain size that markedly increases from 0.6 μm to 5 μm. More results concerning the scanning electron micrograph and elemental mapping of here investi-gated samples, as well as their associated EDX spectrum, are provided in the supplementary information (see Figure S1).

To get insight on the location of CuO, a plate form is made for the composite powder. The elemental analyses of the PSMO-CuO pellet were carried out by EDX micro-analysis and reported in Fig. 3. The latter shows the presence of a large quantity of Cu metal at the grain boundaries, corresponding to about 45% of the atomic ratio. However, the Cu concentration is only about 1% at the grains surface, which is mainly composed of Pr, Sr, Mn, and O chemical elements. These findings reveal that the copper oxide which covers the pure material grains markedly promoted grain growth because of the liquid sintering [START_REF] Neupane | Magnetocaloric study of La 0.45 Nd 0.25 Sr 0.3 MnO 3 /MO (MO= CuO, CoO, and NiO) nanocomposites[END_REF][START_REF] Zhou | Effect of second introduced phase on magnetotransport properties of La 2/3 Sr 1/3 MnO 3 /0[END_REF][START_REF] Liou | Effect of CuO on CaTiO 3 perovskite ceramics prepared using a direct sintering process[END_REF], since the melting point of CuO (1325 °C) is close to the sintering temperature (1200 °C).

Magnetic properties

To better understand the effect of secondary phase and additional heat treatments on the magnetic properties of studied samples, it is necessary to investigate their temperature dependence of magnetization. Thus, in Fig. 4(a), the latter is shown for both pristine and composite samples. The measurements reveal that the magnetization of investigated materials under a magnetic field of 0.1 T rapidly changes near their Curie temperature. The magnetic phase transition from the ordered (ferromagnet) to the disordered (paramagnet) state takes place at high temperatures, which is mainly due to the strong double exchange Mn 3+ (t 3 2g e 1 g )-O 2-(2p)-Mn 4+ (t 3 2g e 0 g ) interaction [START_REF] Coey | Mixed-valence manganites[END_REF]. Furthermore, the Curie temperature can be determined from the lowest value of the dM/dT versus T curves. As shown in Fig. 4(b), the T c of composite materials is around 273 K, which is significantly lower than that of the PSMO sample T c = 293 K. Based on the double exchange model, a lower T c indicates a loss of overlap between the Mn(3d) and O(2p) orbitals [START_REF] Nasri | Impact of CuO phase on magnetocaloric and magnetotransport properties of La 0.6 Ca 0.4 MnO 3 ceramic composites[END_REF], which can be confirmed by the decreased effective one electron bandwidth W Mn-O-Mn for composite samples as reported in Table 1. The inverse susceptibility (χ -1 ) as function of temperature is represented in Fig. 4(c). From the Curie-Weiss fitted plots, we can see that above the Curie temperature, the susceptibility obeys the Curie-Weiss law, defined as,

χ = C T -T θ , (2)
where T θ is the Curie-Weiss temperature, and C is the Curie constant that can be determined by the linear fit of χ -1 . The obtained Curie constants can be further used to deduce the experimental effective magnetic moment, using the following equation [START_REF] Mabrouki | Sintering temperature effect on the magnetic properties of Pr 0.67 Sr 0.33 MnO 3 manganite[END_REF],

μ exp e f f = 3 • C • k B • M m N a μ B , (3)
where μ B is the Bohr magneton, k B is the Boltzmann constant, N a is the Avogadro number, and M m is the molecular formula weight. The obtained experimental values are summarized in Table 2. Moreover, the theoretical effective magnetic moment of Pr 1-x Sr x MnO 3 can be obtained using the following relation [START_REF] Saw | Study of transport, magnetic and magnetocaloric properties in Sr 2+ substituted praseodymium manganite[END_REF],

μ theo e f f = (1 -x) • μ 2 e f f (Pr 3+ ) + (1 -x) • μ 2 e f f (Mn 3+ ) + x • μ 2 e f f (Mn 4+ ), (4)
where μ e f f (Pr 3+ ), μ e f f (Mn 3+ ), and μ e f f (Mn 4+ ) represents the effective magnetic moment of Pr 3+ , Mn 3+ , and Mn 4+ ions, respectively. Pr 2/3 Sr 1/3 MnO 3 has a theoretical effective magnetic moment of 5.44 μ B . The higher experimental value of μ e f f compared with the theoretical value can be related to the presence of the FM state above T c [START_REF] Xu | Magnetocaloric effect and spontaneous magnetization in perovskite manganite Nd 0.55 Sr 0.45 MnO 3[END_REF][START_REF] Martinez | Magnetic field and pressure effects on the magnetic transitions of La 0.9 Ca 0[END_REF]. Furthermore, at 10 K, Fig. 4(d) depicts the magnetization variation as a function of magnetic field. PSMO has a greater magnetization value than the other composite samples, which is primarily due to the decreased FM volume caused by the introduction of CuO. This is confirmed by the magnetization of the composite that is approximately equal to that of the pure sample when removing the contribution of secondary phase, as can be seen in the inset of Fig. 4(d).

The isothermal magnetization M(H) curves recorded in the temperature range going from 250 K to 310 K and under changing magnetic fields up to 7 T for both pristine and composite samples are reported in Fig. 5. At low temperatures, the magnetization curves exhibit a nonlinear pattern with a tendency to saturate in the presence of magnetic fields, indicating a ferromagnetic behavior. However, above T c , the magnetization evolves almost linearly, indicating a paramagnetic state caused by the thermal agitation disrupting the arrangement of magnetic moments.

Magnetocaloric properties

This work also aims to determine the magnetocaloric properties of studied samples. In this way and according to the Arrott plots (M 2 versus H/M) presented in Figure S2, the positive slope of the curves indicates on the occurrence of a secondorder magnetic transition in all studied materials [START_REF] Banerjee | On a generalised approach to first and second order magnetic transitions[END_REF]. Therefore, for materials that show a second-order transition and very low field hysteresis, which is the case of the here investigated Pr 2/3 Sr 1/3 MnO 3 oxide and its composite, the isothermal entropy change resulting from the variation of external magnetic fields from 0 to H can be obtained by using the following Maxwell relation [START_REF] Balli | Advanced materials for magnetic cooling: Fundamentals and practical aspects[END_REF][START_REF] Moya | Caloric materials near ferroic phase transitions[END_REF][START_REF] Balli | The "colossal" magnetocaloric effect in Mn 1-x Fe x As: What are we really measuring?[END_REF],

ΔS (T, 0 → H) = H 0 ∂M ∂T H dH , ( 5
)
where ΔS is determined numerically by using the numerical form of above equation and reported magnetic isotherms in Fig. 5. In this case, ΔS is given by,

ΔS = i M i+1 -M i T i+1 -T i ΔH i , (6)
where M i+1 and M i are the measured magnetization under an external magnetic field at T i+1 and T i , respectively. In Fig. 6(c), the change in magnetic entropy as a function of temperature for all studied samples under some representative magnetic fields is reported. Their entropy change exhibits a maximum close to Curie temperatures and then drops almost symmetrically beyond the phase transition region. The ΔS maximum values increases from 1.32 to 5.77 J kg -1 K -1 and from 1.88 to 6.23 J kg -1 K -1 when increasing the applied magnetic field from 1 T to 7 T for PSMO and PSMO-CuO, respectively, while keeping their location at T c as shown in Fig. 6(a-b). On the other hand, when the magnetic field is varied by 4 T, the highest magnetic entropy change for PSMO-CuO approaches 4.66 J kg -1 K -1 (at 273 K), whereas for PSMO it is roughly equal to 3.7 J kg -1 K -1 (at 293 K). This difference is due mainly to the fact that the magnetic phase transition of the PSMO-CuO compound is broader than that of the pure compound. As mentioned above, the here investigated manganite shows relatively lower ΔS value when compared with gadolinium (Gd), which exhibits a large entropy change value of 2.8 J kg -1 K -1 (at 292 K) under a magnetic field of 1 T [START_REF] Gottschall | Making a cool choice: the materials library of magnetic refrigeration[END_REF]. However, when compared with similar oxides such as La 2/3 (Ca 0.75 Sr 0.25 ) 1/3 MnO 3 , which is considered as the best candidate among manganites for near-room temperature magnetic refrigeration, the PSMO-CuO sample exhibits a similar magnetic entropy change [START_REF] Mira | Drop of magnetocaloric effect related to the change from first-to second-order magnetic phase transition in La 2/3 (Ca 1-x Sr x ) 1/3 MnO 3[END_REF]. Nevertheless, ΔS can be remarkably enhanced for here investigated manganite oxides when an appropriate method of preparation is used. In fact, Phan et al. [START_REF] Phan | Large magnetocaloric effect in single crystal Pr 0.63 Sr 0.37 MnO 3[END_REF] have prepared a single crystal Pr 0.63 Sr 0.37 MnO 3 which exhibits a high magnetic entropy change of 2.57 J kg -1 K -1 (at 305 K) under a magnetic field of 1 T. Furthermore, when the Pr 0.63 Sr 0.37 MnO 3 is shaped into a pellet form with a higher calcination temperature and duration, the magnetic entropy change could also be enhanced according to Ref. [START_REF] Guillou | Development of a new magnetocaloric material used in a magnetic refrigeration device[END_REF]. Fig. 6(d) represent the magnetic entropy changes of PSMO, PSMO-CuO (24h HT) and PSMO-CuO (24h HT), without considering the mass contribution of CuO as a function of temperature under some representative magnetic fields. As can be seen, a slight increase in the entropy change that attains 1.9 J kg -1 K -1 (1 T) is observed. The latter is considered as the real ΔS m value of PSMO magnetic entropy change in the PSMO-CuO composite material, as the CuO secondary phase has no contribution to the total magnetization and accordingly the MCE. In addition to the entropy change, the refrigerant capacity (RC) is an important parameter that describes the efficiency of a magnetocaloric material. In fact, materials with a higher RC are more suitable for practical applications. RC is given by [START_REF] Gschneidner | Magnetocaloric materials[END_REF],

m 0 H = 7 T m 0 H = 1 T -DS m (J kg -1 K -1 ) (c) (d) (a) (b) T (K)
RC = - Thot Tcold ΔS m (T ) dT, ( 7
)
where T cold and T hot are the respective temperatures at the ΔS m peak's full width half maximum, which can be calculated numerically by integrating the area under the ΔS m versus T curve between T cold and T hot . In Fig. 7, the RC values are plotted versus the applied magnetic field. As can be seen, the RC linearly increases with magnetic field reaching 40 J kg -1 and 308 J kg -1 for PSMO, and 32 J kg -1 and 271 J kg -1 for PSMO-CuO under 1 T and 7 T, respectively. The observed decrease in RC regarding the PSMO-CuO is particularly due to the reduction of its working temperature range as the magnetic phase transition is closer to the first order transition character. On the other hand, the resulting RC is nearly similar to that of the near-room temperature refrigerant La 2/3 (Ca 0.75 Sr 0.25 ) 1/3 MnO 3 (RC = 37 J kg -1 ) [START_REF] Mira | Drop of magnetocaloric effect related to the change from first-to second-order magnetic phase transition in La 2/3 (Ca 1-x Sr x ) 1/3 MnO 3[END_REF]. However, when compared to other high-performance magnetic refrigerants such as Gd (RC = 170.17 J kg -1 for 2 T) and LaFe 13-x Si x (RC = 97.05 J kg -1 for 2 T) based compounds, the studied materials exhibit relatively lower values.

Various studies have demonstrated the strong interplay between lattice and spin degrees of freedom in perovskite-type manganese oxides [START_REF] Radaelli | Simultaneous structural, magnetic, and electronic transitions in La 1-x Ca x MnO 3 with x = 0.25 and 0.50[END_REF]. Therefore, a variety of mechanisms may explain the reduction in transition temperature and the variation of PSMO-CuO magnetic entropy. First, the insertion of CuO phase in the PSMO matrix weakens the FM interaction through structural modifications. In fact, the presence of this secondary phase in the grain boundaries leads to an apparent structural deformation in the primary phase where the Mn-O bond distances and Mn-O-Mn bond angles are significantly affected as reported in Table 1. In fact, the loss in FM interactions can be explained by the weaker double exchange mechanism caused by the reduction in the Mn-O-Mn bond angle since the amplitude of charge carriers hopping from one Mn ion to another is reduced when this angle deviates from 180° [START_REF] Kaplan | Physics of manganites[END_REF][START_REF] Goodenough | Theory of the role of covalence in the perovskite-type manganites [La,M(II)]MnO 3[END_REF][START_REF] Dagotto | Colossal magnetoresistant materials: the key role of phase separation[END_REF][START_REF] Salamon | The physics of manganites: Structure and transport[END_REF]. Accordingly, the remarkable decrease in the Curie temperature is expected based on the following relation [START_REF] Lee | Microstructural and magnetotransport properties of La 0.7 Ca 0.3 MnO 3 /BaTiO 3 and La 0.7 Sr 0.3 MnO 3 /BaTiO 3 bilayered films[END_REF],

T c cos(ϕ) d Mn-O , (8)
where d Mn-O is the length of the Mn-O bond and ϕ is the Mn-O-Mn bond angle. The observed modification in ϕ can be explained by a tensile force exerted by the CuO secondary phase, particularly near grain boundaries. Furthermore, the loss in ferromagnetic interactions can be also reinforced by the low magnetic coupling between the grains due to the inter-grain constructed layer (CuO phase), which is magnetically different from the magnetic state of the grains [START_REF] Dey | Effect of grain size modulation on the magneto-and electronic-transport properties of La 0.7 Ca 0.3 MnO 3 nanoparticles: The role of spin-polarized tunneling at the enhanced grain surface[END_REF].

On the other hand, even though there is only an additional 2 hours of heat treatment at 1200 °C to PSMO-CuO, this brief period was good enough to markedly increase the grain size as shown in Fig. 2. This is because of the presence of CuO, which acts as a sintering aid. In Park et al. [START_REF] Park | Magnetic properties at surface boundary of a half-metallic ferromagnet La 0.7 Sr 0.3 MnO 3[END_REF] study, it was demonstrated that the magnetization at the surface boundary of the manganese oxide La 0.7 Sr 0.3 MnO 3 decays much slower than the bulk magnetization. Therefore, the high grain size of the composite samples may promote a more ordered FM phase and, consequently, reduce the magnetic transition width (i.e. more pronounced magnetic phase transition) since the impact from the magnetically disordered surface boundary is reduced, which can explain the slight increase of the magnetic entropy. On the other hand, the near first-order magnetic transition may also be related to the significant decrease in the Curie temperature.

With regards to the effect of additional heat treatments on magnetic and magnetocaloric properties, the obtained results indicate that, in the absence or the presence of secondary phase, the Curie temperature and the magnetic entropy change are almost constant. Therefore, the additional heat treatment has no effect on the magnetic and magnetocaloric properties since the observed modifications in the composite samples are in fact directly arises from the added secondary phase.

Multilayered magnetocaloric refrigerants based on PSMO and PSMO-CuO

As already reported in this study, the investigated manganite oxides show a significant magnetocaloric effect that remains peaked on a relatively narrow temperature range, which is not appreciated from a practical point of view. However, as demonstrated above, the addition of CuO as secondary phase to the pristine PSMO enables us to reduce its Curie temperature to about 273 K without proceeding to doping effects. Taking advantage of such a result, a multilayered refrigerant based on both PSMO and PSMO-CuO is proposed aiming to cover a wide temperature range close to ambient temperature. The proposed multilayered material can be employed as a refrigerant in cooling devices that use AMR and Ericsson cycles to achieve the refrigeration process. The latter cycle particularly requires a constant entropy change over the considered temperature range [START_REF] Balli | Advanced materials for magnetic cooling: Fundamentals and practical aspects[END_REF][START_REF] Balli | Optimization of La(Fe,Co) 13-x Si x based compounds for magnetic refrigeration[END_REF][109][START_REF] Yuan | Numerical study on the multi-layered magnetocaloric regenerators[END_REF][START_REF] Smaıli | Composite materials for ericsson-like magnetic refrigeration cycle[END_REF][START_REF] Balli | Gd1xtbx alloys for ericsson-like magnetic refrigeration cycles[END_REF][START_REF] Diguet | Performance characteristics of a regeneration ericsson refrigeration cycle using a magnetic composite as the working substance[END_REF][START_REF] Xu | A Gd x Ho 1-x -based composite and its performance characteristics in a regenerative ericsson refrigeration cycle[END_REF]. For this purpose, the numerical method proposed in Refs. [109,[START_REF] Smaıli | Thermodynamic investigations of optimum active magnetic regenerators[END_REF] was used to identify the optimum mass ratio, y i , for each material based on the isothermal entropy change represented in Fig. 6. In general, for a refrigerant constituted of n magnetic compounds, the resulting isothermal entropy change can be given by the following relation,

ΔS com = n i=1 y i ΔS i , (9)
where y i is the mass proportion of one of the constituent compounds. If ΔS com of the composite material utilized as the working substance in the Ericsson refrigeration cycle is maintained constant across a specified temperature range, the refrigeration cycle may achieve a complete regeneration [START_REF] Xu | A Gd x Ho 1-x -based composite and its performance characteristics in a regenerative ericsson refrigeration cycle[END_REF]. In this case, Eq. ( 9) can be expressed as follows,

ΔS Com = n i=1 y i ΔS i T j+1 c -ΔS i T j c = 0, (10)
where j = 1,2,..., n-1 and T c is the Curie temperature of ith constituent material. Under a magnetic fields of 1 T and 7 T, the optimum mass ratio of the considered materials is determined using the combination of Eq. ( 10) and the fact that n i=1 y i = 1. The calculated ΔS com versus temperature and magnetic field is shown in Fig. 8(a). As can be observed, ΔS com remains almost constant within the temperature range 273-293 K. Indeed, a nearly perfect plateau is observed in the multilayered isothermal entropy change curve with an average value of 1 J kg -1 K -1 (1 T) and 5.11 J kg -1 K -1 (7 T). Even though the achieved ΔS com is lower than the one of the constituent samples, the broadening of the working temperature makes the proposed composite material a potential working substance in magnetic refrigeration cycles. Similar features were reported by Balli et al. [START_REF] Balli | Gd1xtbx alloys for ericsson-like magnetic refrigeration cycles[END_REF] and Smaili et al. [START_REF] Smaıli | Composite materials for ericsson-like magnetic refrigeration cycle[END_REF] for a composite refrigerant based on Gd 1-x Tb x and Gd 1-x Dy x alloys, where it is observed that ΔS com remains practically constant for near-room temperatures range of (260-300 K) and (210-290 K), respectively, with higher magnetic entropy change when compared to the here investigated materials. Furthermore, as shown in Fig. 8(b), the magnetic refrigeration capacity of the multilayered system is greater than that of the composite material, and approximately equivalent to that of pure PSMO.

Conclusions

To sum up, the structural, magnetic, and magnetocaloric properties of Pr 2/3 Sr 1/3 MnO 3 (PSMO) and Pr 2/3 Sr 1/3 MnO 3 (95%)-CuO(5%) (PSMO-CuO) samples have been studied to particularly assess the real impact of the added CuO secondary phase on the magnetic and magnetocaloric features of PSMO. Therefore, the bond lengths and angles of prepared samples were determined by the Rietveld refinement method of the X-ray data. The latter unveil a significant variation in the Mn-O-Mn bond angles caused by the tensile strain resulting from the secondary phase located at the grain boundaries. This was confirmed by using SEM micrographs. Moreover, such structural deformations weaken the ferromagnetic interactions controlled by double-exchange couplings, which markedly decrease the Curie temperature from 293 K for the pristine PSMO to about 273 K for the PSMO-CuO composite. This is associated with the enhancement of the magnetocaloric effect in the PSMO-CuO. Based on the obtained results, a multilayered refrigerant based on PSMO and PSMO-CuO materials is proposed for room temperature magnetic cooling applications. The resulting entropy change remains nearly constant over the temperature range going from 273 K to 293 K, being a favorable situation for the efficient AMR and Ericsson thermodynamic cycles.

Fig. 1 :

 1 Fig. 1: XRD patterns for (a) PSMO, (b) PSMO (2h HT), (c) PSMO (24h HT), (d) PSMO-CuO (2h HT), and (e) PSMO-CuO (24h HT) polycrystalline.

Fig. 2 :

 2 Fig. 2: Scanning electron micrograph (SEM) of (a) PSMO, (b) PSMO (2h HT), (c) PSMO (24h HT), (d) PSMO-CuO (2h HT), and (e) PSMO-CuO (24h HT) samples.

Fig. 3 :

 3 Fig. 3: Elemental mapping for PSMO-CuO (24h HT) with (a) all present elements and (b) only copper element. (c) SEM micrographs of PSMO-CuO (24h HT) and their associated EDX spectrum at: (d) the grains boundaries and (e) the grains surface.

Fig. 4 :

 4 Fig. 4: (a) Temperature-dependent magnetization under a magnetic field of 0.1 T. (b) Plots of dM/dT versus temperature for 0.1 T. (c) The temperature-dependent inverse susceptibility for 0.1 T of both pristine and composite samples. (d) Isothermal magnetization curve at 10 K. The inset figure represents the magnetization at 10 K for the PSMO, PSMO-CuO (24h HT) and PSMO-CuO (24h HT) without accounting the CuO mass in the calculation.

Fig. 5 :

 5 Fig. 5: Isothermal magnetization curves for the investigated samples at different temperatures close to the magnetic phase transition and up to 7 T.

  HT) PSMO (24h HT) PSMO-CuO (2h HT) PSMO-CuO (24h HT)

Fig. 6 :

 6 Fig. 6: 2D contour plots of the magnetic entropy changes as function of temperature and magnetic field for (a) PSMO and (b) PSMO-CuO (24h HT). (c) Magnetic entropy changes as function of temperature for the investigated samples under a magnetic field of 1 T (open symbols) and 7 T (solid symbols). (d) Magnetic entropy changes as function of temperature for PSMO, PSMO-CuO (24h HT) and PSMO-CuO (24h HT) without accounting the CuO mass in the calculation.

Fig. 7 :

 7 Fig. 7: Refrigerant capacity versus applied magnetic field for PSMO and PSMO-CuO samples.

  Fig. 8: (a) Magnetic entropy changes of PSMO, PSMO-CuO (24h HT), and the multilayered material composed of both PSMO and PSMO-CuO (24h HT) samples as function of temperature under magnetic fields of 1 T and 7 T. (b) Magnetic refrigeration capacity (RC) versus applied magnetic field for PSMO, PSMO-CuO (24h HT) and the multilayered material.

Table 1 :

 1 Rietveld refined lattice constants, unit cell volume, bond angles, bond lengths and MnO 6 octahedron structural characteristics of both pristine and composite samples in case of presence and absence of additional heat treatment. XRD data were obtained at room temperature.

	Compound	a (Å)	b (Å)	c (Å)	c / (Å) √ 2	V (Å 3 )	Mn-O I -Mn Mn-O II -Mn Mn-O I <Mn-O II > (°) (Å) (Å)	<ω>	W Mn-OII-Mn
	PSMO	5.480 5.449 7.701 5.445 229.956	157.964	163.239	1.961	1.953	19.399	0.0950
	PSMO (2h HT)	5.490 5.458 7.714 5.455 231.146	157.440	162.611	1.966	1.958	19.975	0.0941
	PSMO (24h HT)	5.486 5.452 7.704 5.448 230.424	158.930	162.018	1.959	1.958	19.526	0.0940
	PSMO-CuO (2h HT)	5.482 5.449 7.698 5.443 229.950	158.988	156.121	1.957	1.975	22.446	0.0904
	PSMO-CuO (24h HT) 5.486 5.452 7.701 5.445 230.334	160.029	157.838	1.955	1.970	21.067	0.0915

Table 2 :

 2 Curie temperature (T c ), Curie-Weiss temperature (T θ ), Curie constant (C), and experimental magnetic moment μ

			exp e f f obtained from magnetization
	data for all studied samples.
	Sample	T c (K) T θ (K)	C	μ exp e f f (μ B )
	PSMO	293	300	0.0192	5.89
	PSMO (2h HT)	295	299	0.0189	5.84
	PSMO (24h HT)	291	297	0.0205	6.08
	PSMO-CuO (2h HT)	274	276	0.0223	6.35
	PSMO-CuO (24h HT)	270	273	0.0216	6.24

Acknowledgments

O. Chdil and M. Balli acknowledge funding by the International University of Rabat. The authors would like to thank S. Pelletier and B. Rivard for the technical support. We acknowledge the financial support from NSERC (Canada), FQRNT (Québec), CFI, Canada First Research Excellence Fund (Apogée Canada), and Université de Sherbrooke. The authors gratefully acknowledge MAScIR foundation for providing the experimental facilities.