{'fraction_non_alphanumeric': 0.05681236012618977, 'fraction_numerical': 0.02475260873082212, 'mean_word_length': 3.791731266149871, 'pattern_counts': {'":': 0, '<': 19, '<?xml version=': 0, '>': 12, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Fluctuations of a one-dimensional polynuclear growth model in a half space

We consider the multi-point equal time height fluctuations of a one-dimensional polynuclear growth model in a half space. For special values of the nucleation rate at the origin, the multi-layer version of the model is reduced to a determinantal process, for which the asymptotics can be analyzed. In the scaling limit, the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal/symplectic to unitary transition ensemble at soft edge in random matrix theory.


Introduction
The problems of random surface growth have been an important subject of non-equilibrium statistical physics [1,2]. Interestingly, surfaces show universal statistical behaviors, depending on which mechanism plays a major role in growth. Among them is the Kardar-Parisi-Zhang (KPZ) universality class [3]. Though in many cases it is not enough to describe the growth of real materials, the KPZ universality class gives a satisfactory understanding of many growth models such as the Eden model, and hence plays a prominent role in theoretical study of growing surfaces.
It is difficult to study the KPZ universality class analytically for general dimension. In one spatial dimension, however, some exact results had been obtained. For instance, the exponents are already obtained in [3] based on dynamical renormalization group techniques. These have been further confirmed by exact solutions for the models in the KPZ universality class [4,5]. We will also restrict our attention to one spatial dimensional case in this article.
Recently, more refined information about the fluctuation properties of the 1+1 dimensional KPZ universality class have been obtained [6,7,8,9,10,11,12,13,14,15]. For some important quantities such as the height fluctuation, not only the exponents but also the scaling functions are obtained. The whole new developments originate from the work [16] on the statistics of the longest increasing subsequence in a random permutation, which has turned out to be closely related to the random surface growth models [17].
The polynuclear growth (PNG) models are known to belong to the KPZ universality class [1,2]. The PNG models are simple models which describe layer by layer random surface growth. In its standard version, a nucleation of a layer with height one occurs with rate two per unit length. After a nucleation, the layer grows laterally in both directions from the nucleation point with unit speed. In addition, there occur nucleations with rate two on already existing layers. A discrete version of the PNG model was introduced in [18] and studied further in [19]. The discrete PNG model has a close relationship with the asymmetric simple exclusion process (ASEP) [6,13].
In [9,7,8], the height fluctuation of the PNG model at a given time and a given position was studied. It turned out that the fluctuation strongly depends on the geometry of the models. For instance, for the model in the infinite space, the fluctuation at the origin is described by the GUE Tracy-Widom distribution in random matrix theory [20,21]. On the other hand, for the model in a half space with an external source at the origin, it is described by the GOE/GSE Tracy-Widom distributions [22] or by the Gaussian. Models with other geometries are also considered in [7,8]. These are based on the results in [9,23,24,25], in which the longest increasing subsequence problem with some symmetries are considered.
For the models in the infinite space, multi-point equal time height fluctuations are studied by introducing the multi-layer version of the PNG model in [14] for the continuous time case and in [19] for the discrete time case. It is shown that the scaling limit is described by the Airy process, which is closely related to the dynamics of the rightmost particle of the Dyson's Brownian motion model [26].
In this article we study the multi-point equal time height fluctuations of the PNG model in a half space with external source at the origin. As mentioned above, the fluctuation at a single point has already been considered in [25,7]. At the origin, it is given by the GOE/GSE Tracy-Widom distribution whereas at other points, it is the GUE Tracy-Widom distribution. But the multi-point fluctuation has not been studied. Following the ideas of [27,14], we introduce the multi-layer PNG model. In the scaling limit we will show that the crossover between the fluctuation at the origin and that in the bulk corresponds to the orthogonal/symplectic to unitary transition at soft edge in random matrix theory [28].
The outline of the paper is as follows. In the next section, the model we consider is defined. In section 3, a determinantal process is considered. Section 4 deals with the scaling limit of the models. In section 5, the results for the model without external source at the origin are summarized. The standard PNG model in a half space can be considered by taking the limit for the discrete model. This is done in section 6. Some discussions, conjectures and Monte-Carlo results are given in section 7, followed by the conclusion in the last section.

Model
Let r ∈ N = {0, 1, 2, · · · } and t ∈ N denote the discrete space and time coordinates respectively and h(r, t) the height of the surface at position r and at time t. Our model lives in a half space r ∈ N, but we can extend h(r, t) to the whole space by setting h(r, t) = h(−r, t) for r ∈ Z, i.e. the height is symmetric under the reflection with respect to the origin. We can further extend h(r, t) to all r ∈ R by setting h(r, t) = h([r], t). Then the height looks symmetric under the reflection with respect to r = 1/2, i.e. h(r, t) = h(1−r, t) for r ∈ R \ Z. The model considered in this article is described as follows. Initially at time t = 0 we have a flat substrate. At an odd time t = 1, 3, · · · , nucleations occur at even sites r = 0, ±2, · · · , ±t − 1. At the origin, a nucleation of a height k (∈ N) happens with the probability (1 − γ √ q)(γ √ q) k where 0 < q < 1 and 0 ≤ γ < 1/ √ q. For the height to be symmetric, nucleations at other points should be symmetric with respect to the origin; if the nucleation of a height k occurs at r, so does at −r as well. Each independent nucleation with height k occurs with probability (1 − q)q k . At an even time t = 2, 4, cdots, nucleations occur at odd sites r = ±1, ±3, · · · ± t − 1. The nucleations are again synchronous with respect to the origin and a height of each independent nucleation is a geometric random variable with parameter q. In the meantime the steps grow laterally in both directions with unit speed. Sometimes a downstep and an upstep collide with each other, in which case the two steps merge to one with the higher height. See Fig. 1. As is clear, the parameter q is related to the frequency of nucleations in the bulk whereas the parameter γ represents the strength of the external source at the origin. The bigger γ is, the stronger the source is. In particular γ = 0 corresponds to the model without the external source at the origin.
Here ω is the random variable which takes a value in N. ω(r, t) = 0 if t − r is even or if |r| > t, and (i, j) ∈ Z 2 + are geometric random variables. In our model, all w(i, j)'s are not independent as in [19], but a symmetry condition w(i, j) = w(j, i) is imposed. The parameters of the independent geometric random variables are given by q (resp. γ √ q) for off-diagonal (resp. diagonal) points, for k ∈ N. Introduction of w(i, j) above is almost unnecessary for the following discussions, but useful to see the connection to the problems of last passage percolation and ASEP [9,6,25].
In the following, we devote our most discussions to the two special choices of γ where the detailed analysis of the model is possible. One choice is γ = 1, which is known to correspond to a critical point of the model [25]. The other is γ = 0, which corresponds to the model without the external source at the origin. In the following, the γ = 1 (resp. γ = 0) case will sometimes be referred to as the orthogonal (resp. symplectic) case because it will turn out to be related to the orthogonal (resp. symplectic) to unitary transition of symmetries in random matrix theory. The analysis of the two cases are quite analogous, so that we mainly consider the γ = 1 case in the sequel and summarize the results for the γ = 0 case in section 5.
We would like to know the statistical properties of h(r, t). As already mentioned in the introduction, the fluctuation of the height at a single point is already known. It is given by the GOE Tracy-Widom distribution at the origin [25] and by the GUE Tracy-Widom distribution at other points [7]. In this article, we are interested in the crossover between these two. Following [18,14,19], we introduce the multi-layer version of the model which is defined as follows. In the original single-layer discrete PNG model, when two steps meet, the higher step takes over the lower one. The information of the lower step is lost. To keep this information, let us suppose that there was a second layer below the original layer from the outset. The height of the second layer remains −1 until a collision of steps happens at the first layer, at which time the nucleation for the second layer occurs just below the collision. The height of the nucleation is taken to be the height of the overlapped region of the higher and the lower steps which collide at the first layer. See Fig. 1. The steps in the second layer grows laterally in both directions with unit speed as in the first layer. The nucleations at the second layer are only due to collisions of steps at the first layer; there is no additional probabilistic nucleations. We denote the height of the second layer at position r and at time t as h 1 (r, t). One can further suppose that there were infinitely many layers below the first layer, equally spaced at time 0 and successively construct h i (r, t) for i = 2, 3, · · · . We also set h(r, t) = h 0 (r, t). This is the multi-layer PNG model. An example of the whole height configurations is given in Fig. 2.
In the following, we mainly consider an even time, which is fixed to t = M = 2N until the asymptotics is considered. There are infinitely many lines, but for time t = M the number of lines which are not straight is at most N. Moreover, there is a restriction, h i (M, M) = −i(i = 0, 1, · · · , N − 1). For each jump with length k of all layers, we associate a weight ( √ q) k . Since the whole configuration of the layers keep all information of nucleations, the product of all weights of all jumps in the multi-layer PNG at t = M is given by From now on, we change the way of looking at multi-layer heights. The space coordinate r in the original setting will be interpreted as the time coordinate. The height coordinate will be interpreted as the space coordinate and is represented as x. We set h i (r, t) = x r i+1 for i = 0, 1, · · · , N −1 and r = 0, 1, · · · , M, and regard the height lines as paths of particles. Let us denote the collection of the positions of particles at time r as x r = (x r 1 , · · · , x r N ) where x r 1 > · · · > x r N , and a collection of them for all times asx = (x 0 , · · · , x M ). In what follows we use α = √ q. Suppose that, when the the particle picture of heights are employed, the x and r axes are pointing the right and the up respectively. Then at odd times, particles move to the right. The transition weight of a particle from x at time 2u to y at time 2u + 1 is given by At even times, they move to the left. The transition weight for each particle is (2.7) Then, according to the Karlin-McGregor theorem [29,30], the weight of non-intersecting paths of n particles which ends at x M i = 1 − i is given by with the restriction x M i = 1 − i. The weight is written in a determinantal form, and hence the corresponding process is called a determinantal process. Notice that the weight in (2.8) with n = N is slightly different from the weight of the multi-layer PNG model because some configurations contained in (2.8) are not allowed in the multi-layer PNG model where there are infinitely many particles. On the other hand, if we take the limit n → ∞ from the beginning, there would appear another difficulty due to the infinity of the number of particles. Thus, for simplicity of the treatment, we approximate the weight of the multilayer PNG model by (2.8) for the moment and take the limit n → ∞ afterwards when it becomes easy. The simple determinantal expression in (2.8) allows us to perform a detailed analysis of the model. If we took diagonal parameters to be more general as in (2.4), the expression would become more complicated.
For later use, we introduce the generating function of φ r,r+1 , Let us denote the Fourier coefficient of a function a(z) asâ(k) like For instance one sees In addition, for a product, the Fourier coefficient, φ r 1 ,r 2 (x, y) =f r 1 ,r 2 (y − x), (2.15) represents the transition weight of a particle between (r 1 , x) and (r 2 , y). Notice that when r 1 = r 2 = r one has φ r,r (x, y) = δ xy . (2.16) Before going to the next section, a remark is in order. Although we have employed the situation where the geometric random variables are given by a single parameter q, up to some point, it is easy to generalize our discussions to the case where with 0 < a i < 1 for all i. Then (2.9),(2.10) are replaced by with the transition weight of a particle again given by (2.12), (2.13). It should be noted that our free parameters are only a j 's; for the case in [18], there are two sets of free parameters a j , b j .

Determinantal Process and Kernel
Let us consider the weight of non-intersecting paths given by where x M i (i = 1, 2, · · · , n) is fixed. As for x 0 i , there is no restriction. This is the main difference between our model and the model in [19] in which x 0 i is also fixed. The matrix element φ r,r+1 (x r i , x r+1 j ) is the transition weight of a particle between (r, x r i ) and (r+1, x r+1 j ). Our main focus is on the special case in (2.6),(2.7), but the results of this section do not depend on a specific choice of φ. For instance, when particles can hop only to nearest neighbor sites for each time, the determinantal process is nothing but the vicious walks studied in [31,32,33]. Hence our discussions below can immediately be applied to the problems as well.
The partition function is defined as The probability of the non-intersecting paths reads Let g(r, x) (r = 1, · · · , M) be some function. In this section, we show (1 + g(r, x r j ))p n,M (x) = det(1 + K 1 g), (3.4) where the determinant on the right hand side is the Fredholm determinant,

6)
with the matrix elements being Here The expression for general K 1 (r i , x i ; r j , x j ) (i, j = 1, 2, · · · , M) has the same form as K 1 (r 1 , x 1 ; r 2 , x 2 ).
Remark. The subscript 1 refers to the fact that this case is related to the orthogonalunitary transition in random matrix theory. If we take g(r, x) = −χ Jr (x) (r = 1, 2, · · · , M), where χ Jr is the characteristic function of J r , (3.4) gives the probability that there is no particle on J 1 × · · · × J M . It should also be remarked that at this stage the infinite particles limit is taken easily. One only replaces the summation, n i,j=1 , in (3.8),(3.10),(3.11) with ∞ i,j=1 . Proof. We derive (3.4) by generalizing the methods of [19] and [34]. Let us define By repeated use of the Heine identity, one has Now, using another identity [35,34], For the second equality we used φ 0,0 (x, y) = δ x,y . We divide the (i, j) element of the matrix in the above determinant into four parts with or without factors of the form g(r, X r ) or g(r,X r ). We have where (A 1 ) i,j is already defined in (3.12) and (3.25) and (3.28) 1 ) i,j can be rewritten as In the last expression, the dependence on r, x is omitted for notational simplicity. Noticing the antisymmetry of G 1 , we can also rewrite (A 1 ) where we used the notation, Hence we find In view of (3.15), we divide this by In the determinant this corresponds to multiplying by A −1 1 , say from the left. Let One notices that, if one introduces this is equal to Now we use a simple fact The determinant on the left hand side is the determinant of a matrix of finite rank, whereas the one on the right hand side is the Fredholm determinant. We see where we used the notation a ⊗ b for the operator with kernel of the form a(r 1 , The matrix in the determinant may be rewritten as a product of two matrices, leading to The determinant of the right matrix is one, so that this equals Let us remember the definition of ψ i in (3.28) and notice The determinant we are considering is now (3.45) Multiplying from the left by and from the right by

Scaling Limit
In this section we consider the scaling limit, where the universal properties of the model are expected to appear.

Generating Functions
To study the asymptotics of the kernel, we compute the generating functions ofS 1 ,Ĩ 1 , D 1 . First let us recall some basic facts about a Toeplitz matrix [36]. A Toeplitz matrix is a matrix A of the form, It is useful to define a function, which is called the symbol of the Toeplitz matrix A. Conversely, for a given function a(z), set a k =â(k), the Fourier coefficient of the symbol a(z). Then one can define the corresponding Toeplitz matrix (4.1), which will be denoted by T (a). The inverse of A can be given in a compact fashion, if the symbol a(z) has winding number zero and is Wiener-Hopf factorizable as In fact for a factorization (4.3) one has In addition, the inverse of the matrix T (a ± ) is simply given by Hence the inverse of the matrix A is To obtain the generating functions ofS 1 ,Ĩ 1 , D 1 , we need the inverse of the matrix A 1 (3.12). As remarked below (3.13), for the multi-layer PNG model, the matrix A 1 should be taken to be an infinite dimensional one, which turns out to be a Toeplitz matrix. If we set the symbol is computed as where In the second equality of (4.10), we use (2.15) and (3.12). The winding number of a 1 (z) is not zero and hence the formula (4.8) does not hold as it is in this case. But the difficulty can be overcome fairly easily. Let us first notice that s 1 (z) can be written in two ways as 1− (z), (4.12) with s (1) Correspondingly, if we define a (i) Then it is confirmed that the inverse A −1 1 is given by which is the substitute for (4.8). Now one can compute the generating function, which is valid for |z 1 | > |z 2 |. In the first equality we use a fact that 1/a (i) 1+ (k) = 0 for We also introduce the generating function of the function G 1 in (3.13). Note that as a function of x 1 , x 2 , G 1 only depends on the difference x 1 − x 2 , so that we can set (4.20) Then one has With these preparations, calculations of the generating functions ofS 1 ,Ĩ 1 , D 1 are not difficult. We havẽ , (4.22) In the last expressions of these, we have set r i = 2u i for i = 1, 2.

Bulk
We are now in a position to study the asymptotic behaviors of the model. The thermodynamic limit shape is already known to be where 0 < β 0 < 1 is fixed [37]. As for the fluctuation around this limit shape, the KPZ theory tells us that the correlation survives for O(N 1/3 ) along the height direction and O(N 2/3 ) along the r direction. Let us define the scaled height variable as Remark that a(β) just comes from the limit shape (4.25) whereas d(β) and c(β) are taken so that the results become simple. In this subsection, we show that the fluctuation of the model in the bulk is described by the Airy process [14,19]. Namely, we show  The determinant on the right hand side is the Fredholm determinant, where G(τ j , ξ) = −χ (s j ,∞) (x) (j = 1, 2, · · · , m) and K 2 is a scalar kernel, for which a representative element is given by (4.34) Remark. The kernel (4.32) is called the extended Airy kernel and had already appeared in the context of Dyson's Brownian motion model [38,39]. We also remark that some properties of the Airy process have been discussed in [14,19,40,41]. For the special case of a single time, we have where F 2 (ξ) is the GUE Tracy-Widom distribution [20]. In the language of the last passage percolation, (4.35) means that the fluctuation of the last passage time at an off-diagonal point is described by the GUE Tracy-Widom distribution. For the nonsymmetric case, the statement has been already established in [6]. But for our symmetric case, this has not seem to be shown although it was already expected based on a physical argument in [7]. Remember also that the fluctuation at an off-diagonal point has not been handled by the techniques using the Riemann-Hilbert method [25].
Using the generating functions in the previous subsection, we get the double contour integral formula forS 1 ,Ĩ 1 , D 1 , Here C R denotes a contour enclosing the origin anticlockwise with radius R. We have taken R 1 > R 2 to agree with the computation in (4.19). ForĨ 1 , exchanging the radius of the contours of z 1 , z 2 gives In the first equality the second term on the right hand side appears from the contribution of a pole at z 1 = z 2 . Hence one gets the double contour integral representation for I 1 , Let us first consider the asymptotics ofS 1 . Since we can follow the discussions in [19] for a large part, we omit details and state the main steps. If we write we haveS where β 1 = u 1 /N, β 2 = −u 2 /N. µ 1 , µ 2 are arbitrary constants at this stage. We would like to apply the saddle point method to this integral. In general for each value of µ, there are two critical points z ± c = p(µ, β) ± p(µ, β) 2 − q(µ, β) with Since we are considering the scaling in (4.26), we will set with ξ i the scaled space variable. Accordingly we set (4.47) For this special value of µ, the two critical points, z ± c , merge to the double critical point p c (β), where g ′ µc(β),β (p c (β)) = g ′′ µc(β),β (p c (β)) = 0, given by The main contribution to the integral in (4.43) comes from around the double critical points z 1 ∼ p c (β 1 ), z 2 ∼ 1/p c (β 2 ). The paths of integration may be deformed to where w i = ζ i + iη i with −∞ < ζ i < ∞ (i = 1, 2) and η 1 , η 2 > 0 are fixed. For a fixed β 0 , we take where τ i is the scaled time variable. To leading order we have where is used.
Expanding around z 1 ∼ p c (β 1 ) to order O(1), one finds after some computation. Similarly one gets (4.57) Clearly, this is an odd function. Therefore if we expand as we have λ i (β 0 ) = (−1) i+1 λ i (−β 0 ) for i = 0, 1, 2, · · · . Expanding to order O(1), we have Combining (4.49),(4.50) with (4.52),(4.53) gives so that z In addition, to leading order we have to which the second term on the left hand side does not contribute. Finally one obtains We also need the asymptotics of  From (4.52), this can be rewritten as (4.68) Expansion around z ∼ p c (β 1 ) as in (4.55) gives leads to Ng µc(β 2 ),β 2 1 z ∼ Ng µc(β 2 ),β 2 (p c (β 2 )) + 1 3 (τ 2 − iσ) 3 . Hence one gets (4.73) For I 1 , one has where β i = −u i /N. The critical points of z 1 , z 2 are the again given by z 1 = p c (β 1 ), z 2 = 1/p c (β 2 ). To leading order, the paths are so that we have Similarly for D 1 one has where β i = u i /N. The critical points of z 1 , z 2 are z 1 = p c (β 1 ), z 2 = 1/p c (β 2 ). To leading order, the paths are so that we have Now suppose that we substitute the asymptotic expressions (4.65),(4.73),(4.77),(4.81) into (3.6). Notice that the kernel of the form a b c d and αa βb c/β d/α give the same value for the determinant and hence that some of the prefactors in (4.65),(4.73), (4.77),(4.81) have no effect on the value of the determinant. Moreover, in our scaling limit, the offdiagonal elements, I 1 and D 1 , vanish due to the difference of order in N. The diagonal elements are transpose to each other and hence one finds

Near the Origin
In the last subsection, we saw that the fluctuation in the bulk is described by the Airy process. On the other hand, as already mentioned, we know that the height fluctuation at the origin is given by the GOE Tracy-Widom distribution. In this subsection, we are interested in the crossover between them. Let us define the scaled height variable near the origin as The second term in (4.83) comes from the expansion of a(β) in (4.27) around β = 0. We show that the fluctuation of the model near the origin is described by the process which gives the orthogonal-unitary transition in random matrix theory. Namely, we show where G(τ j , ξ) = −χ (s j ,∞) (ξ) (j = 1, 2, · · · , m) and K 1 is the 2 × 2 matrix kernel, for which a representative element is given by with the matrix elements being (4.90) Definition of Φ τ 1 ,τ 2 (ξ 1 , ξ 2 ) is already given in (4.34).
Remark. The same kernel appeared in the context of the orthogonal-unitary transition in [28] and the problem of vicious walks in [33]. For the special case of a fluctuation at τ = 0, we have lim where F 1 (s) is the GOE Tracy-Widom distribution [22]. It is remarked that this result was anticipated in [6] based on the Meixner orthogonal ensemble representation and was shown in [25]. On the other hand, for τ → ∞, I 1 and D 1 goes to zero. We recover the Airy process, which is consistent with the results of the previous subsection.
Let us first consider the asymptotics ofS 1 . Basically one can follow the same route as in the last subsection with β 0 = 0, but there appear some simplifications and differences. For instance, (4.61),(4.62) reduce to (4.101) Then we have Compare this with (4.64), where the second term on the left hand side was negligible. Expansion of G(β) in (4.58) is now simply so that one has instead of (4.59),(4.60). Asymptotics of φ, which is already given in [19], can be obtained by setting β 0 = 0 in (4.73);

Model without Source at the Origin
In this section we study the case where γ = 0 in (2.4). As already mentioned in the introduction, this case corresponds to the PNG model without the source at the origin. The analysis proceeds in a fairy parallel way to the orthogonal case. The main difference is that there is a new restriction on the height lines h i ; h 2i (0, t) = h 2i+1 (0, t) + 1 (i = 0, 1, · · · , N/2 − 1) at time t = M = 2N. Here and hereafter we assume N is even. This condition might look somewhat strange at first sight, but in fact can be understood after a short reflection. An example of a multi-layer heights is given in Fig. 3, in which one sees that h 2i and h 2i+1 form a pair at r = 0. As in the orthogonal case, we change the way of looking at multi-layer heights. The space coordinate r will be interpreted as the time coordinate. The height coordinate will be interpreted as the space coordinate and is represented as x. Then the restrictions read x M i = 1 − i for i = 1, 2, · · · , N and x 0 2i = x 0 2i−1 − 1 for i = 1, 2, · · · , N/2.

Determinantal Process and Kernel
We consider the weight of non-intersecting paths of n particles given by where n is assumed to be even and x M i (i = 1, 2, · · · , n) is fixed. As in the orthogonal case, this is slightly different from the weight of the multi-layer PNG model even if we take n = N, but we employ this for the moment and take the infinite particles limit when it becomes easy. The partition function and the probability of the non-intersecting paths are given by (3.2),(3.3) respectively. Again, our main focus is on the special case in (2.6),(2.7), but the results of this section do not depend on a specific choice of φ. For instance, when particles can hop only to nearest neighbor sites for each time, the determinantal process is a variant of the vicious walks with a condition that walkers start in pairs and end at neighboring sites [42].
Under this arrangement, we can show Proposition 5.1. (1 + g(r, x r j ))p n,M (x) = det(1 + K 4 g), where g is some function. The determinant on the right hand side is the Fredholm determinant, where K 4 is a 2 × 2 matrix kernel, K 4 (r 1 , x 1 ; r 2 , x 2 ) = S 4 (r 1 , x 1 ; r 2 , x 2 ) D 4 (r 1 , x 1 ; r 2 , x 2 ) I 4 (r 1 , x 1 ; r 2 , x 2 ) S 4 (r 2 , x 2 ; r 1 , x 1 ) , with the matrix elements being Here Remark. The subscript 4 refers to the fact that this case is related to the symplecticunitary transition in random matrix theory. It should also be remarked that at this stage the infinite particles limit is taken easily. One only replaces the summation, n i,j=1 , in (5.5),(5.7),(5.8) with ∞ i,j=1 . Proof. As in the orthogonal case, we derive (5.2) by generalizing the methods of [19,34]. Using the Heine identity (3.16), the partition function Z n,M [g] is written as Here In the second equality we have used ψ j defined in (3.28). From (5.12), (5.13) and the identity [22,35], one finds where A 4 is defined in (5.9) and (A We should notice that the forms of A 4 and A 4 are the same as the corresponding ones in the orthogonal case, (3.29), (3.31) and (3.32), respectively. Only difference is that sgn in (3.12) and (3.13) is replaced by s, (5.11), in (5.9) and (5.10). Thus we can calculate the kernel along the line in section 3. The result is Recalling the definitions of φ i and (G 4 ) i , we see that (5.2) holds.

Generating Functions
The symbol a 4 (z) of the matrix A 4 is computed as where The winding number of a 4 (z) is again not zero, but the difficulty can be overcome as in the orthogonal case. s 4 (z) can be expressed in two ways (5.31)

Bulk
In the bulk region, the asymptotic behavior is just the same as in the orthogonal case.
Proposition 5.2. For the scaled height variable defined by (4.26), (4.30) holds in the symplectic case as well.
Proof. To show this, we need the asymptotics ofS 4 ,Ĩ 4 and D 4 in the bulk region. Applying the same reasoning as in 4.2, one finds Then discussions similar to ones below (4.81) lead to (4.30).
where F 4 (s) is the GSE Tracy-Widom distribution [22]. Notice a notational difference in [22] and [25]. We follow the convention in the latter; our F 4 (s) is F 4 (s) = F BR 4 (s) = F TW 4 ( √ 2s). The result, (5.41), was shown in [25]; it would also be possible to prove this by using 42) and the skew orthogonal polynomials techniques [43,44,45]. The Meixner symplectic ensemble representation, (5.42), was not given in [6] but can be proved similarly if one notices that a symmetric N × N matrix with zero elements on diagonal and non-negative integer elements on off-diagonal is mapped to a semistandard Young tableau with all columns of even length through Knuth correspondence [46].
Proof. Derivation of (5.35) is analogous to that of (4.87). Using the integral representation of the Airy function, (4.97), we have, forS 4 , where η 1 + η 2 + τ 1 − τ 2 > 0. For I 4 , where there is no additional condition on τ 1 , τ 2 , η 1 , η 1 . Then applying the same method as in 4.3, we can show where 0 < β 0 < 1 is fixed [1]. To consider the scaling limit in the bulk, define the scaled height variable as where a(β) = 2 1 − β 2 , (6.6) The proof of the Proposition 6.1 is parallel to that in 4.2. The main difference is that the function g µ,β in (4.42) is replaced by a simpler for which the double critical point is µ c (β) = 2 1 − β 2 , (6.10) As for the fluctuation near the origin, if we define the scaled height variable as

Discussions
In the preceding sections, we have given a detailed analysis of the height fluctuation of the model for two special values of the strength of the external source at the origin, γ = 1 and γ = 0. Our results show that the fluctuation near the origin is described by the orthogonal/symplectic to unitary transition in random matrix theory.
This implies, in particular, that the height fluctuation of the PNG model at a single point near the origin is equivalent to that of the largest eigenvalue of the transition ensemble. To check this, we performed Monte-Carlo simulations of the PNG model and the transition random matrix. In Figs. 4 and 5, we have shown the fluctuations of the height of the PNG model and the largest eigenvalue of the transition ensemble. We see an excellent agreement between them.
Unfortunately properties of the transition ensemble have not been well studied compared to the ensembles with a specified symmetry. For example, it seems difficult to numerically compute the probability distribution function of the largest eigenvalue . This is sharply contrasted to the situation for the GUE/GOE/GSE, for which the Painlevé representation allows us to plot the probability distribution function with high accuracy. It would be desirable to study the transition ensemble to better understand the statistical properties of the PNG model.
Now we argue what happens for other values of γ. For the fluctuation at the origin, the results have been obtained in [25] where a strong universality is observed. The GSE Tracy-Widom distribution describes the fluctuation not only for γ = 0 but also for all values in 0 ≤ γ < 1. The fluctuation becomes the Gaussian for all values in γ > 1. On the other hand, the GOE Tracy-Widom distribution appears only at γ = 1. The Gaussian fluctuation for γ > 1 is stated only for the continuous model in [25], but is expected to persist for the discrete model as well. In Fig. 6, typical shapes of the droplet are shown for three cases where γ < 1, γ = 1, γ > 1. From these one should be able to guess that the shape looks similar for all values in γ ≤ 1 but a cusp-like piece appears near the origin when γ becomes greater than unity.
This suggests that the fluctuation near the origin also has a similar universality. For all values in 0 ≤ γ < 1, it is expected to be described by the symplectic-unitary transition. This is a conjecture because we do not know how to prove this at present, but is supported by a Monte-Carlo simulation. See Fig. 7, where a good agreement is observed.
For γ > 1, we can also expect some universal behavior for the fluctuation near the origin. We have not found a compact formula for general γ, but the situation becomes quite simple in the limiting case where α → 0 with 0 < γα < 1 fixed. In this limit, the nucleations in the bulk are so rare that the only those at the origin are important. Then the height at a position r and at time t would be almost the same as that at a position 0 and at time t − r. Therefore the height fluctuation at a single point is given by the Gaussian. If we set r = ρt with 0 < ρ < 1 fixed, we have where a G = γα 1−γα and d G = 1 1−γα . A result of a Monte-Carlo simulation corresponding to this case is shown in Fig. 8. In addition, since nucleations at the origin are independent for each time, the multi-point equal time height fluctuation would be described by the one-dimensional Brownian motion. The situation would be somewhat more difficult for a smaller γ, but we expect that the same Gaussian fluctuation is observed in an appropriate limit.

Conclusion
In this paper, we have studied fluctuation properties of the one-dimensional polynuclear growth (PNG) model in a half space with an external source at the origin. We have mainly considered the model in a discrete space and time, but have also given results for the model in a continuous setting. The results in the scaling limit are the same for both cases.
For two special values of the strength of the external source, γ, we have performed a detailed analysis. The γ = 1 case corresponds to a critical point, which we call the orthogonal case. The γ = 0 case corresponds to the model without the external source, which we call the symplectic case. The main results are (4.87) for the orthogonal case, and (5.35) for the symplectic case. According to these the height fluctuation of the model near the origin is equivalent to those of the largest eigenvalue of the orthogonal/symplectic to unitary transition ensemble at soft edge in random matrix theory. We have also shown that the height fluctuation at bulk is described by the Airy process. For other values of γ, we have conjectured that the fluctuation is the symplectic-unitary type for 0 ≤ γ < 1, whereas it is the Gaussian type for γ > 1. Some Monte-Carlo simulation results are also presented to confirm our results and conjectures. Once the nucleation occurs, it grows laterally toward right and left with unit speed. (b) When two steps collide, the one with higher height swallow the one with lower height. In the multi-layer version, a nucleation occurs in the lower layer at the position above which the collision occurs in the upper layer. The height is equal to the height of the swallowed region in the upper layer.   (c) r = 1000 (each 10000 samples). The first two cases correspond to (a) τ = 0 (b) τ = 1, in which the scaling (4.83) is used. The case (c) is better fitted to the bulk scaling limit; the scaling (4.26) is used. In the figures, they are represented as circles. For the transition random matrix ensemble, the data are for N = 500, (a) τ = 0 (b) τ = 1 (c) τ = 10 (each 1000 samples). In the figures, they are represented as +. For (a)(resp. (c)), the GOE (resp. GUE) Tracy-Widom distribution is also shown as a solid line.  Notice that the appearance of the shape changes drastically when γ gets greater than one.