Generalizations of TASEP in Discrete and Continuous Inhomogeneous Space

Abstract

We investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems can be thought of as new exactly solvable examples of tandem queues, directed first- or last-passage percolation models, or Robinson–Schensted–Knuth type systems with random input. One of the novel features of the particle systems is the presence of spatial inhomogeneity which can lead to the formation of traffic jams. For systems with special step-like initial data, we find explicit limit shapes, describe hydrodynamic evolution, and obtain asymptotic fluctuation results which put the systems into the Kardar–Parisi–Zhang universality class. At a critical scaling around a traffic jam in the continuous space TASEP, we observe deformations of the Tracy–Widom distribution and the extended Airy kernel, revealing the finer structure of this novel type of phase transitions. A homogeneous version of a discrete space system we consider is a one-parameter deformation of the geometric last-passage percolation, and we obtain extensions of the limit shape parabola and the corresponding asymptotic fluctuation results. The exact solvability and asymptotic behavior results are powered by a new nontrivial connection to Schur measures and processes.

Appendices

Equivalent Models

Here we discuss a number of equivalent combinatorial formulations of our discrete DGCG model. For simplicity we consider only fully homogeneous models with $a_i\equiv a$, $\nu _j\equiv \nu$, $\beta _t\equiv \beta$. In Appendix A.4 we also describe an equivalent formulation of the (homogeneous) continuous space TASEP.

1.1 Parallel TASEP with geometric-Bernoulli jumps

Let us interpret the doubly geometric corner growth $H_T(N)$ as a TASEP-like particle system.

Definition A.1

The geometric-Bernoulli random variable ${\mathsf {g}}\in {\mathbb {Z}}_{\ge 0}$ (gB variable, for short; notation ${\mathsf {g}}\sim \mathrm {gB}(a\beta ,\nu )$) is a random variable with distribution

$$\begin{aligned} {\mathrm {Prob}}({\mathsf {g}}=j) := \frac{{\mathbf {1}}_{j=0}}{1+a\beta } + \frac{a\beta \,{\mathbf {1}}_{j\ge 1}}{1+a\beta } \left( \frac{\nu +a\beta }{1+a\beta } \right) ^{j-1} \frac{1-\nu }{1+a\beta } , \qquad j\in {\mathbb {Z}}_{\ge 0}. \end{aligned}$$

Definition A.2

The geometric-Bernoulli Totally Asymmetric Simple Exclusion Process (gB-TASEP, for short) is a discrete time Markov chain $\{\vec {G}(T)\}_{T\in {\mathbb {Z}}_{\ge 0}}$ on the space of particle configurations $\vec {G}=(G_1>G_2>\ldots )$ in ${\mathbb {Z}}$, with at most one particle per site allowed, and the step initial condition $G_i(0)=-i$, $i=1,2,\ldots$.

The dynamics of gB-TASEP proceeds as follows. At each discrete time step, each particle $G_j$ with an empty site to the right (almost surely there are finitely many such particles at any finite time) samples an independent random variable ${\mathsf {g}}_j\sim \mathrm {gB}(a\beta ,\nu )$, and jumps by $\min ({\mathsf {g}}_j,G_{j-1}-G_{j}-1)$ steps (with $G_0=+\infty$ by agreement). See Fig. 4 (in the Introduction) for an illustration.

Proposition A.3

Let $H_T(N)$ be the DGCG height function. Then for all $T\in {\mathbb {Z}}_{\ge 0}$ and $N\in {\mathbb {Z}}_{\ge 1}$ we have

$$\begin{aligned} H_T(N)=\#\{i\in {\mathbb {Z}}_{\ge 1}:G_{i}(T)+i+1\ge N\}, \end{aligned}$$

where $\{G_i(T)\}$ is the gB-TASEP with the step initial configuration.

Remark A.4

Replacing particles by holes and vice versa in gB-TASEP one gets a stochastic particle system of zero range type. It is called the generalized TASEP in [DPP15].

1.2 Directed last-passage percolation like growth model

Let us present another equivalent formulation of DGCG as a variant of directed last-passage percolation. For each $N\in {\mathbb {Z}}_{\ge 2}$ and $H\in {\mathbb {Z}}_{\ge 1}$, sample two families of independent identically distributed geometric random variables:

• $W_{N,H}\in {\mathbb {Z}}_{\ge 1}$ has the geometric distribution with parameter $w:=a \beta /(1+a\beta )$, that is, ${\mathrm {Prob}}(W_{N,H}=j)=w^{j}(1-w)$, $j\ge 1$.

• $U_{N,H}\in {\mathbb {Z}}_{\ge 0}$ has the geometric distribution

  $$\begin{aligned} {\mathrm {Prob}}(U_{N,H}=j)= \frac{1-\nu }{1+a\beta }\left( \frac{\nu +a\beta }{1+a\beta } \right) ^{j}, \qquad j\ge 0, \end{aligned}$$

  which is the homogeneous version of (1.3)–(1.4).

Fig. 22

Directed last-passage percolation formulation of DGCG. The independent random variables $W_{N,H}$ and $U_{N,H}$ are written in rectangular boxes in each cell, and the variables $L_{N,H}$ (times at which each cell is covered by the growing interface) are circled. Shaded are the cells which are covered instantaneously during the growth

Define a family of random variables $L_{N,H}\in {\mathbb {Z}}_{\ge 1}$, $N\ge 2$, $H\ge 1$, depending on the W’s and the U’s via the recurrence relation

$$\begin{aligned} \begin{aligned} L_{N,H}&:= \max (L_{N-1,H},L_{N,H-1}) +W_{N,H} \\&\quad -W_{N,H}{\mathbf {1}}_{L_{N-1,H}>L_{N,H-1}} \sum _{j=1}^{N-2}{\mathbf {1}}_{L_{N-1,H} =\cdots =L_{N-j,H}>L_{N-j-1,H} } {\mathbf {1}}_{U_{N-j,H}\ge j} , \end{aligned} \end{aligned}$$

(A.1)

together with the boundary conditions

$$\begin{aligned} L_{1,H}=L_{N,0}=0,\qquad H\ge 0,\quad N\ge 1. \end{aligned}$$

(A.2)

An example is given in Fig. 22.

Proposition A.5

The time-dependent formulation $\{H_T(N)\}$ (with homogeneous parameters) and the last-passage formulation $\{L_{N,H}\}$ are equivalent in the sense that

$$\begin{aligned} L_{N,H}=\min \left\{ T:H_{T}(N)=H \right\} \end{aligned}$$

for all $H\ge 1$, $N\ge 2$.

Proof

In $\{H_T(N)\}$ a cell (N, H) in the lattice can be covered by the growing interface at the step $T\rightarrow T+1$ in two cases:

• it was an inner corner, and event (1.2) occurred;

• it was added to the covered inner corner instantaneously according to the probabilities (1.3)–(1.4).

Here $W_{N,H}$ is identified with the waiting time to cover (N, H) once this cell becomes an inner corner. The coefficient by $W_{N,H}$ in the second line in (A.1) is the indicator of the event that the cell (N, H) is covered instantaneously by a covered inner corner at some $(N-j,H)$. The random variable $U_{N-j,H}$ is precisely the random number of boxes which are instantaneously added when $(N-j,H)$ is covered, and it has to be at least j to cover (N, H). Moreover, it must be $L_{N-1,H}>L_{N,H-1}$, this corresponds to the truncation in (1.3). When (N, H) is covered instantaneously (so that the indicator is equal to 1), we have $L_{N,H}=L_{N-1,H}$, and $W_{N,H}$ is not added. $\quad \square$

Remark A.6

The first line in (A.1) corresponds to the usual directed last-passage percolation model with geometric weights. Denote it by ${\widetilde{L}}_{N,H}$, i.e., ${\widetilde{L}}_{N,H}=\max ({\widetilde{L}}_{N-1,H},{\widetilde{L}}_{N,H-1})+W_{N,H}$ (with the same boundary conditions (A.2)). Almost surely we have ${\widetilde{L}}_{N,H}\ge L_{N,H}$ for all N, H. Limit shape and fluctuation results for ${\widetilde{L}}_{N,H}$ were obtained in [Joh00] (for the homogeneous case $a_N\equiv a$). In Sect. 6 we compare our limit shape with the one for ${\widetilde{L}}_{N,H}$.

1.3 Strict-weak first-passage percolation

Any TASEP with parallel update and step initial configuration can be restated in terms of the First-Passage Percolation (FPP) on a strict-weak lattice. Let us define the FPP model. Take a lattice $\{(T,j):T\ge 0,\, j\ge 1\}$, and draw its elements as $(1,0)T+(1,1)j\subset {\mathbb {R}}^2$, see Fig. 23. Assign random weights to the edges of the lattice: put weight zero at each diagonal edge, and independent random weights with gB distribution (Definition A.1) at all horizontal edges. This model (with the gB distributed weights) appeared in [Mar09] together with a queuing interpretation, see Remark A.8 below. Its limit shape was described in [Mar09] in terms of a Legendre dual.

Fig. 23

Interpreting $G_j(T)$ as first-passage percolation times

We consider directed paths on our lattice, i.e., paths which are monotone in both T and j. For any path, define its weight to be the sum of weights of all its edges. Let the first passage time$F_j(T)$ from (0, 0) to (T, j) to be the minimal weight of a path over all directed paths from (0, 0) to (T, j).

Proposition A.7

We have $F_j(T)=G_j(T+j-1)+j$ for all j, T (equality in distribution of families of random variables), where $G_j(T)$ is the coordinate of the j-th particle in the gB-TASEP started from the step initial configuration.

Proof

The first passage times satisfy the recurrence:

$$\begin{aligned} F_j(T)=\min (F_{j-1}(T),F_j(T-1)+w_{j,T}), \end{aligned}$$

where $w_{j,T}$ is the gB random variable at the horizontal edge connecting $(j,T-1)$ and (j, T). At the same time, the gB-TASEP particle locations satisfy

$$\begin{aligned} G_j(T)=\min (G_{j-1}(T-1)-1,G_j(T-1)+{\tilde{w}}_{j,T}), \end{aligned}$$

where ${\tilde{w}}_{j,T}$ is the gB random variable corresponding to the desired jump of the j-th particle at time step $T-1\rightarrow T$. One readily sees that the boundary conditions for these recurrences also match, which completes the proof. $\quad \square$

The FPP times $F_j(T)$ have an interpretation in terms of column Robinson–Schensted–Knuth (RSK) correspondence. We refer to [Ful97, Sag01, Sta01] for details on the RSK correspondences. Applying the column RSK to a random integer matrix of size $j\times (T+j-1)$ with independent gB entries, one gets a random Young diagram $\lambda =(\lambda _1\ge \cdots \ge \lambda _j\ge 0 )$ of at most j rows. The FPP time is related to this diagram as $F_j(T)=\lambda _j$. The full diagram $\lambda$ can also be recovered with the help of Greene’s theorem [Gre74] by considering minima of weights over nonintersecting directed paths in the strict-weak lattice with edge weights coming from the integer matrix.

To the best of our knowledge, the gB distribution presents a new family of random variables for which the corresponding oriented FPP times (obtained by applying the column RSK to a random matrix with independent entries) can be analyzed to the point of asymptotic fluctuations. Other known examples of random variables with tractable (to the point of asymptotic fluctuations) behavior of the FPP times consist of the pure geometric and Bernoulli distributions. Under a Poisson degeneration, the question of oriented FPP fluctuations can be reduced to the Ulam’s problem on asymptotics of the longest increasing subsequence in a random permutation. Tracy–Widom fluctuations in the latter case were obtained in the celebrated work [BDJ99].

Remark A.8

The oriented FPP model (as well as the TASEP with parallel update) is equivalent to a tandem queuing system. For our models, the service times in the queues have the gB distribution. We refer to [Bar01, O’C03a, Mar09] for tandem queue interpretation of the usual TASEP as well as of the column RSK correspondence. See also the end of Sect. 1.6 for a similar interpretation of the continuous space TASEP.

1.4 Continuous space TASEP and semi-discrete directed percolation

The homogeneous version (i.e., with $\xi (\chi )\equiv 1$) of the continuous space TASEP with no roadblocks possesses an interpretation in the spirit of directed First-Passage Percolation (FPP). This construction is very similar to a well-known interpretation of the usual continuous time TASEP on ${\mathbb {Z}}$ via FPP. We are grateful to Jon Warren for this observation.

Fix $M\in {\mathbb {Z}}_{\ge 1}$ and consider the space ${\mathbb {R}}_{\ge 0}\times \{1,\ldots ,M \}$ in which each copy of ${\mathbb {R}}_{\ge 0}$ is equipped with an independent standard Poisson point process of rate 1. See Fig. 24 for an illustration. Let us first recall the connection to the usual continuous time, discrete space TASEP $({\tilde{X}}_1(t)>{\tilde{X}}_2(t)>\ldots )$, ${\tilde{X}}_i(t)\in {\mathbb {Z}}$, $t\in {\mathbb {R}}_{\ge 0}$, started from the step initial configuration ${\tilde{X}}_i(0)=-i$, $i=1,2,\ldots$. In this TASEP each particle has an independent exponential clock with rate 1, and when the clock rings it jumps to the right by one provided that the destination is unoccupied. Fix $t\in {\mathbb {R}}_{>0}$. For each $m=1,\ldots ,M$ consider up-right paths from (0, 1) to (t, m) as in Fig. 24. The energy of an up-right path is, by definition, the total number of points in the Poisson processes lying on this path.

Fig. 24

A minimal energy up-right path from (0, 1) to (t, 4) in the semi-discrete Poisson environment. We have ${\tilde{X}}_4(t)+4=3$

Proposition A.9

For each m and t, the minimal energy of an up-right path from (0, 1) to (t, m) in the Poisson environment has the same distribution as the displacement ${\tilde{X}}_m(t)+m$ of the m-th particle in the usual TASEP.

For the continuous space TASEP consider a variant of this construction by putting an independent exponential random weight with mean $L^{-1}$ at each point of each of the Poisson processes as in Fig. 24. That is, let now the weight of each point be random instead of 1. One can say that we replace the Poisson processes on ${\mathbb {R}}_{\ge 0}\times \left\{ 1,\ldots ,M \right\}$ by marked Poisson processes. This environment corresponds to the continuous space TASEP $(X_1(t)\ge X_2(t)\ge \ldots )$, $X_i(t)\in {\mathbb {R}}_{\ge 0}$:

Proposition A.10

For each $t>0$ and $m=1,\ldots ,M$ the minimal energy of an up-right path from (0, 1) to (t, m) in the marked Poisson environment has the same distribution as the coordinate $X_m(t)$ of the m-th particle in the continuous space TASEP with mean jumping distance $L^{-1}$.

Both Propositions A.9 and A.10 are established similarly to Proposition A.7 while taking into account the continuous horizontal coordinate. The interpretation via minimal energies of up-right paths also allows to define random Young diagrams depending on the Poisson or marked Poisson processes, respectively, by minimizing over collections of nonintersecting up-right paths. Utilizing Greene’s theorem [Gre74], (see also [Ful97, Sag01], or [Sta01]) one sees that in the case of the usual TASEP the distribution of this Young diagram is the Schur measure $\propto s_\lambda (1,\ldots ,1 )s_\lambda (\vec {0};\vec {0};t)$. It would be very interesting to understand the distribution and asymptotics of random Young diagrams arising from the marked Poisson environment.

Hydrodynamic Equations for Limiting Densities

Here we present informal derivations of hydrodynamic partial differential equations which the limiting densities and height functions of the DGCG and continuous space TASEP should satisfy. These equations follow from constructing families of local translation invariant stationary distributions of arbitrary density for the corresponding dynamics. The argument could be made rigorous if one shows that these families exhaust all possible (nontrivial) translation invariant stationary distributions (as, e.g., it is for TASEP [Lig05] or PushTASEP [Gui97, AG05]). We do not pursue this classification question here.

1.1 Hydrodynamic equation for DGCG

Consider the discrete DGCG model in the asymptotic regime described in Sect. 6. Locally around every scaled point $\eta$ the distribution of the process should be translation invariant and stationary under the homogeneous version of DGCG on ${\mathbb {Z}}$ (recall that it depends on the three parameters $a,\beta ,\nu$). The existence (for suitable initial configurations) of the homogeneous dynamics on ${\mathbb {Z}}$ can be established similarly to [Lig73, And82].

A supply of translation invariant stationary distributions on particle configurations on ${\mathbb {Z}}$ is given by product measures. That is, let us independently put particles at each site of ${\mathbb {Z}}$ with the gB probability (cf. Definition A.1)

$$\begin{aligned} \pi (j):={\mathrm {Prob}}(j \text { particles at a site}) = {\left\{ \begin{array}{ll} \displaystyle \frac{1-c}{1-c\nu }, &{} j=0; \\ \displaystyle c^j\,\frac{(1-c)(1-\nu )}{1-c\nu },&{} j\ge 1. \end{array}\right. } \end{aligned}$$

(B.1)

Proposition B.1

The product measure $\pi ^{\otimes {\mathbb {Z}}}$ on particle configurations in ${\mathbb {Z}}$ corresponding to the distribution $\pi$ (B.1) at each site is invariant under the homogeneous DGCG on ${\mathbb {Z}}$ with any values of the parameters a and $\beta$.

Proof

Let us check directly that $\pi$ is invariant, i.e., satisfies

$$\begin{aligned}&\pi (k+1)P(k+1\rightarrow k)+\pi (k-1)P(k-1\rightarrow k)+\pi (k)P(k\rightarrow k)=\pi (k),\nonumber \\&\qquad k=0,1,2,\ldots , \end{aligned}$$

(B.2)

where $P(k\rightarrow l)$ are the one-step transition probabilities of the homogeneous DGCG restricted to a given site (say, we are looking at site 0). The probability that a particle coming from the left crosses the bond $-1\rightarrow 0$ is equal to⁹

$$\begin{aligned} u:=\sum _{n=0}^{\infty }\pi (0)^{n}(1-\pi (0)) \frac{a\beta }{1+a\beta }\left( \frac{\nu +a\beta }{1+a\beta } \right) ^n = \frac{ac\beta }{1+ac\beta }, \end{aligned}$$

where we sum over the number of empty sites to the left of 0, multiply by the probability that a particle leaves a stack, and then travels distance n. We have

$$\begin{aligned} P(k+1\rightarrow k)=\frac{a\beta }{1+a\beta }(1-u), \end{aligned}$$

the probability that a particle leaves the stack at 0, and another particle does not join it from the left. Moreover, for $k\ge 1$ we have

$$\begin{aligned} P(k-1\rightarrow k)= \frac{u(1-\nu )}{1+a\beta }\,{\mathbf {1}}_{k=1}+ \frac{u}{1+a\beta }\,{\mathbf {1}}_{k\ge 2}, \end{aligned}$$

where for $k=1$ we require that the moving particle stops at site 0, and for $k\ge 2$ we need the stack at 0 not to emit a particle. Finally,

$$\begin{aligned} P(k\rightarrow k)=\left( 1-\frac{u(1-\nu )}{1+a\beta } \right) {\mathbf {1}}_{k=0} + \left( \frac{a \beta u}{1+a\beta }+\frac{1-u}{1+a\beta } \right) {\mathbf {1}}_{k\ge 1}, \end{aligned}$$

where we require that no particle has stopped at site 0 for $k=0$ and sum over two possibilities to preserve the number of particles at 0 for $k\ge 1$. With these probabilities written down, checking (B.2) is straightforward. $\quad \square$

The density of particles under the product measure $\pi ^{\otimes {\mathbb {Z}}}$ is $\rho (c)=\frac{c(1-\nu )}{(1-c)(1-c\nu )}$, and the current (i.e., the average number of particles crossing a given bond) is equal to the quantity u from the proof of Proposition B.1, that is, $j(c)=\frac{c a\beta }{1+ca\beta }$. Thus, the dependence of the current on the density has the form (where we recall that the parameters $a,\nu$ depend on the space coordinate $\eta$)

$$\begin{aligned} j(\rho )= \frac{2 a \beta \rho }{2 a \beta \rho +\nu (\rho -1)+\rho +1+ \sqrt{(\nu (\rho -1)+\rho +1)^2-4 \nu \rho ^2}}. \end{aligned}$$

(B.3)

The partial differential equation for the limiting density $\rho (\tau ,\eta )$ expressing the continuity of the hydrodynamic flow has the form [AK84, Rez91, Lan96, GKS10]

$$\begin{aligned} \frac{\partial }{\partial \tau }\rho (\tau ,\eta ) + \frac{\partial }{\partial \eta }j\bigl (\rho (\tau ,\eta )\bigr )=0. \end{aligned}$$

(B.4)

One can readily verify that the limit shape (6.1) satisfies this equation. Equation (B.4) should also hold for the scaling limit of the inhomogeneous DGCG, when the parameters $a,\beta ,\nu$ of the homogeneous dynamics on the full line ${\mathbb {Z}}$ depend on the spatial coordinate $\eta$. That is, one should replace $j(\rho (\tau ,\eta ))$ (B.3) by $j(\rho (\tau ,\eta );\eta )$ with $a=a(\eta ),a=\beta (\eta ),a=\nu (\eta )$ being the scaled values of the parameters.

1.2 Hydrodynamic equation for continuous space TASEP

Assume that the set of roadblocks ${\mathbf {B}}$ is empty. Then locally at every point $\chi >0$ the behavior of the continuous space TASEP should be homogeneous. Locally the parameters can be chosen so that the mean waiting time to jump is $1/\xi \equiv 1/\xi (\chi )$ and the mean jumping distance is 1.

The local distribution (on the full line ${\mathbb {R}}$) should be invariant under space translations, and stationary under our homogeneous Markov dynamics. The existence (for suitable initial configurations) of the dynamics on ${\mathbb {R}}$ can be established similarly to [Lig73, And82].

A supply of translation invariant stationary distributions of arbitrary density may be constructed as follows. Fix a parameter $0<c<1$ and consider a Poisson process on ${\mathbb {R}}$ with rate (i.e., mean density) $\frac{c}{1-c}$. Put a random geometric number of particles at each point of this Poisson process, independently at each point, with the geometric distribution

$$\begin{aligned} \mathrm {Prob}(j\ge 1 \text { particles})=(1-c)c^{j-1}. \end{aligned}$$

Thus we obtain a so-called marked Poisson process — a distribution of stacks of particles on ${\mathbb {R}}$. It is clearly translation invariant. The stationarity of this process under the dynamics (for any $\xi$) follows by setting $q=0$ in [BP18b, Appendix B] so we omit the computation here.

The density of particles under this marked Poisson process is

$$\begin{aligned} \rho =\frac{c}{(1-c)^2}. \end{aligned}$$

One can check that the current of particles (that is, the mean number of particles passing through, say, zero, in a unit of time) has the form

$$\begin{aligned} j=\xi c=\xi \,\frac{1+2 \rho -\sqrt{1+4 \rho }}{2 \rho }. \end{aligned}$$

The partial differential equation for the limiting density $\rho (\theta ,\chi )$ (under the scaling described in Sect. 4.1) expressing the continuity of the hydrodynamic flow has the form $\rho _\theta +(j(\rho ))_{\chi }=0$, or

$$\begin{aligned} \frac{\partial }{\partial \theta }\rho (\theta ,\chi )+ \frac{\partial }{\partial \chi } \biggl [ \xi (\chi ) \,\frac{1+2 \rho (\theta ,\chi ) -\sqrt{1+4 \rho (\theta ,\chi )}}{2 \rho (\theta ,\chi ) } \biggr ] =0,\quad \rho (0,\chi )= +\infty \, {\mathbf {1}}_{\chi =0}.\nonumber \\ \end{aligned}$$

(B.5)

The density is related to the limiting height function as $\rho (\theta ,\chi )=-\frac{\partial }{\partial \chi }{\mathfrak {h}}(\theta ,\chi )$, and so ${\mathfrak {h}}$ should satisfy

$$\begin{aligned} {\mathfrak {h}}_{\chi }(\theta ,\chi )= - \frac{\xi (\chi ){\mathfrak {h}}_{\theta }(\theta ,\chi )}{\bigl (\xi (\chi )-{\mathfrak {h}}_{\theta }(\theta ,\chi )\bigr )^2},\qquad {\mathfrak {h}}(0,\chi )=+\infty \,{\mathbf {1}}_{\chi =0}. \end{aligned}$$

(B.6)

The passage from (B.5) to (B.6) is done via integrating from $\chi$ to $+\infty$ followed by algebraic manipulations. One can check that the limit shape in the curved part

$$\begin{aligned} {\mathfrak {h}}(\theta ,\chi )= \theta {\mathfrak {w}}^\circ (\theta ,\chi )- \int \limits _{0}^{\chi } \dfrac{\xi (u) {\mathfrak {w}}^\circ (\theta ,\chi ) du}{(\xi (u) - {\mathfrak {w}}^\circ (\theta ,\chi ))^{2}} \end{aligned}$$

from Definition 4.3 indeed satisfies (B.6) whenever all derivatives make sense. Such a check is very similar to the one performed in the discrete case in Appendix B.1 (and also corresponds to setting $q=0$ in [BP18b, Appendix B]), so we omit it for the continuous model.

Fluctuation Kernels

1.1 $\hbox {Airy}_2$ kernel and GUE Tracy–Widom distribution

Let $\mathsf {Ai}(x):=\frac{1}{2\pi }\int e^{i\sigma ^3/3+i\sigma x}d\sigma$ be the Airy function, where the integration is over a contour in the complex plane from $e^{{{\mathbf {i}}}\frac{5\pi }{6}}\infty$ through 0 to $e^{{{\mathbf {i}}}\frac{\pi }{6}}\infty$. Define the extended Airy kernel¹⁰ [Mac94, FNH99, PS02] on ${\mathbb {R}}\times {\mathbb {R}}$ by

$$\begin{aligned} \begin{aligned}&{\mathsf {A}}^{\mathrm {ext}}(s,x;s',x')= {\left\{ \begin{array}{ll} \int _{0}^{\infty }e^{-\mu (s-s')}\mathsf {Ai}(x+\mu )\mathsf {Ai}(x'+\mu )d\mu ,&{} \text{ if } s\ge s';\\ -\int _{-\infty }^{0}e^{-\mu (s-s')}\mathsf {Ai}(x+\mu )\mathsf {Ai}(x'+\mu )d\mu ,&{} \text{ if } s<s' \end{array}\right. } \\&\quad = - \frac{{\mathbf {1}}_{s<s'}}{\sqrt{4\pi (s'-s)}} \exp \left( - \frac{(x-x')^2}{4(s'-s)} -\frac{1}{2}(s'-s)(x+x') +\frac{1}{12}(s'-s)^3 \right) \\&\qquad + \frac{1}{(2\pi {{\mathbf {i}}})^2} \iint \exp \Big ( s x-s'x' -\frac{1}{3}s^3+\frac{1}{3}s'^3 -(x-s^2)u+(x'-s'^2)v \\&\qquad -s u^2+s' v^2 + \frac{1}{3}(u^3-v^3) \Big ) \frac{du\,dv}{u-v}. \end{aligned} \end{aligned}$$

(C.1)

In the double contour integral expression, the v integration contour goes from $e^{-{{\mathbf {i}}}\frac{2\pi }{3}}\infty$ through 0 to $e^{{{\mathbf {i}}}\frac{2\pi }{3}}\infty$, and the u contour goes from $e^{-{{\mathbf {i}}}\frac{\pi }{3}}\infty$ through 0 to $e^{{{\mathbf {i}}}\frac{\pi }{3}}\infty$, and the integration contours do not intersect. This expression for the extended Airy kernel which is most suitable for our needs appeared in [BK08, Section 4.6], see also [Joh03].

We also use the following gauge transformation of the extended Airy kernel:

$$\begin{aligned} \begin{aligned}&\widetilde{{\mathsf {A}}}^{\mathrm {ext}}(s,x;s',x'): = e^{-sx+s'x'+\frac{1}{3} s^3-\frac{1}{3}s'^3} {\mathsf {A}}^{\mathrm {ext}}(s,x;s',x') \\&\quad = - \frac{{\mathbf {1}}_{s<s'}}{\sqrt{4\pi (s'-s)}} \exp \left( - \frac{(s^2-x-s'^2+x')^2}{4(s'-s)} \right) \\&\qquad + \frac{1}{(2\pi {{\mathbf {i}}})^2} \iint \exp \Big ( -(x-s^2)u+(x'-s'^2)v -s u^2+s' v^2 + \frac{1}{3}(u^3-v^3) \Big ) \frac{du\,dv}{u-v}. \end{aligned} \end{aligned}$$

(C.2)

When $s=s'$, ${\mathsf {A}}^{\mathrm {ext}}(s,x;s',x')$ becomes the usual Airy kernel (independent of s):

$$\begin{aligned} \begin{aligned} {\mathsf {A}}(x;x') := {\mathsf {A}}^{\mathrm {ext}}(s,x;s,x')&= \frac{1}{(2\pi {{\mathbf {i}}})^2} \iint \frac{e^{u^3/3-v^3/3-xu+x'v}du\,dv}{u-v} \\&=\frac{\mathsf {Ai}(x)\mathsf {Ai}'(x')-\mathsf {Ai}'(x)\mathsf {Ai}(x')}{x-x'}, x,x'\in {\mathbb {R}}. \end{aligned} \end{aligned}$$

(C.3)

The GUE Tracy–Widom distribution function [TW94] is the following Fredholm determinant of (C.3):

$$\begin{aligned} F_{GUE}(r)=\det \left( {\mathbf {1}}-{\mathsf {A}} \right) _{(r,+\infty )}, \qquad r\in {\mathbb {R}}, \end{aligned}$$

(C.4)

defined analogously to (3.15) with sums replaced by integrals over $(r,+\infty )$.

1.2 BBP deformation of the $\hbox {Airy}_2$ kernel

Fix m and a vector ${\mathbf {b}}=(b_1,\ldots ,b_m )\in {\mathbb {R}}^{m}$. Define the extended BBP kernel on ${\mathbb {R}}\times {\mathbb {R}}$ by

$$\begin{aligned}&\widetilde{{\mathsf {B}}}^{\mathrm {ext}}_{m,{\mathbf {b}}}(s,x;s',x'): = - \frac{{\mathbf {1}}_{s<s'}}{\sqrt{4\pi (s'-s)}} \exp \left( - \frac{(s^2-x-s'^2+x')^2}{4(s'-s)} \right) \nonumber \\&\quad + \frac{1}{(2\pi {{\mathbf {i}}})^2} \iint \prod _{j=1}^{m}\frac{v-b_i}{u-b_i} \exp \Big ( -(x-s^2)u+(x'-s'^2)v -s u^2+s' v^2 + \frac{1}{3}(u^3-v^3) \Big ) \frac{du\,dv}{u-v}.\nonumber \\ \end{aligned}$$

(C.5)

The integration contours are as in the Airy kernel (C.1) with the additional condition that they both must pass to the left of the poles $b_i$.

For $s=s'=0$ this kernel (denote it by $\widetilde{{\mathsf {B}}}_{m,{\mathbf {b}}} (x,x')$) was introduced in [BBP05] the context of spiked random matrices. The extended version appeared in [IS07]. In this paper we are using the gauge transformation similar to (C.2), hence the tilde in the notation. Denote for ${\mathbf {b}}=(0,\ldots ,0 )$ the corresponding distribution function by

$$\begin{aligned} F_m(r):= \det \bigl ( {\mathbf {1}}-\widetilde{{\mathsf {B}}}_{m,{\mathbf {b}}} \bigr ) _{(r,+\infty )}, \qquad r\in {\mathbb {R}}. \end{aligned}$$

Remark C.1

Note that in several other papers, e.g., [BCF14, Bar15, BP18b] the kernel like (C.5) has the reversed product $\prod _{i=1}^{m}\frac{u-b_i}{v-b_i}$, but the contours pass to the right of the poles. Such a form is equivalent to (C.5). In [BP08] a common generalization with poles on both sides of the contours is considered.

1.3 Deformation of the $\hbox {Airy}_2$ kernel arising at a traffic jam

For $\delta >0$ introduce the following deformation of the extended $\hbox {Airy}_2$ kernel (C.2):

$$\begin{aligned}&\widetilde{{\mathsf {A}}}^{\mathrm {ext,\delta }}(s,x;s',x') = - \frac{{\mathbf {1}}_{s<s'}}{\sqrt{4\pi (s'-s)}} \exp \left( - \frac{(s^2-x-s'^2+x')^2}{4(s'-s)} \right) \nonumber \\&\quad + \frac{1}{(2\pi {{\mathbf {i}}})^2} \iint \exp \bigg ( \frac{\delta }{v}-\frac{\delta }{u} -(x-s^2)u+(x'-s'^2)v -s u^2+s' v^2 + \frac{1}{3}(u^3-v^3) \bigg ) \frac{du\,dv}{u-v}\nonumber \\ \end{aligned}$$

(C.6)

with the same integration contours as in the Airy kernel with the additional condition that they both pass to the left of 0. This kernel can be related to certain random matrix and percolation models considered in [BP08], see Sect. 5.5.4 for details. A Fredholm determinant at $s=s'$ of this kernel is a deformation of the GUE Tracy–Widom distribution (C.4):

$$\begin{aligned} F_{GUE}^{(\delta ,s)}(r) = \det \bigl ( {\mathbf {1}} - \widetilde{{\mathsf {A}}}^{\mathrm {ext},\delta }(s,\cdot ;s,\cdot ) \bigr )_{(r,+\infty )}, \qquad r\in {\mathbb {R}}. \end{aligned}$$

(C.7)

Note that this deformation additionally depends on s in contrast with the undeformed case, so the deformation breaks translation invariance of the kernel and the process. When $\delta =0$, both the extended kernel (C.6) and the deformed Tracy–Widom GUE distribution turn into the corresponding undeformed objects.

One can show by a change of variables in the integral in (C.6) that $F_{GUE}^{(\delta ,0)}(r+2\delta ^{\frac{1}{2}})\rightarrow F_{GUE}(2^{-\frac{2}{3}}r)$ as $\delta \rightarrow +\infty$. This explains why the deformed distribution $F_{GUE}^{(\delta ,0)}$ arises at a phase transition between two GUE Tracy–Widom laws. We are grateful to Guillaume Barraquand for this observation.

1.4 Fluctuation kernel in the Gaussian phase

Let $m\in {\mathbb {Z}}_{\ge 1}$ and $\gamma >0$ be fixed. Define the kernel on ${\mathbb {R}}$ as follows:

$$\begin{aligned} \begin{aligned} \widetilde{{\mathsf {G}}}^{\mathrm {ext}}_{m,\gamma }(h;h')&:= -{\mathbf {1}}_{\gamma >1} \frac{\exp \left\{ -\frac{(h-h'\gamma )^2}{2(\gamma ^2-1)} \right\} }{\sqrt{2\pi (\gamma ^2-1)}} \\&\quad + \frac{1}{(2\pi {{\mathbf {i}}})^2} \iint \exp \left\{ -\tfrac{1}{2} w^2+\tfrac{1}{2} z^2-hw+h'z \right\} \left( \frac{z}{w\gamma } \right) ^{m}\frac{dz\,dw}{z-w\gamma }. \end{aligned} \end{aligned}$$

(C.8)

The z contour is a vertical line in the left half-plane traversed upwards which crosses the real line to the left of $-\gamma$. The w contour goes from $e^{-{{\mathbf {i}}}\frac{\pi }{6}}\infty$ to $-1$ to $e^{{{\mathbf {i}}}\frac{\pi }{6}}$.

For $\gamma =1$ a Fredholm determinant of this kernel describes the distribution of the largest eigenvalue of an $m\times m$ GUE random matrix $H=[H_{ij}]_{i,j=1}^{m}$, $H^*=H$, ${\mathrm {Re}}H_{ij}\sim {\mathcal {N}}\bigl (0,\frac{1+{\mathbf {1}}_{i=j}}{2}\bigr )$, $i\ge j$, ${\mathrm {Im}}H_{ij}\sim {\mathcal {N}}\bigl (0,\frac{1}{2}\bigr )$, $i>j$. That is, the distribution function of the largest eigenvalue is

$$\begin{aligned} G_m(r)= \det \bigl ( {\mathbf {1}}-\widetilde{{\mathsf {G}}}_{m,1}^{\mathrm {ext}} \bigr )_{(r,+\infty )}. \end{aligned}$$

The extended version (C.8) appeared in [EM98, IS05] (see also [IS07]).