We think of AdS2 in embedding space, defined by

$Y^{2}=Y^{a}\cdot Y^{a}\equiv-(Y^{-1})^{2}-(Y^{0})^{2}+(Y^{1})^{2}=-1.$ (135)

The embedding coordinates are related to global and Poincare coordinates by

$\displaystyle Y^{a}$ $\displaystyle=\left(\frac{\cos
t_{gl}}{\sin\sigma},\frac{\sin
t_{gl}}{\sin\sigma},-\cot\sigma\right)=\left(\frac{1-t_{P}^{2}+z^{2}}{z},\frac{t_{P}}{z},\frac{1+t_{P}^{2}-z^{2}}{z}\right)$
(136)

In the absence of matter, the solution for the dilaton can be written as

$\phi=-Q\cdot Y.$ (137)

$Q^{a}$ is an $SL(2,\mathbb{R})$ charge that measures the charge of the cutout
under the three isometries of $AdS_{2}$. Because the boundary of the cutout is
given by a fixed value of the dilaton, see (3), the trajectory of the boundary
satisfies

$\displaystyle Q\cdot X$ $\displaystyle=-\phi_{r},\qquad
X\equiv\lim_{\epsilon\to 0}\epsilon Y,\text{ and }X^{2}=0$ $\displaystyle
Q^{2}$ $\displaystyle=-4\phi_{r}M,$ (138)

where $M$ is the ADM energy. For a Schwarzchild black hole of temperature $T$,
$M=(\pi T)^{2}\phi_{r}$.

The real power of this formalism is that it is also a way to package the
solutions of semi-classical JT gravity. To bulk matter, we assign a charge

$Q_{m}^{a}=\int(\zeta^{a})^{\mu}n^{\nu}T_{\mu\nu},$ (139)

where the $\zeta^{a}$s are the Killing vectors of $AdS_{2}$. Then, invariance
under the $SL(2,\mathbb{R})$ gauge symmetry means that

$(Q_{l}+Q_{r}+Q_{m})^{a}=0,$ (140)

where $Q_{l,r}$ are the charges for the left and right boundary particles
respectively. The choice of gauge is the freedom in different ways of solving
this equation. So, by calculating $Q_{m}$ and choosing an appropriate gauge,
we can solve for $Q_{l,r}$, which using (138) allows us a full solution.

For the eternal black hole, we fix gauge so that

$Q_{r}^{a}=-Q_{l}^{a}=-\frac{2\pi\phi_{r}}{\beta}(1,0,0).$ (141)

The calculation of $Q_{m}$ simplifies for point particles of mass $m$. The
trajectory of a massive particle $Y^{a}(s)$ satisfies

$\displaystyle d_{s}^{2}Y+m^{2}Y$ $\displaystyle=0.$ (142)

Its $SL(2,\mathbb{R})$ charge is also an integral of motion,

$\displaystyle Q_{a}=\varepsilon_{abc}Y^{b}d_{s}Y^{c},\quad$ $\displaystyle
Q\cdot Y=0\text{ by antisymmetry, }$ $\displaystyle Q^{2}$
$\displaystyle=m^{2}.$ (143)

Massless particles satisfy the $m\to 0$ limit of this equation. For a massless
particle passing through the point

$Y^{a}=(\sqrt{1+y^{2}},0,s_{1}y),\quad y>0,s_{1}=\pm 1,$ (144)

we find in the limit in which the initial position goes towards the boundary,

$\lim_{y\to\infty}Q^{a}_{s_{1}}=\Delta E(s_{1},0,1).$ (145)

Finally, let us use the above equations to calculate a boundary particle
trajectory in AdS-Vaidya. We will do this in global coordinates explicitly,
since this is the one in which the answer is hard to guess. The strategy is to
take the relation between global and embedding coordinates in (136) and plug
it into (138) to find a differential equation for the trajectory. It is a
differential equation because the definition (3) of $u$ means that

$\sin\sigma_{bd}(u)=\epsilon t_{gl}^{\prime}(u)\quad\Rightarrow\quad
X=\frac{(\cos t_{gl},\sin t_{gl},-1)}{t_{gl}^{\prime}},$ (146)

The first equation of (138) is a linear equation in $X$ and therefore a
differential equation for $t_{gl}(u)$.

For a TFD with a shock on the right, we have from (141), (145) and the
$SL(2,\mathbb{R})$ gauge condition $Q_{r}+Q_{l}+Q_{m}=0$,

$Q_{r,V}^{a}=-\mu\left(1+\alpha,0,\alpha\right),\quad\mu=\frac{2\pi\phi_{r}}{\beta},\
\alpha=\frac{\Delta E}{\mu}.$ (147)

As a check, we can calculate the ADM mass after the shock using (138); we get
$[\pi T\sqrt{1+2\alpha}]^{2}\phi_{r}$, which is a well-known answer. Plugging
the value (147) of the charge and the expression (146) for $X$ into (138)
gives for the post-shock solution

$\displaystyle t_{gl}(u)$ $\displaystyle=2\tan^{-1}\frac{\tanh\left(\pi
T\sqrt{1+2\alpha}\;u\right)}{\sqrt{1+2\alpha}}.$ (148)

Following a similar procedure in Poincare coordinates gives

$t_{P}(u)=\frac{1}{\pi T\sqrt{1+2\alpha}}\tanh(\pi T\sqrt{1+2\alpha}\;u)$
(149)

after the shock.

##  Appendix B Exact Analysis of the Microcanonical dPSSY Model

In section 3, we focused on replica-symmetric saddle point geometries. But
following Penington:2019kki , we can in fact also perform the full
gravitational path integral by summing over geometries which contribute in the
planar limit. In this section only, we will set $\phi_{r}=2\pi$, and we will
also restrict to the microcanonical ensemble in this section for simplicity.

We start with the state of (26), that is:

$|\Psi\rangle=\frac{1}{\sqrt{\mathcal{N}}}\sum_{i,j=1}^{D}M_{ij}|\ell,i\rangle_{1}\otimes|\ell,j\rangle_{2}^{*},$
(150)

where we have taken $\ell_{1}=\ell_{2}$ for simplicity, and once more

$\mathcal{N}=\mathrm{Tr}(M^{\dagger}M)\,Z_{1}^{2}+\mathrm{Tr}(M)\mathrm{Tr}(M^{\dagger})\,Z_{2}.$

The reduced density matrix on, say, the first factor is given by

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{N}}\sum_{i,i^{\prime}=1}^{D}\sum_{j,j^{\prime}=1}^{D}M_{ij}M^{*}_{i^{\prime}j^{\prime}}\langle\ell,j^{\prime}|\ell,j\rangle^{*}_{2}\;|\ell,i\rangle\langle\ell,i^{\prime}|_{1}$
(151) $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{N}}\sum_{i,i^{\prime}=1}^{D}\sum_{j,j^{\prime}=1}^{D}M_{ij}\,M^{\dagger}_{j^{\prime}i^{\prime}}\langle\ell,j|\ell,j^{\prime}\rangle_{2}\;|\ell,i\rangle\langle\ell,i^{\prime}|_{1}.$

From here, we get

$\mathrm{Tr}\,\rho^{n}=\frac{1}{\mathcal{N}^{n}}\sum_{i_{1},j_{1}}\sum_{i^{\prime}_{1},j^{\prime}_{1}}\cdots\sum_{i_{n},j_{n}}\sum_{i^{\prime}_{n},j^{\prime}_{n}}M_{i_{1}j_{1}}\langle
i_{2}^{\prime}|i_{1}\rangle M^{\dagger}_{j_{2}^{\prime},i_{2}^{\prime}}\langle
j_{2}|j_{2}^{\prime}\rangle M_{i_{3},j_{2}}\cdots\langle
j_{1}|j_{n}^{\prime}\rangle.$ (152)

To further simplify our analysis, we will now restrict the matrix $M_{ij}$ to
be an orthogonal projection, i.e., we require that $M^{2}=M$, and
$M^{\dagger}=M$, with $k=\mathrm{Tr}\,M$. In order to proceed, we introduce
the resolvent following Penington:2019kki :

$R(x)=\mathrm{Tr}\frac{1}{x^{2}-\rho}=\sum_{n=0}^{\infty}\frac{1}{x^{2(n+1)}}\mathrm{Tr}\,\rho^{n}.$
(153)

We can equivalently write

$R(x)=\int
d\lambda\;D(\lambda)\frac{1}{x^{2}-\lambda}|\lambda\rangle\langle\lambda|=\int
d\lambda\,\frac{D((\lambda)}{2\sqrt{\lambda}}\left[\frac{1}{x-\sqrt{\lambda}}-\frac{1}{x+\sqrt{\lambda}}\right]|\lambda\rangle\langle\lambda|.$
(154)

This resolvent will thus have two branch cuts, one along the positive real $x$
axis and one along the negative real $x$ axis, symmetric under reflection
about the imaginary axis. The eigenvalue density can be obtained from the
resolvent as

$D(\lambda)=\frac{2\sqrt{\lambda}}{2\pi
i}\;\text{disc}_{x=+\sqrt{\lambda}}R(x).$ (155)

The resolvent defined in this way is closely related to the trace of the
resolvent defined and computed in Penington:2019kki , which we will call
$R_{(0)}(\lambda)$. The main difference is that our resolvent only involves
gravitational amplitudes with an even number of boundaries, and in addition,
the normalization factors are different. But we can restrict the PSSY
resolvent to an even number of boundaries by simply taking its anti-symmetric
part in $\lambda$. So, we find

$R(x)=\frac{\mathrm{Tr}\,1-k}{x^{2}}+\frac{\alpha}{2x}\left[R_{(0)}(\alpha
x)-R_{(0)}(-\alpha x)\right],$ (156)

where

$\alpha=\left(\frac{1}{k}+\frac{Z_{2}}{Z_{1}^{2}}\right)^{1/2}.$

We now restrict to the microcanonical ensemble. In this case, the PSSY
resolvent is given by

$R_{(0)}(\lambda)=-\frac{(\varrho-k)-\varrho
k\lambda}{2\lambda}\pm\frac{\varrho
k}{2\lambda}\sqrt{\left(\lambda-\left(\frac{1}{\sqrt{\varrho}}-\frac{1}{\sqrt{k}}\right)^{2}\right)\left(\lambda-\left(\frac{1}{\sqrt{\varrho}}+\frac{1}{\sqrt{k}}\right)^{2}\right)}$
(157)

where

$\varrho=e^{\mathbf{S}},\;\;\;\;\alpha=\left(\frac{1}{k}+\frac{1}{\varrho}\right)^{1/2}.$

Here $\mathbf{S}$ is the microcanonical entropy at chosen energy $E$ and
energy window $\Delta E$:

$\varrho=e^{\mathbf{S}}=e^{S_{0}}\rho(E)\Delta
E,\;\;\;\rho(E)=\frac{E}{2\pi^{2}}\sinh(2\pi E),$ (158)

and note that the dependence on the brane tension $\mu$ drops out in the
microcanonical ensemble. The PSSY resolvent $R_{(0)}$ only has a branch cut
along the positive real axis, but the combination in equation (156) has the
right structure to give us branch cuts along both the positive and negative
real axes, as we expected. Using equations (156), (155) and (157), we thus
obtain the eigenvalue density

$D(\lambda)=\frac{\alpha\varrho
k}{4\pi}\sqrt{\left[\sqrt{\lambda}-\left(\frac{1}{\sqrt{\alpha\varrho}}-\frac{1}{\sqrt{\alpha
k}}\right)^{2}\right]\left[\left(\frac{1}{\sqrt{\alpha\varrho}}+\frac{1}{\sqrt{\alpha
k}}\right)^{2}-\sqrt{\lambda}\right]},$ (159)

for

$\left(\frac{1}{\sqrt{\alpha\varrho}}-\frac{1}{\sqrt{\alpha
k}}\right)^{4}\leq\lambda\leq\left(\frac{1}{\sqrt{\alpha\varrho}}+\frac{1}{\sqrt{\alpha
k}}\right)^{4}.$

When $k\ll\varrho$, the density is highly peaked around $\lambda=\frac{1}{k}$
with a width $\frac{\Delta\lambda}{\lambda}\sim\sqrt{\frac{k}{\varrho}}$.
Thus, the entropy is approximately given by $\log\,k$. On the other hand when
$k\gg\varrho$, the density is highly peaked around $\lambda=\frac{1}{\varrho}$
with a width $\frac{\Delta\lambda}{\lambda}\sim\sqrt{\frac{\varrho}{k}}$.
Finally, we can also obtain the Rényi entropies by expanding the resolvent in
equation (156) in $x$. The first few Rényi entropies are shown in figure 19.

Figure 19: The first few Rényi entropies as a function of $k$.

##  Appendix C Details of the Schwarz-Christoffel map

In this appendix, we want to derive equation (109) directly using the doubly-
connected Schwarz-Christoffel (SC) map. Recall from equation (92), that the
derivative of the relevant SC map is given by

$f^{\prime}(z)=C\frac{\Theta(\mu,z)\Theta(\mu,-z)\Theta(\mu,\frac{z}{\mu})}{\Theta^{3}(\mu,-\frac{z}{\mu})},$
(160)

where $C$ is a constant, and recall that $\mu=e^{-\tau}$ is the conformal
modulus. The theta functions are

$\Theta(\mu,z)=\frac{1}{\prod_{k=1}^{\infty}(1-\mu^{2k})}\sum_{n=-\infty}^{\infty}\mu^{2n}(-z)^{n}.$
(161)

This form is related to that in dep-sc ; driscoll2002schwarz  by the so-called
Jacobi triple product identity. In the present case, we can conveniently re-
write $f^{\prime}$ in terms of the more standard Jacobi theta function:

$f^{\prime}(z)=C\frac{\vartheta(\frac{1}{2i}\log(z);\mu)\vartheta(\frac{1}{2i}\log(-z);\mu)\vartheta(\frac{1}{2i}\log(-z/\mu);\mu)}{\vartheta^{3}(\frac{1}{2i}\log(z/\mu);\mu)},$
(162)

where

$\vartheta(u;q)=\sum_{n=-\infty}^{\infty}q^{n^{2}}e^{2inu}.$ (163)

We are interested in the limit $\mu\to 1$; we will later show that this is
indeed the correct limit by relating $\mu$ to the Euclidean cutoff parameter
$\delta$. In the $\mu\to 1$ limit, it is helpful to use the following modular
transformation property for the elliptic function:

$\vartheta(u;q=e^{-\tau})=\frac{1}{(\tau/\pi)^{1/2}\exp(\frac{u^{2}}{\tau})}\vartheta\left(u^{\prime}=\frac{\pi
u}{i\tau};q^{\prime}=e^{-\pi^{2}/\tau}\right).$ (164)

Writing $z=re^{i\theta}$, we find that one of the relevant theta function
becomes:

$\displaystyle\vartheta\left(\frac{1}{2i}\log\left(z\right);\mu\right)$
$\displaystyle=$
$\displaystyle\vartheta\left(\frac{1}{2i}\left(\log\left(r\right)+i\theta\right);\mu\right)$
(165) $\displaystyle=$
$\displaystyle\frac{1}{\left(\tau/\pi\right)^{1/2}\exp\left(-\frac{\left(\log\,r+i\theta\right)^{2}}{4\tau}\right)}\vartheta\left(u^{\prime}=-\frac{\pi\left(\log\,r+i\theta\right)}{2\tau};q^{\prime}=e^{-\pi^{2}/\tau}\right)$
$\displaystyle=$
$\displaystyle\frac{\exp\left(\frac{\left(\log\,r\right)^{2}-\theta^{2}+2i\theta\log\,r}{4\tau}\right)}{\left(\tau/\pi\right)^{1/2}}\sum_{n=-\infty}^{\infty}e^{-\frac{\pi^{2}n^{2}}{\tau}+\frac{n\pi\theta}{\tau}-\frac{i\pi
n}{\tau}\log r}$ $\displaystyle=$
$\displaystyle\frac{\exp\left(\frac{\left(\log\,r\right)^{2}+2i\theta\log\,r}{4\tau}\right)}{\left(\tau/\pi\right)^{1/2}}\sum_{n=-\infty}^{\infty}e^{-\frac{\pi^{2}}{\tau}\left(n-\frac{\theta}{2\pi}\right)^{2}-\frac{i\pi
n}{\tau}\log r}.$

We will always work in the domain $0\leq\theta<2\pi$. Note from above that
when $\theta<\pi,$ the $n=0$ term dominates in the sum, when $\theta=\pi$ the
$n=0$ and $n=1$ terms are degenrate, and when $\pi<\theta<2\pi$, the $n=1$
term dominates. So as long as we stick to the domain $0\leq\theta<2\pi$, we
need only keep the $n=0$ and $n=1$ terms in our analysis:

$\displaystyle\vartheta\left(\frac{1}{2i}\log\left(z=re^{i\theta}\right);\mu\right)$
$\displaystyle=$
$\displaystyle\frac{\exp\left(\frac{\left(\log\,z\right)^{2}}{4\tau}\right)}{\left(\tau/\pi\right)^{1/2}}\left(1+e^{-\frac{\pi^{2}}{\tau}-\frac{i\pi}{\tau}\log
z}\right)$ (166) $\displaystyle=$
$\displaystyle\frac{\exp\left(\frac{\left(\log\,r\right)^{2}-\theta^{2}+2i\theta\log\,r}{4\tau}\right)}{\left(\tau/\pi\right)^{1/2}}\left(1+e^{-\frac{\pi\left(\pi-\theta\right)}{\tau}-\frac{i\pi}{\tau}\log
r}\right).$

In order to further understand the SC map, we will analyse its derivative in
the complex $z$-plane in two coordinate patches: (i) the far from quench
region $-\varepsilon<(\pi-\theta)<\varepsilon$, where $\tau\ll\varepsilon\ll
1$, and (ii) the near quench region, which is the complement of patch (i).
(The terminology far-from-quench or near-quench will become clear shortly.)

###  C.1 Far from quench

When $z$ is in patch (i), i.e., the far-from-quench region, we get

$\vartheta\left(\frac{1}{2i}\log\left(z=re^{i\theta}\right);\mu\right)=\frac{\exp\left(\frac{\left(\log\,r\right)^{2}-\theta^{2}+2i\theta\log\,r}{4\tau}\right)}{\left(\tau/\pi\right)^{1/2}}\left(1+e^{-\frac{\pi\left(\pi-\theta\right)}{\tau}-\frac{i\pi}{\tau}\log
r}+\mathcal{O}\left(e^{-\pi^{2}/\tau}\right)\right).$ (167)

The other theta function (with $z\to-z$) relevant for us in this patch is then
given by

$\vartheta\left(\frac{1}{2i}\log\left(-z\right);\mu\right)=\frac{\exp\left(\frac{\left(\log\,r\right)^{2}-\left(\pi-\theta\right)^{2}-2i\left(\pi-\theta\right)\log\,r}{4\tau}\right)}{\left(\tau/\pi\right)^{1/2}}\left(1+\mathcal{O}\left(e^{-\pi^{2}/\tau}\right)\right).$
(168)

Putting things together, we get

$\displaystyle f^{\prime}\left(z\right)$ $\displaystyle=$ $\displaystyle-
iC\,e^{-\frac{\pi}{\tau}\left(\frac{\pi}{2}-\theta\right)-\frac{i\pi\log
r}{\tau}-\log\,r-i\theta-\tau/2}\frac{\left(1+e^{-\frac{\pi}{\tau}\left(\pi-\theta\right)-\frac{i\pi\log
r}{\tau}}\right)}{\left(1-e^{-\frac{\pi}{\tau}\left(\pi-\theta\right)-\frac{i\pi\log
r}{\tau}}\right)^{3}}$ (169) $\displaystyle=$
$\displaystyle-\frac{iCe^{\frac{\pi^{2}}{2\tau}-\frac{\tau}{2}}}{4z}\;\frac{\cosh\left(\frac{\pi}{2\tau}\left(\pi-\theta\right)+\frac{i\pi\log
r}{2\tau}\right)}{\sinh^{3}\left(\frac{\pi}{2\tau}\left(\pi-\theta\right)+\frac{i\pi\log
r}{2\tau}\right)}.$

We can integrate this to obtain

$f\left(z\right)=A+C\frac{\tau
e^{\frac{\pi^{2}}{2\tau}-\frac{\tau}{2}}}{4\pi}\frac{1}{\sinh^{2}\left(\frac{\pi^{2}}{2\tau}+\frac{i\pi}{2\tau}\log\,z\right)},$
(170)

where $A$ and $C$ are both constants. Now, we impose the boundary conditions
that $f\left(z=-\mu\right)=0$, and $f\left(z=-\sqrt{\mu}\right)=-1$. These
conditions fix the constants of integration to be

$A=1,\;\;\;C=\frac{4\pi}{\tau}e^{-\frac{\pi^{2}}{2\tau}+\frac{\tau}{2}},$

and thus the function becomes

$f\left(z\right)=1+\frac{1}{\sinh^{2}\left(\frac{\pi^{2}}{2\tau}+\frac{i\pi}{2\tau}\log\,z\right)}=\frac{\cosh^{2}\left(\frac{\pi^{2}}{2\tau}+\frac{i\pi}{2\tau}\log\,z\right)}{\sinh^{2}\left(\frac{\pi^{2}}{2\tau}+\frac{i\pi}{2\tau}\log\,z\right)}.$
(171)

When $r=\sqrt{\mu}$, then it is a simple matter to check that $|f|=1$, i.e.,
the time-reflection symmetric slice in the complex $z$ plane maps to the time-
reflection symmetric slice in the $f$-plane, as expected. Writing
$z=e^{-\tau/2-i\zeta}$ and $f=e^{2i\mathfrak{s}}$, we can now
straightforwardly solve for $\zeta\left(\mathfrak{s}\right)$ from the above
formula, and we find

$\zeta=\pi+\frac{\tau}{\pi}\log\,\tan\frac{\mathfrak{s}}{2},$ (172)

which agrees with equation (109) derived in the main text using the Schwarzian
method.

###  C.2 Near quench

Next, we wish to understand the SC map $f$ in the near quench region. Our main
goal is to relate the conformal modulus $\mu=e^{-\tau}$ to the regulator
$\delta$. In order to do so, it is sufficient to focus on the real axis
$\theta=0$ and $\mu\leq r\leq 1$. In this case, the relevant theta functions
are given by

$\vartheta\left(\frac{1}{2i}\log\left(r\right);\mu\right)=\frac{\exp\left(\frac{\left(\log\,r\right)^{2}}{4\tau}\right)}{\left(\tau/\pi\right)^{1/2}}\left(1+\mathcal{O}\left(e^{-\frac{\pi^{2}}{\tau}}\right)\right),$
(173)
$\vartheta\left(\frac{1}{2i}\log\left(-r\right);\mu\right)=\frac{2\exp\left(-\frac{\left(\pi^{2}-\left(\log\,r\right)^{2}\right)}{4\tau}\right)}{\left(\tau/\pi\right)^{1/2}}\left[\cos\left(\frac{\pi}{2\tau}\log\,r\right)+\mathcal{O}\left(e^{-\frac{2\pi^{2}}{\tau}}\right)\right].$
(174)

Putting everything together, we get on the real axis:

$f^{\prime}\left(r\right)=-2Ce^{-\frac{\pi^{2}}{2\tau}-\frac{1}{2}\left(\tau+2\log\,r\right)}\sin\left(\frac{\pi}{\tau}\log\,r\right)\left[1+\mathcal{O}\left(e^{-\frac{\pi^{2}}{\tau}}\right)\right].$
(175)

Therefore, we get

$\displaystyle f\left(r\right)$ $\displaystyle=$ $\displaystyle
f\left(r_{0}\right)+\int_{r_{0}}^{r}dr^{\prime}\,f^{\prime}\left(r^{\prime}\right)$
(176) $\displaystyle=$ $\displaystyle
f\left(r_{0}\right)+\frac{2\tau}{\pi}Ce^{-\frac{\pi^{2}}{2\tau}-\frac{\tau}{2}}\left[\cos\left(\frac{\pi}{\tau}\log
r\right)-\cos\left(\frac{\pi}{\tau}\log r_{0}\right)\right]+\cdots.$

Now we impose boundary conditions. Taking $r_{0}=\mu$ and
$f\left(r_{0}\right)=e^{-2\delta}$, we get

$f\left(r\right)=e^{-2\delta}+\frac{4\tau}{\pi}Ce^{-\frac{\pi^{2}}{2\tau}-\frac{\tau}{2}}\cos^{2}\left(\frac{\pi}{2\tau}\log
r\right)+\cdots.$ (177)

Next, setting $f\left(\sqrt{\mu}\right)=1$ and $f\left(1\right)=e^{2\delta}$
gives:

$1=e^{-2\delta}+\frac{2\tau}{\pi}Ce^{-\frac{\pi^{2}}{2\tau}-\frac{\tau}{2}}+\cdots$
(178)
$e^{2\delta}=e^{-2\delta}+\frac{4\tau}{\pi}Ce^{-\frac{\pi^{2}}{2\tau}-\frac{\tau}{2}}+\cdots$
(179)

Using the value of $C$ obtained previously, both the above equations are
solved if we make the identification

$\delta=4e^{-\frac{\pi^{2}}{\tau}}+\cdots,$ (180)

where the $\cdots$ denote terms suppressed by more powers of
$e^{-\frac{\pi^{2}}{\tau}}$. Note that as $\tau\to 0$, we find $\delta\to 0$.
This justifies working in the $\tau\to 0$ limit.

Finally, we can now also obtain the function $f$ on the time-reflection
symmetric slice in the near quench region. As before, setting
$f=e^{2i\mathfrak{s}}$ and $z=e^{-\frac{\tau}{2}-i\zeta}$, we find

$(1-e^{2i\mathfrak{s}})=8e^{-\frac{\pi^{2}}{\tau}}\sin\frac{i\pi\zeta}{\tau},\;\;\Rightarrow\zeta(\mathfrak{s})=-i\frac{\tau}{\pi}\sin^{-1}\left(\frac{1-e^{2i\mathfrak{s}}}{8e^{-\frac{\pi^{2}}{\tau}}}\right).$
(181)

Note that a finite interval around $\zeta=0$ gets mapped to an infinitesimal
neighbourhood of $\mathfrak{s}=0$, which is the near quench region. We can
compute the stress tensor on the $\mathfrak{s}$-plane from here, and we find

$T(\mathfrak{s})=\frac{c}{24\pi}\left[-2\frac{e^{\frac{4\pi^{2}}{\tau}}(e^{2i\mathfrak{s}}-1)^{4}-64e^{\frac{2\pi^{2}}{\tau}}(5e^{4i\mathfrak{s}}-4e^{2i\mathfrak{s}}+2)+4096}{(e^{\frac{2\pi^{2}}{\tau}}(e^{2i\mathfrak{s}}-1)^{2}-64)^{2}}\right].$
(182)

We see that the stress tensor is highly peaked at $\mathfrak{s}=0$, with

$T(\mathfrak{s}=0)\sim\frac{c}{24\pi}\frac{3}{2\delta^{2}}.$ (183)

## References

  * (1) S. Raju, _Lessons from the information paradox_ , 2012.05770.
  * (2) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, _The entropy of hawking radiation_ , 2006.06872.
  * (3) J.M. Maldacena, _Eternal black holes in anti-de sitter_ , _JHEP_ 04 (2003) 021 [hep-th/0106112].
  * (4) S.D. Mathur, _The information paradox: A pedagogical introduction_ , _Class. Quant. Grav._ 26 (2009) 224001 [0909.1038].
  * (5) A. Almheiri, D. Marolf, J. Polchinski and J. Sully, _Black holes: Complementarity or firewalls?_ , _JHEP_ 02 (2013) 062 [1207.3123].
  * (6) A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, _An apologia for firewalls_ , _JHEP_ 09 (2013) 018 [1304.6483].
  * (7) L. Susskind, L. Thorlacius and J. Uglum, _The stretched horizon and black hole complementarity_ , _Phys. Rev. D_ 48 (1993) 3743 [hep-th/9306069].
  * (8) E. Verlinde and H. Verlinde, _Black hole entanglement and quantum error correction_ , _JHEP_ 10 (2013) 107 [1211.6913].
  * (9) K. Papadodimas and S. Raju, _Remarks on the necessity and implications of state-dependence in the black hole interior_ , _Phys. Rev. D_ 93 (2016) 084049 [1503.08825].
  * (10) Y. Nomura, J. Varela and S.J. Weinberg, _Complementarity endures: No firewall for an infalling observer_ , _JHEP_ 03 (2013) 059 [1207.6626].
  * (11) L. Susskind, _Black hole complementarity and the harlow-hayden conjecture_ , 1301.4505.
  * (12) J. Maldacena and L. Susskind, _Cool horizons for entangled black holes_ , _Fortsch. Phys._ 61 (2013) 781 [1306.0533].
  * (13) M. Van Raamsdonk, _Evaporating firewalls_ , _JHEP_ 11 (2014) 038 [1307.1796].
  * (14) G. Penington, _Entanglement wedge reconstruction and the information paradox_ , _JHEP_ 09 (2020) 002 [1905.08255].
  * (15) A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, _The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole_ , _JHEP_ 12 (2019) 063 [1905.08762].
  * (16) A. Almheiri, R. Mahajan and J. Maldacena, _Islands outside the horizon_ , 1910.11077.
  * (17) A. Almheiri, R. Mahajan and J.E. Santos, _Entanglement islands in higher dimensions_ , _SciPost Phys._ 9 (2020) 001 [1911.09666].
  * (18) A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, _The page curve of hawking radiation from semiclassical geometry_ , _JHEP_ 03 (2020) 149 [1908.10996].
  * (19) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, _Replica wormholes and the entropy of hawking radiation_ , _JHEP_ 05 (2020) 013 [1911.12333].
  * (20) M. Rozali, J. Sully, M. Van Raamsdonk, C. Waddell and D. Wakeham, _Information radiation in bcft models of black holes_ , _JHEP_ 05 (2020) 004 [1910.12836].
  * (21) H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, _Information flow in black hole evaporation_ , _JHEP_ 03 (2020) 152 [1911.03402].
  * (22) H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, _Evaporating Black Holes Coupled to a Thermal Bath_ , 2007.11658.
  * (23) H.Z. Chen, R.C. Myers, D. Neuenfeld, I.A. Reyes and J. Sandor, _Quantum extremal islands made easy, part i: Entanglement on the brane_ , _JHEP_ 10 (2020) 166 [2006.04851].
  * (24) H.Z. Chen, R.C. Myers, D. Neuenfeld, I.A. Reyes and J. Sandor, _Quantum extremal islands made easy, part ii: Black holes on the brane_ , _JHEP_ 12 (2020) 025 [2010.00018].
  * (25) J. Hernandez, R.C. Myers and S.-M. Ruan, _Quantum extremal islands made easy, partiii: Complexity on the brane_ , 2010.16398.
  * (26) T.J. Hollowood and S.P. Kumar, _Islands and page curves for evaporating black holes in jt gravity_ , _JHEP_ 08 (2020) 094 [2004.14944].
  * (27) T.J. Hollowood, S. Prem Kumar and A. Legramandi, _Hawking radiation correlations of evaporating black holes in jt gravity_ , _J. Phys. A_ 53 (2020) 475401 [2007.04877].
  * (28) I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi and Z. Wei, _Entanglement entropy in holographic moving mirror and page curve_ , 2011.12005.
  * (29) H. Geng and A. Karch, _Massive islands_ , _JHEP_ 09 (2020) 121 [2006.02438].
  * (30) V. Balasubramanian, A. Kar, O. Parrikar, G. Sárosi and T. Ugajin, _Geometric secret sharing in a model of Hawking radiation_ , _JHEP_ 01 (2021) 177 [2003.05448].
  * (31) V. Balasubramanian, A. Kar and T. Ugajin, _Entanglement between two disjoint universes_ , 2008.05274.
  * (32) T. Hartman, E. Shaghoulian and A. Strominger, _Islands in asymptotically flat 2d gravity_ , _JHEP_ 07 (2020) 022 [2004.13857].
  * (33) T. Hartman, Y. Jiang and E. Shaghoulian, _Islands in cosmology_ , _JHEP_ 11 (2020) 111 [2008.01022].
  * (34) V. Balasubramanian, A. Kar and T. Ugajin, _Islands in de sitter space_ , 2008.05275.
  * (35) T. Anegawa and N. Iizuka, _Notes on islands in asymptotically flat 2d dilaton black holes_ , _JHEP_ 07 (2020) 036 [2004.01601].
  * (36) K. Hashimoto, N. Iizuka and Y. Matsuo, _Islands in schwarzschild black holes_ , _JHEP_ 06 (2020) 085 [2004.05863].
  * (37) X. Dong, X.-L. Qi, Z. Shangnan and Z. Yang, _Effective entropy of quantum fields coupled with gravity_ , _JHEP_ 10 (2020) 052 [2007.02987].
  * (38) A. Laddha, S.G. Prabhu, S. Raju and P. Shrivastava, _The holographic nature of null infinity_ , 2002.02448.
  * (39) A. Mousatov and E. Silverstein, _Recovering infalling information via string spreading_ , 2002.12377.
  * (40) S. Ryu and T. Takayanagi, _Holographic derivation of entanglement entropy from ads/cft_ , _Phys. Rev. Lett._ 96 (2006) 181602 [hep-th/0603001].
  * (41) S. Ryu and T. Takayanagi, _Aspects of holographic entanglement entropy_ , _JHEP_ 08 (2006) 045 [hep-th/0605073].
  * (42) V.E. Hubeny, M. Rangamani and T. Takayanagi, _A covariant holographic entanglement entropy proposal_ , _JHEP_ 07 (2007) 062 [0705.0016].
  * (43) N. Engelhardt and A.C. Wall, _Quantum extremal surfaces: Holographic entanglement entropy beyond the classical regime_ , _JHEP_ 01 (2015) 073 [1408.3203].
  * (44) M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, _Causality & holographic entanglement entropy_, _JHEP_ 12 (2014) 162 [1408.6300].
  * (45) D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, _Relative entropy equals bulk relative entropy_ , _JHEP_ 06 (2016) 004 [1512.06431].
  * (46) X. Dong, D. Harlow and A.C. Wall, _Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality_ , _Phys. Rev. Lett._ 117 (2016) 021601 [1601.05416].
  * (47) T. Faulkner and A. Lewkowycz, _Bulk locality from modular flow_ , _JHEP_ 07 (2017) 151 [1704.05464].
  * (48) J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle and M. Walter, _Entanglement wedge reconstruction via universal recovery channels_ , _Phys. Rev. X_ 9 (2019) 031011 [1704.05839].
  * (49) D. Harlow, _The Ryu–Takayanagi Formula from Quantum Error Correction_ , _Commun. Math. Phys._ 354 (2017) 865 [1607.03901].
  * (50) B. Swingle, _Entanglement renormalization and holography_ , _Phys. Rev. D_ 86 (2012) 065007 [0905.1317].
  * (51) P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, _Holographic duality from random tensor networks_ , _JHEP_ 11 (2016) 009 [1601.01694].
  * (52) N. Bao, G. Penington, J. Sorce and A.C. Wall, _Beyond toy models: Distilling tensor networks in full ads/cft_ , _JHEP_ 11 (2019) 069 [1812.01171].
  * (53) P. Caputa, J. Kruthoff and O. Parrikar, _Building tensor networks for holographic states_ , 2012.05247.
  * (54) H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas et al., _Information transfer with a gravitating bath_ , 2012.04671.
  * (55) D. Marolf and A.C. Wall, _Eternal black holes and superselection in ads/cft_ , _Class. Quant. Grav._ 30 (2013) 025001 [1210.3590].
  * (56) K. Papadodimas and S. Raju, _Local operators in the eternal black hole_ , _Phys. Rev. Lett._ 115 (2015) 211601 [1502.06692].
  * (57) D.L. Jafferis, _Bulk reconstruction and the Hartle-Hawking wavefunction_ , 1703.01519.
  * (58) R. Jackiw, _Lower dimensional gravity_ , _Nucl. Phys. B_ 252 (1985) 343.
  * (59) C. Teitelboim, _Gravitation and hamiltonian structure in two space-time dimensions_ , _Phys. Lett. B_ 126 (1983) 41.
  * (60) A. Almheiri and J. Polchinski, _Models of ads 2 backreaction and holography_, _JHEP_ 11 (2015) 014 [1402.6334].
  * (61) J. Maldacena, D. Stanford and Z. Yang, _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ , _PTEP_ 2016 (2016) 12C104 [1606.01857].
  * (62) K. Jensen, _Chaos in $ads_{2}$ holography_, _Phys. Rev. Lett._ 117 (2016) 111601 [1605.06098].
  * (63) J. Engelsöy, T.G. Mertens and H. Verlinde, _An investigation of $ads_{2}$ backreaction and holography_, _JHEP_ 07 (2016) 139 [1606.03438].
  * (64) I. Kourkoulou and J. Maldacena, _Pure states in the syk model and nearly- $ads_{2}$ gravity_, 1707.02325.
  * (65) G. Penington, S.H. Shenker, D. Stanford and Z. Yang, _Replica wormholes and the black hole interior_ , 1911.11977.
  * (66) S.W. Hawking and D.N. Page, _Thermodynamics of black holes in anti-de sitter space_ , _Commun. Math. Phys._ 87 (1983) 577.
  * (67) J.A. Wheeler, _On the nature of quantum geometrodynamics_ , _Annals Phys._ 2 (1957) 604.
  * (68) G.V. Lavrelashvili, V.A. Rubakov and P.G. Tinyakov, _Disruption of quantum coherence upon a change in spatial topology in quantum gravity_ , _JETP Lett._ 46 (1987) 167.
  * (69) S.R. Coleman, _Black holes as red herrings: Topological fluctuations and the loss of quantum coherence_ , _Nucl. Phys. B_ 307 (1988) 867.
  * (70) S.B. Giddings and A. Strominger, _Axion induced topology change in quantum gravity and string theory_ , _Nucl. Phys. B_ 306 (1988) 890.
  * (71) A. Adams, X. Liu, J. McGreevy, A. Saltman and E. Silverstein, _Things fall apart: Topology change from winding tachyons_ , _JHEP_ 10 (2005) 033 [hep-th/0502021].
  * (72) P. Saad, S.H. Shenker and D. Stanford, _A semiclassical ramp in syk and in gravity_ , 1806.06840.
  * (73) P. Saad, S.H. Shenker and D. Stanford, _Jt gravity as a matrix integral_ , 1903.11115.
  * (74) P. Saad, _Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity_ , 1910.10311.
  * (75) U. Moitra, S.K. Sake and S.P. Trivedi, _Jackiw-teitelboim gravity in the second order formalism_ , 2101.00596.
  * (76) Y. Chen, V. Gorbenko and J. Maldacena, _Bra-ket wormholes in gravitationally prepared states_ , 2007.16091.
  * (77) T. Anous, J. Kruthoff and R. Mahajan, _Density matrices in quantum gravity_ , _SciPost Phys._ 9 (2020) 045 [2006.17000].
  * (78) D. Harlow and D. Jafferis, _The Factorization Problem in Jackiw-Teitelboim Gravity_ , _JHEP_ 02 (2020) 177 [1804.01081].
  * (79) D. Stanford, _More quantum noise from wormholes_ , 2008.08570.
  * (80) Z. Yang, _The Quantum Gravity Dynamics of Near Extremal Black Holes_ , _JHEP_ 05 (2019) 205 [1809.08647].
  * (81) T. Faulkner, _The holographic map as a conditional expectation_ , 2008.04810.
  * (82) A. Belin and J. de Boer, _Random statistics of ope coefficients and euclidean wormholes_ , 2006.05499.
  * (83) N. Engelhardt, S. Fischetti and A. Maloney, _Free energy from replica wormholes_ , 2007.07444.
  * (84) C. Murthy and M. Srednicki, _Structure of chaotic eigenstates and their entanglement entropy_ , _Phys. Rev. E_ 100 (2019) 022131 [1906.04295].
  * (85) X. Dong and H. Wang, _Enhanced corrections near holographic entanglement transitions: a chaotic case study_ , _JHEP_ 11 (2020) 007 [2006.10051].
  * (86) D. Marolf, S. Wang and Z. Wang, _Probing phase transitions of holographic entanglement entropy with fixed area states_ , _JHEP_ 12 (2020) 084 [2006.10089].
  * (87) P. Gao, D.L. Jafferis and A.C. Wall, _Traversable wormholes via a double trace deformation_ , _JHEP_ 12 (2017) 151 [1608.05687].
  * (88) P. Calabrese and J. Cardy, _Entanglement entropy and conformal field theory_ , _J. Phys. A_ 42 (2009) 504005 [0905.4013].
  * (89) A. Lewkowycz and J. Maldacena, _Generalized gravitational entropy_ , _JHEP_ 08 (2013) 090 [1304.4926].
  * (90) X. Dong, _The gravity dual of renyi entropy_ , _Nature Commun._ 7 (2016) 12472 [1601.06788].
  * (91) X. Dong and A. Lewkowycz, _Entropy, extremality, euclidean variations, and the equations of motion_ , _JHEP_ 01 (2018) 081 [1705.08453].
  * (92) S.-J. Rey and V. Rosenhaus, _Scanning tunneling macroscopy, black holes, and ads/cft bulk locality_ , _JHEP_ 07 (2014) 050 [1403.3943].
  * (93) J. Maldacena and X.-L. Qi, _Eternal traversable wormhole_ , 1804.00491.
  * (94) P. Calabrese and J. Cardy, _Quantum quenches in 1 + 1 dimensional conformal field theories_ , _J. Stat. Mech._ 1606 (2016) 064003 [1603.02889].
  * (95) J. Dubail and J.-M. Stéphan, _Universal behavior of a bipartite fidelity at quantum criticality_ , _Journal of Statistical Mechanics: Theory and Experiment_ 2011 (2011) L03002 [1010.3716].
  * (96) T.K. DeLillo, A.R. Elcrat and J.A. Pfaltzgraff, _Schwarz–christoffel mapping of the annulus_ , _SIAM review_ 43 (2001) 469.
  * (97) T.A. Driscoll and L.N. Trefethen, _Schwarz-christoffel mapping_ , vol. 8, Cambridge University Press (2002), 10.1017/CBO9780511546808.
  * (98) C. Hu, _Algorithm 785: a software package for computing schwarz-christoffel conformal transformation for doubly connected polygonal regions_ , _ACM Transactions on Mathematical Software (TOMS)_ 24 (1998) 317. Code at http://www.netlib.org/toms-2014-06-10/785 (can also be found at internet archive).
  * (99) https://github.com/ronakmsoni/2d-cft-single-strip-joining-quench.
  * (100) T. Takayanagi, _Holographic dual of bcft_ , _Phys. Rev. Lett._ 107 (2011) 101602 [1105.5165].
  * (101) M. Fujita, T. Takayanagi and E. Tonni, _Aspects of ads/bcft_ , _JHEP_ 11 (2011) 043 [1108.5152].
  * (102) S. Cooper, M. Rozali, B. Swingle, M. Van Raamsdonk, C. Waddell and D. Wakeham, _Black Hole Microstate Cosmology_ , _JHEP_ 07 (2019) 065 [1810.10601].
  * (103) T. Shimaji, T. Takayanagi and Z. Wei, _Holographic Quantum Circuits from Splitting/Joining Local Quenches_ , _JHEP_ 03 (2019) 165 [1812.01176].
  * (104) A. Almheiri, A. Mousatov and M. Shyani, _Escaping the interiors of pure boundary-state black holes_ , 1803.04434.
  * (105) A. Castro, S. Detournay, N. Iqbal and E. Perlmutter, _Holographic entanglement entropy and gravitational anomalies_ , _JHEP_ 07 (2014) 114 [1405.2792].
  * (106) J. Maldacena and A. Milekhin, _Syk wormhole formation in real time_ , 1912.03276.
  * (107) Y.D. Lensky and X.-L. Qi, _Rescuing a black hole in the large- $q$ coupled syk model_, 2012.15798.
  * (108) D. Harlow, _Wormholes, emergent gauge fields, and the weak gravity conjecture_ , _JHEP_ 01 (2016) 122 [1510.07911].
  * (109) A. Almheiri, X. Dong and D. Harlow, _Bulk locality and quantum error correction in ads/cft_ , _JHEP_ 04 (2015) 163 [1411.7041].
  * (110) D. Berenstein and A. Miller, _Superposition induced topology changes in quantum gravity_ , _JHEP_ 11 (2017) 121 [1702.03011].
  * (111) D. Marolf and H. Maxfield, _Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information_ , _JHEP_ 08 (2020) 044 [2002.08950].
  * (112) D. Marolf and H. Maxfield, _Observations of hawking radiation: the page curve and baby universes_ , 2010.06602.
  * (113) J. Maldacena, D. Stanford and Z. Yang, _Diving into traversable wormholes_ , 1704.05333v1.
  * (114) K. Bulycheva, _Semiclassical correlators in Jackiw-Teitelboim gravity_ , _JHEP_ 11 (2019) 023 [1905.05692].


between neurons $j$ and $i$.

Figure 11:  Figure showing the excitatory and inhibitory pre-synaptic neurons
with excitatory and inhibitory spikes respectively incident on the post-
synaptic neuron. We use this model to model the nonlinear interacting Hawkes
process with inhibition.

To simplify the notation, we can rewrite Eq. 50 as

$\lambda_{i}(t)=\Phi_{i}\left(\sum_{k\in
I}\int_{(0,t)}h_{ki}(t-s)dZ_{k}(s)\right).$ (51)

where $h_{ik}(t-s)$ measures the influence of neuron $k$ on neuron $i$ and how
this influence vanishes with the time. More precisely, $h_{ik}(t-s)$ describes
how a spike of neuron $k$ lying back $t-s$ time units in the past influences
the present spiking rate at time $t$.

The goal of using heterogeneity in the STDP dynamics is to get better
orthogonalization among the recurrent network states to lower higher-order
correlations in spike trains. Studies have shown that the correlation of
higher order progressively decreases the information available through neural
population (Montani et al., 2009; Abbott & Dayan, 1999). Since we are trying
to engineer a spike-efficient model, we leverage the heterogeneity in the STDP
dynamics to reduce the higher-order correlations. The hypothesis is that using
heterogeneity in STDP helps us orthogonalize the recurrent layer that can help
us achieve an  efficient representation of the input spike patterns with fewer
spikes. This may be interpreted as the recurrent layer acting as an orthogonal
bases function where inputs are projected onto these bases. Thus, having
orthogonal bases can efficiently map inputs without much loss. While
heterogeneous LIF neurons help us increase the number of principal components,
thereby enabling us to store a greater subclass of features, heterogeneous
STDP helps us efficiently encode this orthogonalization of the recurrent
layer, resulting in fewer spikes compared to a homogeneous RSNN. Thus, in
effect, heterogeneous STDP parameters can learn the output more precisely,
which is projected back into the recurrent network. One of the primary reasons
why heterogeneous STDP helps project the input to orthogonal activations of
the recurrent network can be attributed to the distribution of LTD dynamics,
as this increases the competition and helps distribute the input projection to
multiple principal components. We discuss that the heterogeneous LTP/LTD
dynamics in STDP lead to fewer spikes in the transmission of information.

Lemma 3.2.1:  If the neuronal firing rate of the HRSNN network with only
heterogeneity in LTP/LTD dynamics of STDP is represented as $\Phi_{R}$ and
that of MRSNN represented as $\Phi_{M}$, then the HRSNN model promotes
sparsity in the neural firing which can be represented as
$\displaystyle\Phi_{R}<\Phi_{M}$.

Proof: In this lemma, we show that the average firing rate of the model with
heterogeneous STDP (LTP/LTD) dynamics (averaged over the population of
neurons) is lesser than the corresponding average neuronal activation rate for
a model with homogeneous STDP dynamics. We prove this by taking a sub-network
of the HRSNN model as illustrated by Fig. 11. Now, we model the input spike
trains of the pre-synaptic neurons using a multivariate interactive, nonlinear
Hawkes process with multiplicative inhibition (Duval et al., 2022).

We consider a population of neurons of size $N$ that is divided into
population $A$ (excitatory) with size $N_{A}:=\alpha N$ and a population $B$
(inhibitory) with size $N_{B}=(1-\alpha)N$. A particular instance of the model
is then given in terms of a family of counting processes
$\left(Z_{t}^{1},\ldots,Z_{t}^{N_{A}}\right.$ ) (population $\left.A\right)$
and $\left(Z_{t}^{N_{A}+1},\ldots,Z_{t}^{N}\right)$ (population $B$ ) with
coupled conditional stochastic intensities given respectively by $\lambda^{A}$
and $\lambda^{B}$. Consider on a filtered probability space
$\left(\Omega,\mathcal{F},\left(\mathcal{F}_{t}\right)_{t\geq
0},\mathbf{P}\right)$ an independent family of i.i.d. Poisson measures
$\left(\pi_{i}(\mathrm{~{}d}s,\mathrm{~{}d}z),i\in\\{1,\ldots,N\\}\right)$
with intensity measure $\mathrm{d}s\times\mathrm{d}z$ on
$[0,\infty)\times[0,\infty)$. Let $(x,y)\mapsto F(x,y)$ and $(x,y)\mapsto
G(x,y)$ two nonnegative functions defined on $(0,\infty)^{2}$. We assume that
$F$ and $G$ satisfy

$F(x,y)=\Phi_{A}(x)\Phi_{B\rightarrow
A}(y),G(x,y)=\Phi_{B}(x)+\Phi_{A\rightarrow B}(y),$

where $\Phi_{A},\Phi_{B\rightarrow A},\Phi_{B}$ and $\Phi_{A\rightarrow B}$
are nonnegative functions, each of them globally Lipschitz with
$\Phi_{B\rightarrow A}$ bounded (and with no loss of generality we assume
$0\leq\Phi_{B\rightarrow A}\leq 1$ ).

Let us consider the family of càdlàg $\left(\mathcal{F}_{t}\right)_{t\geq 0}$
point processes $\left(Z_{t}^{i}\right)_{t\geq 0,i=1,\ldots,N}$ given by

$Z_{t}^{i}=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant\lambda_{s}^{i}}\pi_{i}(\mathrm{~{}d}s,\mathrm{~{}d}z),i=1,\ldots,N,$

where the intensity $\lambda^{i},i=1,\ldots,N$, is given as:

$\displaystyle\lambda_{t}^{A,N}:=\Phi_{A}\left(\frac{1}{N}\sum_{j\in
A}\int_{0}^{t^{-}}h_{1}(t-u)\mathrm{d}Z_{u}^{j}\right)\Phi_{B\rightarrow
A}\left(\frac{1}{N}\sum_{j\in
B}\int_{0}^{t^{-}}h_{2}(t-u)\mathrm{d}Z_{u}^{j}\right)$ (52)
$\displaystyle\lambda_{t}^{B,N}:=\Phi_{B}\left(\frac{1}{N}\sum_{j\in
B}\int_{0}^{t^{-}}h_{3}(t-u)\mathrm{d}Z_{u}^{j}\right)+\Phi_{A\rightarrow
B}\left(\frac{1}{N}\sum_{j\in
A}\int_{0}^{t^{-}}h_{4}(t-u)\mathrm{d}Z_{u}^{j}\right)$ (53)

, where $A\&B$ are the populations of the excitatory and inhibitory neurons,
respectively.

The dynamics given by Eq. 53 is of Hawkes type: each particle’s intensity
depends on the whole system’s history, through memory kernels
$h_{i},i=1,\ldots,4$ and firing rate functions $\Phi_{A}$ and $\Phi_{B}$. The
multiplicative influence of inhibitory population $B$ onto population $A$, is
represented using the inhibition kernel $\Phi_{B\rightarrow A}$ which is a
decreasing nonnegative function on $[0,+\infty)$, with $\Phi_{B\rightarrow
A}(0)=1$ and $\Phi_{B\rightarrow
A}(x)\underset{x\rightarrow\infty}{\longrightarrow}0$\- i.e., activity of
population $A$ should decrease as activity of population $B$ rises. The model
secondly incorporates retroaction from population $A$ onto population $B$,
which is supposed to be mostly additive, although possibly modulated by a
nonlinear feedback kernel $\Phi_{A\rightarrow B}$.

Now, without loss of generality we assume that $\Phi_{A}$ and $\Phi_{B}$ are
linear - i.e., $\displaystyle\Phi_{A}(x)=\mu_{A}+x,\Phi_{B}(x)=\mu_{B}+x,x\geq
0,$ where $\mu_{A},\mu_{B}\geq 0$, and $h_{i}\geq 0$ for $i=1,\ldots,4$.

Hence, Eq. 53 becomes

$\left\\{\begin{array}[]{l}\lambda_{t}^{A}=\left(\mu_{A}+\alpha\int_{0}^{t}h_{1}(t-u)\lambda_{u}^{A}\mathrm{~{}d}u\right)\Phi_{B\rightarrow
A}\left((1-\alpha)\int_{0}^{t}h_{2}(t-u)\lambda_{u}^{B}\mathrm{~{}d}u\right),\\\
\lambda_{t}^{B}=\mu_{B}+(1-\alpha)\int_{0}^{t}h_{3}(t-u)\lambda_{u}^{B}\mathrm{~{}d}u+\Phi_{A\rightarrow
B}\left(\alpha\int_{0}^{t}h_{4}(t-u)\lambda_{u}^{A}\mathrm{~{}d}u\right).\end{array}\right.$
(54)

For heterogeneous neuron populations, there exists an asymmetry of the
weights. Based on balanced spiking neural networks with heterogeneous
connection strengths, previous works have revealed that such heterogeneous
networks possess heavy-tailed Lévy fluctuations (Shlesinger et al., 1987;
Mantegna & Stanley, 1995; Cossell et al., 2015). The heterogeneous heavy-
tailed distributions of synaptic weights have been fitted to lognormal
distributions (Buzsáki & Mizuseki, 2014; Kuśmierz et al., 2020). We model the
inputs to neuron $i\in E$ as:

$\displaystyle l_{i}(t)$ $\displaystyle=W_{EX}\tau_{mE}\sum_{j\in
X}c_{ij}s_{j}(t)+W_{EE}\tau_{mE}\sum_{j\in
E}c_{ij}s_{j}(t)-W_{EI}\tau_{mE}\sum_{j\in I}c_{ij}s_{j}(t)$ (55)
$\displaystyle=\mu_{1}E+\Delta\mu_{i}+\eta_{i}(t)$ (56)

where $\mu$ denotes the mean inputs such that
$\displaystyle\mu_{E}=K\tau_{mE}\left(W_{EX}r_{X}+W_{EE}r_{E}-W_{EI}r_{l}\right)$;
$\Delta\mu_{i}=$ ’quenched’ fluctuations (from neuron to neuron) with variance
$\displaystyle\left\langle\Delta\mu^{2}\right\rangle_{E}=K\tau_{mE}^{2}\left(W_{EX}^{2}\left(r_{X}^{2}+\Delta
r_{X}^{2}\right)+W_{EE}^{2}\left(r_{E}^{2}+\Delta
r_{E}^{2}\right)+W_{EI}^{2}\left(r_{I}^{2}+\Delta r_{I}^{2}\right)\right)$ due
to random connectivity. Finally, $\eta_{i}$ denotes temporal fluctuations due
to spiking activity. We assume that the pre-synaptic neurons fire as using the
interactive Hawkes process described above.

Consider a case where $\mu_{A}\gg 1,\mu_{B}=0$ and $h=1_{[0,\theta]}$

$\left\\{\begin{array}[]{l}\lambda_{t}^{A}:=\left(\mu_{A}+\alpha\int_{0}^{t}h_{1}(t-u)\mathrm{d}\lambda_{u}^{A}\right)\Phi_{B\rightarrow
A}\left((1-\alpha)\int_{0}^{t}h_{2}(t-u)\mathrm{d}\lambda_{u}^{B}\right),\\\
\lambda_{t}^{B}:=(1-\alpha)\int_{0}^{t}h_{3}(t-u)\mathrm{d}\lambda_{u}^{B}+\alpha\int_{0}^{t}h_{4}(t-u)\mathrm{d}\lambda_{u}^{A}.\end{array}\right.$
(57)

In a normal case, the excitatory and inhibitory populations follow the
following steps: (1) $t\approx 0,\lambda_{t}^{A}\approx\mu_{A}$ is high and
$\lambda_{t}^{B}\approx 0$ is small (2) Feedback from $A$ to
$B:\lambda_{t}^{B}$ increases (3) Inhibition of $B$ to $A$ : when
$\lambda_{t}^{B}$ gets high, $\Phi_{B\rightarrow A}$ reduces $\lambda_{t}^{A}$
(4) $h_{4}$ has compact support: after a time $\theta_{4},B$ no longer feels
the influence of $A$ : intensity of $B$ is back to $\mu_{B}\approx 0$ and $A$
to its normal high activity $\mu_{A}$ (State 1)

This leads to oscillations which lead to spikes. However, heterogeneity in the
synaptic dynamics increases the stochasticity of the pre-synaptic spike
arrival. Thus, due to the heterogeneity, $\Phi_{B\rightarrow A}$ promotes the
system in the inhibition state (state 3) and inhibits the system’s movement to
system 4 and system 1, thereby creating a spike. Hence,
$\Phi_{R}^{A}<\Phi_{M}^{A}$. Similarly, for the inhibitory neurons, we can
show that $\Phi_{R}^{B}<\Phi_{M}^{B}$. Thus, we get $\Phi_{R}<\Phi_{M}$
$\blacksquare$

This lemma might be interpreted as the heterogeneous STDP dynamics increasing
the synaptic noise, which reduces the number of spikes of the post-synaptic
neuron. A heterogeneous STDP leads to a non-uniform scaling of correlated
spike trains leading to de-correlation. Hence, we can say that heterogeneous
STDP models have learned a better-orthogonalized subspace representation,
leading to a better encoding of the input space with fewer spikes.

It is to be mentioned here that the synaptic noise might be thought of as
analogous to the stochasticity in the gradient descent algorithm. As recently
proved by Simsekli et al. (Simsekli et al., 2020; 2019), stochasticity plays
an important role in the generalization ability of the model. We might
interpret the synaptic noise in the heterogeneous STDP to play a similar role
and helps in better generalizability of the HRSNN model. This hypothesis is
empirically proven in Supplementary Section A. However, a detailed theoretical
analysis would be a very interesting direction for future work.

Theorem 2: For a given number of neurons $N_{\mathcal{R}}$, the spike
efficiency of the model $\mathcal{E}=\frac{\mathcal{C}(N_{\mathcal{R}})}{S}$
for HRSNN ($\mathcal{E}_{R}$) is greater than MRSNN ($\mathcal{E}_{M}$) i.e.,
$\mathcal{E}_{R}\geq\mathcal{E}_{M}$

Proof:  To study the effect of the spike time when the weight $w_{k}$ changes,
we look into the expected value of the time difference in the post-synaptic
spikes, which is given as:

$\mathbb{E}\left[\Delta t_{\text{post }}\right]=\mathbb{E}\left[t_{\text{post
}}-t_{\text{post }}\right]=\left(\mathbb{E}\left[t_{\text{post
}}\right]-t_{\text{post }}\right)\operatorname{Pr}[s]$ (58)

where $\operatorname{Pr}[\exists s]$ is the probability of occurrence of the
post-synaptic spike. Thus, the expected input to the neuron at time
$t(\mathbb{E}\left[i(t)\right])$, which comprises of its excitatory and
inhibitory components
$\mathbb{E}\left[i_{e}(t)\right],\mathbb{E}\left[i_{i}(t)\right]$ can be
expressed as:

$\displaystyle\mathbb{E}\left[\Delta
i(t)\right]=\Delta\mathbb{E}\left[i_{e}(t)\right]-\Delta\mathbb{E}\left[i_{i}(t)\right]\quad\text{
for }t<t_{\text{post }}$ (59) where
$\displaystyle\mathbb{E}\left[i_{e}(t)\right]=\rho_{e}\int_{0}^{\infty}\mu_{w_{e}}(w,t)dw\quad;\quad\quad\mathbb{E}\left[i_{i}(t)\right]=\rho_{i}\int_{0}^{\infty}\mu_{w_{i}}(w,t)dw$
(60)

where $\rho_{e},\rho_{i}$ are the rates of incoming spikes and
$\mu_{w_{e}}(w,t),\mu_{w_{i}}(w,t)$ the probabilities of the weights
associated to time $t$. Now, considering the case for RSNNs with homogeneous
STDP ($M$) and with heterogeneous STDP ($R$), the difference in the variances
of the two populations is given as:

$\Delta\text{Var}[V_{M}]-\Delta\text{Var}[V_{R}]=\Delta\int_{-\infty}^{t}[\mathbb{E}\left[i_{M}^{2}(t)\right]-\left(\mathbb{E}\left[i_{R}^{2}(t)\right]-\mathbb{E}[i_{R}(t)]^{2}\right)]dt$
(61)

Since $t<t_{\text{post }}$, STDP potentiates both inhibitory and excitatory
synapses, so
$\Delta\mathbb{E}\left[i_{i}^{2}(t)\right]>0,\Delta\mathbb{E}\left[i_{e}^{2}(t)\right]>0$.
The term $\mathbb{E}[i_{M}(t)]^{2}=0$ by the symmetry of the weights, and it
is maintained at zero by the symmetry of the STDP. But for heterogeneous
neuron populations, as described above, there exists an asymmetry of the
weights. Based on balanced spiking neural networks with heterogeneous
connection strengths, previous works have revealed that such heterogeneous
networks possess heavy-tailed, Lévy fluctuations (Shlesinger et al., 1987;
Mantegna & Stanley, 1995; Cossell et al., 2015). This implies
$\mathbb{E}[i_{R}(t)]^{2}>0\Rightarrow\Delta\text{Var}[V_{R}]<\Delta\text{Var}[v(t)_{M}]$
We calculate the number of post-synaptic spikes triggered when the stimulus is
present. Now, representing the spike rate of the HRSNN and the MRSNN as
$\Phi_{R},\Phi_{M}$ resp.,

$\displaystyle\int_{0}^{t}\Phi_{R}(t)dt\leq\int_{0}^{t}\Phi_{M}(t)\Rightarrow
S_{R}=N_{\mathcal{R}}\frac{T}{\hat{t}_{R}^{ISI}}\leq
N_{\mathcal{R}}\frac{T}{\hat{t}_{M}^{ISI}}=S_{M}$ (62)

Thus, spikes decrease when we use heterogeneity in the LTP/LTD Dynamics.
Hence, we compare the efficiencies of the HRSNN with that of MRSNN as follows:

$\displaystyle\frac{\mathcal{E}_{R}}{\mathcal{E}_{M}}=\frac{M_{R}(N_{\mathcal{R}})\times
S_{M}}{S_{R}\times
M_{M}(N_{\mathcal{R}})}=\frac{\sum_{\tau=1}^{N_{\mathcal{R}}}\frac{\operatorname{Cov}^{2}\left(x(t-\tau),\mathbf{a}^{R}_{\tau}\mathbf{r}_{R}(t)\right)}{\operatorname{Var}\left(\mathbf{a}^{R}_{\tau}\mathbf{r}_{R}(t)\right)}\times\int\limits_{t_{ref}}^{\infty}t\Phi_{R}dt}{\sum_{\tau=1}^{N_{\mathcal{R}}}\frac{\operatorname{Cov}^{2}\left(x(t-\tau),\mathbf{a}^{M}_{\tau}\mathbf{r}_{M}(t)\right)}{\operatorname{Var}\left(\mathbf{a}^{M}_{\tau}\mathbf{r}_{M}(t)\right)}\times\int\limits_{t_{ref}}^{\infty}t\Phi_{M}dt}$
(63)

Since $S_{R}\leq S_{M}$ and also,the covariance increases when the neurons
become correlated, and as neuronal correlation decreases, $\mathcal{H}$
increases (Theorem 1), we see that
$\frac{\mathcal{E}_{R}}{\mathcal{E}_{M}}\geq
1\Rightarrow\mathcal{E}_{R}\geq\mathcal{E}_{M}\quad\blacksquare$

###  Appendix C Supplementary Section C

####  C.1 Higher Order Correlation

In this paper, we took inspiration from results in reservoir computing, which
show that we can maximize memory capacity using orthogonalization among
reservoir states in the case of reservoir computers (Farkaš & Gergel’, 2017;
Farkaš et al., 2016). The goal of using heterogeneous STDP dynamics is to get
better orthogonalized recurrent network states to achieve more efficient
information transfer with lower higher-order correlations in spike trains.
Recent studies (Montani et al., 2009; Abbott & Dayan, 1999) have shown that
the correlation of higher order progressively decreases the information
available through the neural population. The decrease in information becomes
larger as the interaction order grows. Since we are trying to engineer a
spike-efficient model, we leverage the heterogeneity in neuronal parameters to
reduce the higher-order correlations. The hypothesis is that an orthogonal
recurrent layer can help us efficiently represent the input spike patterns
with fewer spikes. This may be interpreted as the recurrent layer acting as an
orthogonal bases function where the inputs are projected onto these bases.
Thus, having orthogonal bases can efficiently map the inputs without much
loss. The heterogeneous STDP helps us efficiently achieve this
orthogonalization of the recurrent layer, resulting in a lesser voltage
variance across the neuron population. This leads to fewer spikes (since the
mean is constant) compared to a homogeneous RSNN. Thus, in effect,
heterogeneous STDP parameters can learn the output more precisely, which is
projected back into the recurrent network. Hence, using heterogeneous STDP
parameters leads to a better orthogonalization among the neuronal states and
hence, a higher $\mathcal{C}$.

In this paper, we show that using a distribution of LTP/LTD dynamics in the
STDP parameters helps us in mappings the input onto the orthogonal activations
of the recurrent network to capture the principal components of the input
signal. The LTD dynamics play an important role in determining the
orthogonality of neuronal activations. LTD windows of the STDP rules enable
robust sequence learning amid background noise in cooperation with a large
signal transmission delay between neurons and a theta rhythm (Hayashi &
Igarashi, 2009). The LTD window in the range of positive spike-timing plays an
important role in preventing noise influences with sequence learning. Oja
(Oja, 1982; 1989) showed that the LIF neuron’s time constant is very fast
compared to the time constant of learning in which the weights $w_{ji}$
change. The learning is assumed to take place according to the STDP type
conjunction of the inputs $\xi_{i}$ and the integrated effect of the inputs,
$\nu_{j}$, with an additional forgetting term attributed to the LTD dynamics:
$\frac{dw_{ji}}{dt}=\alpha\nu_{j}\xi_{i}-f\left(\nu_{j},\xi_{i},w_{ji}\right)$
In the case of homogeneous STDP, $f(.)$ is a constant; hence, the model can
only efficiently learn the first principal component of the input. However,
quite interesting functions emerge when considering STDP to have a
distribution. This also helps us determine the next principal components other
than the first one. Hence the diversity in the different LTD dynamics
increases the competition and helps that not all inputs are mapped to the
first principle component. Thus, the diversity in the LTD dynamics helps in
projecting the input to orthogonal activations of the recurrent network.

Table 11: Table Showing the estimated highest order of correlation for HRSNN
vs. MRSNN using CuBIC

| $\hat{\xi}$ | p-value |  | Estimated Highest-
---
order of correlation
($\hat{\xi}+1$)
HRSNN | 2 | 0.423 | 3
MRSNN | 5 | 0.358 | 6

Now, for homogeneous RSNNs, several higher-order correlations, which according
to our hypothesis, arise because of the poor orthogonalization among the
network states. This results in the redundancies of spikes for encoding the
same information. In this paper, we use heterogeneous STDP dynamics to learn
an efficient orthogonal representation of the state space, which result in the
network learning the same patterns but using fewer spikes. (theorem: 3) We
also show that heterogeneity in the neuronal parameters decreases the neuronal
correlation (theorem 1 And fig 2a). Thus, since heterogeneity results in
better orthogonalization among the neuronal states, it results in fewer
higher-order correlations. Moreover, recent studies have shown that the
correlation of higher order progressively decreases the information available
through the neural population, and the decrease in information becomes larger
as the interaction order grows. Since we are trying to engineer an efficient
model, we aim to reduce the higher-order correlations using heterogeneity in
neuronal parameters (as shown in Theorem 1). In addition to this, to verify
this, we used CuBIC (Staude et al., 2010), a cumulant-based inference of
higher-order correlations in massively parallel spike trains. The details of
the experimental methodology are given in Supplementary Section C. The outcome
of CuBIC is a lower bound $\hat{\xi}$ on the order of correlation in the
spiking activity of large groups of simultaneously recorded neurons. CuBIC can
provide statistical evidence for large correlated groups without the
discouraging requirements on a sample size that direct tests for higher-order
correlations have to meet. This is achieved by exploiting constraining
relations among correlations of different orders. However, it must be noted
that CuBIC is not designed to estimate the order of correlation directly; the
inferred lower bound might not always correspond to the maximal order of
correlation present in a given data set.


#  Lower bound for large local transversal fluctuations of Geodesics in Last
Passage Percolation

Pranay Agarwal  Pranay Agarwal, Department of Mathematics, Indian Institue of
Technology, Kanpur, India<EMAIL_ADDRESS>
###### Abstract.

For exactly solvable models of planar last passage percolation, it is known
that geodesics of length $n$ exhibit transversal fluctuations at scale
$n^{2/3}$ and matching (up to exponents) upper and lower bounds for the tail
probabilities are available. The local transversal fluctuations near the
endpoints are expected to be much smaller; it is known that the transversal
fluctuation up to distance $r\ll n$ is typically of the order $r^{2/3}$ and
the probability that the fluctuation is larger than $tr^{2/3}$ is at most
$Ce^{-c{t}^{3}}$. In this note, we provide a short argument establishing a
matching lower bound for this probability.

##  1\. Introduction

We consider the Exponential Last Passage Percolation model on the integer
lattice ${\mathbb{Z}}^{2}$, where one assigns independently sampled values
from the exponential distribution to each vertex of the lattice. We study up-
right paths from the origin to the point $(n,n)$ in this model, the paths with
the maximal weight sum are called the geodesic $\Gamma_{n}$, which are almost
surely unique in this model. It was proved in [Joh00b] that $\Gamma_{n}$ shows
transversal fluctuations of the order $n^{2/3+o(1)}$. Later the result was
improved in [BSS14, Theorem 11.1] and subsequently in [BGZ21, Proposition
C.9], which showed that the geodesic $\Gamma_{n}$ exhibits transversal
fluctuations at scale $n^{2/3}$, and the probability that these fluctuations
are greater than $tn^{2/3}$ is at most $Ce^{-c{t}^{3}}$. One can establish a
matching lower bound from the arguments of [HS18, Proposition 1.4], even
though their arguments focused on the Poissonian LPP model. As one would
expect, the transversal fluctuations are much smaller near the endpoints of
the geodesic. At a distance $r$ from the endpoints, where $r\ll n$, the
fluctuations are typically of the order $r^{2/3}$.

Now we define transversal fluctuations formally as

$\mathrm{TF}^{n}(r):=\max\left\\{{\left|{x-y}\right|:(x,y)\in\Gamma_{n},0\leq
x+y\leq 2r}\right\\}.$

Thus, $\mathrm{TF}^{n}(n)$ is the global transversal fluctuation of
$\Gamma_{n}$ and $\mathrm{TF}^{n}(r)$ are the local transversal fluctuations.
It was shown in [BBB23, Proposition 2.1] that for large $t$, for some
constants $c,C,$

$\underset{}{{\mathbb{P}}}\left({{\mathrm{TF}^{n}(r)>tr^{2/3}}}\right)\leq
C\exp\left({-ct^{3}}\right).$

We will give a more precise statement later as Theorem 2.2. Our main result
gives a matching lower bound for this probability.

######  Theorem 1.1.

There exist positive constants $c_{1},C_{1},r_{0},t_{0}$ such that for all
$\epsilon>0$, $r>r_{0}$, $n\geq r$ and $(1-\epsilon)r^{1/3}>t>t_{0}$

$\underset{}{{\mathbb{P}}}\left({{\mathrm{TF}^{n}(r)>tr^{2/3}}}\right)\geq
C_{1}\exp\left({-c_{1}t^{3}}\right).$

This completes the picture for local transversal fluctuations of finite
geodesics. The main tools we use to prove the bound are moderate deviation
estimates from integrable probability along with the ordering of geodesics. We
shall simplify the arguments presented in [HS18, Proposition 1.4] for
Exponential LPP and use them to prove Theorem 3.1. The resulting construction
that we get from Theorem 3.1 will then be used to prove Theorem 1.1.

Some related expressions have appeared in the literature before as well. A
similar lower bound for the transversal fluctuation of semi-infinite geodesics
was shown in [SS20, Theorem 2.8] by utilizing stationary exponential LPP. A
formula to calculate the probability of the geodesic to pass through a given
point was derived in [Liu22, Theorem 1.1]. However, as remarked by the author,
the complicated expression of their formula makes it seem intangible to use
their result to derive any useful asymptotics.

##  2\. Notation and Preliminaries

For a point $u\in{\mathbb{Z}}^{2}$, we use $X_{u}$ to denote the exponential
random variable associated with $u$, also known as the passage time at $u$. We
use $\mathbf{o}$ to denote the origin and $\mathbf{n}$ to denote the point
$(n,n)$ for $n\in{\mathbb{N}}$. We represent the line $x+y=T$ by
${\mathcal{L}}_{T}$. Let $\gamma$ be an up-right path from $u$ to $v$, we
define passage time of the path $\gamma$ as

$\ell(\gamma)=\sum_{w\in\gamma\setminus\left\\{{v}\right\\}}X_{w}.$

We will use $T(u,v)$ to denote the maximal passage time over all up-right
paths from $u$ to $v$, also known as the last passage time form $u$ to $v$.
$\Gamma_{u}$ denotes the path achieving maximal length from $\mathbf{o}$ to
$u$. As a special case, we will use $\Gamma_{n}$ to denote the geodesic from
$\mathbf{o}$ to $\mathbf{n}$. We define
$f:{\mathbb{R}}^{2}\times{\mathbb{R}}^{2}\to{\mathbb{R}}$ as

$f(u,v)=\begin{cases}\left({\sqrt{v_{1}-u_{1}}+\sqrt{v_{2}-u_{2}}}\right)^{2}&\text{if
}u\preceq v,\\\ 0,&\text{otherwise},\end{cases}$

where $u\preceq v$ means that $u$ is coordinate wise smaller than $v$.
Similarly, we define the functions,

$\displaystyle\phi(u)=u_{2}+u_{1}$ $\displaystyle\mathrm{and,}$
$\displaystyle\psi(u)=u_{2}-u_{1}.$

We use $C,c$ to denote arbitrary constants which may change from equation to
equation. We have avoided using floor and ceiling functions in many places to
make notations less cumbersome. Adding the floor/ceiling functions will not
change our arguments in a non-trivial way.

###  2.1. Estimates for temporal fluctuations

Johansson proved [Joh00a, Remark 1.5] that the last passage time is equal in
distribution to the top eigenvalue of the Laguerre Unitary Ensemble, and upper
bounds on the upper and lower tails on this eigenvalue were proved in [LR10,
Theorem 2]. Their arguments gave us matching bounds for the upper tail and an
upper bound for the lower tail. A matching lower bound for the lower tail was
then proved in [BGHK21, Theorem 2]. We summarize the results in the following
theorem.

######  Theorem 2.1.

Let $n\in{\mathbb{N}},h\in(0,1]$. For $u=(u_{1},u_{2})\in{\mathbb{Z}}^{2}$
such that $\phi(u)=n$ and $h\leq\frac{u_{1}}{u_{2}}\leq 1/h$. Then there exist
positive constants
$c_{1},C_{1},c_{2},C_{2},c_{3},C_{3},c_{4},C_{4},n_{0},t_{0}$ depending on $h$
such that for $n>n_{0}$ and $t_{0}<t<n^{2/3}$,

(2.1)
$C_{1}\exp\left({-c_{1}t^{3/2}}\right)\leq\underset{}{{\mathbb{P}}}\left({{T(\mathbf{o},u)-f(\mathbf{o},u)>tn^{1/3}}}\right)\leq
C_{2}\exp\left({-c_{2}t^{3/2}}\right),$ (2.2)
$C_{3}\exp\left({-c_{3}t^{3}}\right)\leq\underset{}{{\mathbb{P}}}\left({{T(\mathbf{o},u)-f(\mathbf{o},u)\leq-
tn^{1/3}}}\right)\leq C_{4}\exp\left({-c_{4}t^{3}}\right).$

###  2.2. Estimates for transversal fluctuations

Exponential decay for the upper tail of global transversal fluctuation was
shown in [BSS14, Theorem 11.1]. Later, the optimal bound was found in [BGZ21,
Proposition C.9]. Upper bound for large local transversal fluctuations for
$r\ll n$ was shown in [BSS19, Theorem 3] and was improved in [BBB23,
Proposition 2.1] . We summarize their results as follows.

######  Theorem 2.2.

There exist positive constants $c_{1},C_{1},r_{0},n_{0},t_{0}$ such that for
$r>r_{0},n>r\vee n_{0}$ and $t>t_{0}$,

$\underset{}{{\mathbb{P}}}\left({{\mathrm{TF}^{n}(r)>tr^{2/3}}}\right)\leq
C_{1}\exp\left({-c_{1}t^{3}}\right).$

##  3\. Proof of Main Result

To motivate our arguments, we provide a proof for the global transversal
fluctuation of the geodesic $\Gamma_{r}$. The arguments essentially follows
from [HS18, Proposition 1.4], see Remark 2.

######  Theorem 3.1.

There exist constants $r_{0},t_{0},c_{1},C_{1}$ such that for all
$\epsilon>0$, $r>r_{0}$ and $(1-\epsilon)r^{1/3}>t>t_{0}$,

$\underset{}{{\mathbb{P}}}\left({{\mathrm{TF}^{r}(r)\geq tr^{2/3}}}\right)\geq
C_{1}e^{-c_{1}t^{3}}.$

Consider a rectangle in our integer lattice formed by the lines $x-y=0$,
$x+y=0$, $y=x+r^{2/3}$ and $x+y=r/3$. Let the midpoints of the smaller sides
be denoted by $u,v$. Let $A$ be the event that a path $\gamma$ from $u$ to $v$
lies inside this rectangle and has a passage time greater than
$4r+10t^{2}r^{1/3}$.

######  Lemma 3.2.

There exist constants $c,C,r_{0},t_{0}$ such that for $r>r_{0}$ and $t>t_{0}$,

$\underset{}{{\mathbb{P}}}\left({{A}}\right)\geq Ce^{-ct^{3}}.$

######  Remark 1.

Our proof is a combination of arguments that are known in the literature. They
can be found in [BSS14, Lemma 12.1] and [GH23, Theorem 2]. For completeness,
we provide a short self-contained proof.

###### Proof.

Figure 1. The proof of Lemma 3.2 is illustrated here; the blue paths indicate
a path with large passage time between the midpoints that lie inside the
rectangle. Connecting these blue paths gives us a long path between the points
$x_{0}$ and $x_{N}$.

Let $\delta>0$, we shall later choose $\delta$ depending on $t$. We partition
the given rectangle into $N:=\delta^{-1}$ pieces using equidistant lines
parallel to the smaller side; without loss of generality we assume that
$\delta^{-1}$ is a natural number. We denote the midpoint of the $i^{th}$ line
by $x_{i}$ and define $x_{0}=u$ and $x_{N}=v$. Refer to Figure 1 for an
illustration of the setup.

Let $A_{i}$ be the event that there is a path lying inside the rectangle from
$x_{i-1}$ to $x_{i}$ which has a passage time greater than $4\delta
r+10t\left({\delta r}\right)^{1/3}$. By Theorem 2.2 and Theorem 2.1, we have
constants $c_{1},c_{2},C_{1},C_{2}$ such that

$\displaystyle\underset{}{{\mathbb{P}}}\left({{A_{i}}}\right)$
$\displaystyle\geq C_{1}e^{-c_{1}t^{3/2}}-C_{2}e^{-c_{2}(\delta^{-2/3})^{3}}$
$\displaystyle\geq C_{1}e^{-c_{1}t^{3/2}}-C_{2}e^{-c_{2}(\delta^{-1})}.$

For the right-hand side to be positive, we require $\delta^{-1}\geq
c_{0}t^{3/2}$ for a suitable choice of $c_{0}$. Thus we choose
$\delta^{-1}=c_{0}t^{3/2}$ and assume $t$ to be large enough. This allows us
to write

$\underset{}{{\mathbb{P}}}\left({{A_{i}}}\right)\geq Ce^{-ct^{3/2}}\geq
e^{-ct^{3/2}}.$

Now, since all the events $A_{i}$ are independent of each other, we have that

$\underset{}{{\mathbb{P}}}\left({{\bigcap_{1\leq i\leq
N}A_{i}}}\right)\geq\left({e^{-ct^{3/2}}}\right)^{\delta^{-1}}\geq
e^{-ct^{3}}.$

But if the event $A_{i}$ holds for all $i\in\left\\{{1,2,3\dots N}\right\\}$,
then the concatenation of these paths will have passage time greater than
$4r+10t\delta^{-2/3}r^{1/3}=4r+10ct^{2}r^{1/3}$. Thus, we have proved our
claim that,

$\underset{}{{\mathbb{P}}}\left({{A}}\right)\geq Ce^{-ct^{3}}.$

∎

###### Proof of Theorem 3.1.

Figure 2. Illustration for the proof of Theorem 3.1. The blue path corresponds
to event $A$, and the green paths connect the good blue path to the origin and
the point $\mathbf{r}$. Events $B$ and $C$ ensure that the green paths are not
too bad. On $D$, any path contained in $S$ will not have a very large passage
time. Thus on $A\cap B\cap C\cap D$, the geodesic from $\mathbf{o}$ to
$\mathbf{r}$ must exit the region $S$.

Consider a rectangle $R$ obtained by translating the rectangle described in
Lemma 3.2. We specify the midpoints of the shorter sides to be

$u:=\left({\frac{r}{3}-(t+1)r^{2/3},\frac{r}{3}+(t+1)r^{2/3}}\right)\text{ and
}v:=\left({\frac{2r}{3}-(t+1)r^{2/3},\frac{2r}{3}+(t+1)r^{2/3}}\right).$

We define the set

$S=\left\\{{v:\psi\left({v}\right)\leq
tr^{2/3},v\in{\mathbb{Z}}^{2}}\right\\}.$

By our construction, the rectangle $R$ and set $S$ do not intersect. We define
the event $A$ as we did for Lemma 3.2. Furthermore, we define the events

$\displaystyle B$ $\displaystyle:=\left\\{{T(\mathbf{o},u)\geq
f(\mathbf{o},u)-tr^{1/3}}\right\\},$ $\displaystyle C$
$\displaystyle:=\left\\{{T(v,\mathbf{r})\geq
f(v,\mathbf{r})-tr^{1/3}}\right\\}.$

Let $D$ be the event that all paths from $\mathbf{o}$ to $\mathbf{r}$ that
stay inside the region $S$ have passage time less than $4r+t^{2}r^{1/3}$.
Clearly,

$\underset{}{{\mathbb{P}}}\left({{D}}\right)\geq\underset{}{{\mathbb{P}}}\left({{T(\mathbf{o},\mathbf{r})\leq
4r+t^{2}r^{1/3}}}\right).$

By Theorem 2.1, following (2.2) we have that for all $\epsilon>0$ and
$t\leq(1-\epsilon)r^{1/3}$

(3.1)
$\displaystyle\underset{}{{\mathbb{P}}}\left({{B}}\right),\underset{}{{\mathbb{P}}}\left({{C}}\right)\geq
1-Ce^{-ct^{3}}.$

Similarly by (2.1), we have that

(3.2) $\displaystyle\underset{}{{\mathbb{P}}}\left({{D}}\right)\geq
1-Ce^{-c{t}^{3}}.$

On the event $A\cap B\cap C,$ the geodesic from $\mathbf{o}$ to $u$, the path
in $R$ from Lemma 3.2 and the geodesic from $v$ to $\mathbf{r}$ form a path
with passage time

$\displaystyle\geq
4r/3+10t^{2}r^{1/3}+f(\mathbf{o},u)+f(v,\mathbf{r})-2tr^{1/3}$
$\displaystyle\geq
4r/3+10t^{2}r^{1/3}+8r/3-6(t+1)^{2}r^{1/3}-2tr^{1/3}-t^{2}r^{1/3}$
$\displaystyle\geq 4r+2t^{2}r^{1/3}$

for $t$ large enough, by using Taylor expansion of $f$ and dominating the
error terms by $t^{2}r^{1/3}$.

Thus, on the event $A\cap B\cap C\cap D$, we have a path from $\mathbf{o}$ to
$\mathbf{r}$ that is not contained in $S$, which has a higher passage time
than any path in $S$. This prevents the geodesic $\Gamma_{r}$ from being
contained in the region $S$. Thus, there exists a point $w\in\Gamma_{r}$ such
that $\psi\left({w}\right)>tr^{2/3}$ causing $\mathrm{TF}^{r}(r)\geq
tr^{2/3}$. Furthermore, the event $A$ is independent of the event $B\cap C\cap
D$. Thus by Lemma 3.2, (3.1) and (3.2), we have that

$\underset{}{{\mathbb{P}}}\left({{\mathrm{TF}^{r}(r)\geq
tr^{2/3}}}\right)\geq\underset{}{{\mathbb{P}}}\left({{A\cap B\cap C\cap
D}}\right)\geq Ce^{-ct^{3}}\left({1-3Ce^{-ct^{3}}}\right).$

Now, by taking $t$ to be large enough, we have that

$\underset{}{{\mathbb{P}}}\left({{\mathrm{TF}^{r}(r)\geq tr^{2/3}}}\right)\geq
Ce^{-ct^{3}}.$

∎

######  Remark 2.

The proof of Theorem 3.1 uses a simplified version of the arguments made in
[HS18, Proposition 1.4]. Instead of dealing with both cases of High and Low as
in [HS18, Proposition 1.4], we showed that case High occurs by Lemma 3.2 and
modified the arguments accordingly for exponential LPP.

We are now in a position to present the proof of Theorem 1.1. We shall use the
arguments we made in Theorem 3.1 along with the ordering of geodesics to prove
our result.

###### Proof of Theorem 1.1.

(a) $\Gamma_{r}$ forcing $\Gamma_{n}$ to have large fluctuation

(b) The event $E^{R}$ implying event $E$

Figure 3. Illustrations for Theorem 1.1:(A) Since two geodesics cannot
intersect twice, the condition that $\Gamma_{n}$ has non-negative fluctuation
at ${\mathcal{L}}_{r}$ forces it to be above $\Gamma_{r}$. (B) If the event
$E$ does not happen while $E^{\prime}$ does, then $\Gamma_{n}$ must look like
either the green or blue paths. The maximality of $\Gamma^{R}$ implies that
the blue path cannot be a geodesic. Meanwhile, suppose the green path is a
geodesic. In that case, the path we get from following $\Gamma^{R}$ initially
and then shifting to the green path after, the last intersection will
contradict the maximality of $\Gamma^{R}$.

As in the proof of Theorem 3.1, define the sets $A,B,C,D$ and the rectangle
$R$. Thus, as previously shown, on the event $A\cap B\cap C\cap D$, we will
have that $\mathrm{TF}^{r}(r)\geq tr^{2/3}$. In fact, we have shown that there
exists a point $w\in\Gamma_{r}$ such that

$\psi\left({w}\right)>tr^{2/3}.$

We shall now use $\Gamma_{n}(r)$ to denote the intersection point of the
geodesic $\Gamma_{n}$ and the line ${\mathcal{L}}_{r}$. We say that the
geodesic has negative fluctuation at ${\mathcal{L}}_{r}$ if
$\psi\left({\Gamma_{n}(r)}\right)<0$ and call the fluctuation non-negative
otherwise. Define the event $E=\left\\{{\psi\left({\Gamma_{n}(r)}\right)\geq
0}\right\\}$. Now on the event $A\cap B\cap C\cap D\cap E$, $\Gamma_{n}$ will
be forced to be to the left of $\Gamma_{r}$, since any two geodesics can not
intersect twice. Thus, the point $w$ on $\Gamma_{r}$ will force
$\mathrm{TF}^{n}(r)\geq tr^{2/3}$.

$\displaystyle\underset{}{{\mathbb{P}}}\left({{A\cap B\cap C\cap D\cap
E}}\right)$
$\displaystyle=\underset{}{{\mathbb{P}}}\left({{A}}\right)\left({\underset{}{{\mathbb{P}}}\left({{B\cap
C\cap D\cap E\>|A}}\right)}\right)$
$\displaystyle\geq\underset{}{{\mathbb{P}}}\left({{A}}\right)\left({\underset{}{{\mathbb{P}}}\left({{E|A}}\right)-\underset{}{{\mathbb{P}}}\left({{\left({B\cap
C\cap D}\right)^{c}|A}}\right)}\right)$
$\displaystyle=\underset{}{{\mathbb{P}}}\left({{A}}\right)\left({\underset{}{{\mathbb{P}}}\left({{E|A}}\right)-\underset{}{{\mathbb{P}}}\left({{\left({B\cap
C\cap D}\right)^{c}}}\right)}\right).$

Our aim now will be to lower bound the term
$\underset{}{{\mathbb{P}}}\left({{E|A}}\right)$. Let $T^{R}(\gamma)$ be the
passage time for the upright path $\gamma$ to go from $\mathbf{o}$ to
$\mathbf{n}$ after discounting the weights in $R$. Therefore,

$T^{R}(\gamma)=\sum_{u\in\gamma,u\notin R}X_{u}.$

We use $\Gamma^{R}$ to denote the path from $\mathbf{o}$ to $\mathbf{n}$ with
maximal $T^{R}$. Since our random variables are continuous, such a path is
almost surely unique. Let $\Gamma^{R}(r)$ denote the intersection point of
${\mathcal{L}}_{r}$ and $\Gamma^{R}$. We define

$E^{R}=\left\\{{\Gamma^{R}\text{ lies below }R\text{ and
}\psi\left({\Gamma^{R}(r)}\right)\geq 0\text{ at
}{\mathcal{L}}_{r}}\right\\}.$

Notice that the event $E^{R}\subseteq E$, as the path $\Gamma^{R}$ forces the
path $\Gamma_{n}$ to have non-negative fluctuation at ${\mathcal{L}}_{r}$.
This is because increasing the weights from 0 in $R$ can only increase the
transversal fluctuation of the geodesic at ${\mathcal{L}}_{r}$.

At the same time, on the event

$E^{\prime}=\left\\{{\Gamma_{n}\text{ lies below }R\text{ and
}\psi\left({\Gamma_{n}(r)}\right)\geq 0}\right\\},$

the path $\Gamma_{n}$ and $\Gamma_{R}$ will be the same. Thus
$E^{\prime}\subseteq E^{R}$ and for large enough $t$, by Theorem 2.2

$\underset{}{{\mathbb{P}}}\left({{E^{R}}}\right)\geq\underset{}{{\mathbb{P}}}\left({{E^{\prime}}}\right)\geq\frac{1}{2}-Ce^{-c{t}^{3}}\geq\frac{1}{4}.$

$E^{R}$ is also independent of the event $A$ since $E^{R}$ does not depend on
the passage time of the vertices inside $R$ while $A$ depends only on the
weights of those vertices. Thus, we have that

$\displaystyle\underset{}{{\mathbb{P}}}\left({{A\cap B\cap C\cap D\cap
E}}\right)$
$\displaystyle\geq\underset{}{{\mathbb{P}}}\left({{A}}\right)\left({\underset{}{{\mathbb{P}}}\left({{E^{R}|A}}\right)-\underset{}{{\mathbb{P}}}\left({{\left({B\cap
C\cap D}\right)^{c}}}\right)}\right)$
$\displaystyle=\underset{}{{\mathbb{P}}}\left({{A}}\right)\left({\underset{}{{\mathbb{P}}}\left({{E^{R}}}\right)-\underset{}{{\mathbb{P}}}\left({{\left({B\cap
C\cap D}\right)^{c}}}\right)}\right).$

Now by (3.1), (3.2) and Lemma 3.2, we can claim that

$\displaystyle\underset{}{{\mathbb{P}}}\left({{A}}\right)\left({\underset{}{{\mathbb{P}}}\left({{E^{R}}}\right)-\underset{}{{\mathbb{P}}}\left({{\left({B\cap
C\cap D}\right)^{c}}}\right)}\right)$ $\displaystyle\geq
Ce^{-c{t}^{3}}\left({1/4-3Ce^{-c{t}^{3}}}\right)$ $\displaystyle\geq
Ce^{-c{t}^{3}}.$

∎

######  Remark 3.

The result directly puts us in a position to comment on the fluctuations of
semi-infinite geodesics. Theorem 1.1 will give us a similar lower bound for
large fluctuations of the semi-infinite geodesics in the direction of $x=y$
since the finite geodesics converge subsequentially to the semi-infinite
geodesic. Such a lower bound for semi-infinite geodesics was proven in [SS20,
Theorem 2.8]. Our result also holds for finite geodesics while their result
hold for all semi-infinite geodesics in non-axial directions.

######  Remark 4.

The proof uses the event $E^{R}$, which is independent of the rectangle $R$
and the symmetry of the problem. While considering other non-axial directions,
one can still produce similar constructions for the event $E^{R}$, but the
symmetry is lost. However, it is to be noted that the symmetry is used to
claim that the $\underset{}{{\mathbb{P}}}\left({{E^{R}}}\right)$ is greater
than some constant; this should be achievable even in the non-axial
directions. It is also to be noted that we did not use any explicit property
of exponential LPP. All our arguments follow from the moderate deviation
result from integrable probability. Thus, the same arguments should hold for
other solvable models like the Poissonian LPP and Geometric LPP.

######  Remark 5.

We have proven a lower bound for the maximal transversal fluctuation of the
geodesic up to the line ${\mathcal{L}}_{r}$. The same lower bound (up to
exponents) should also hold for large transversal fluctuation of the geodesic
at the line ${\mathcal{L}}_{r}$. However, this requires some refinement of the
arguments we make.

### Acknowledgement

I thank my supervisor, Riddhipratim Basu, for the useful discussions and
comments.

## References

  * [BBB23] Márton Balázs, Riddhipratim Basu, and Sudeshna Bhattacharjee.  Geodesic trees in last passage percolation and some related problems.  arXiv, August 2023.
  * [BGHK21] Riddhipratim Basu, Shirshendu Ganguly, Milind Hegde, and Manjunath Krishnapur.  Lower deviations in $\beta$-ensembles and law of iterated logarithm in last passage percolation.  Israel J. Math., 242(1):291–324, April 2021.
  * [BGZ21] Riddhipratim Basu, Shirshendu Ganguly, and Lingfu Zhang.  Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison.  Commun. Math. Phys., 383(3):1805–1888, May 2021.
  * [BSS14] Riddhipratim Basu, Vladas Sidoravicius, and Allan Sly.  Last Passage Percolation with a Defect Line and the Solution of the Slow Bond Problem.  arXiv, August 2014.
  * [BSS19] Riddhipratim Basu, Sourav Sarkar, and Allan Sly.  Coalescence of geodesics in exactly solvable models of last passage percolation.  J. Math. Phys., 60(9), September 2019.
  * [GH23] Shirshendu Ganguly and Milind Hegde.  Optimal tail exponents in general last passage percolation via bootstrapping & geodesic geometry.  Probab. Theory Related Fields, 186(1):221–284, June 2023.
  * [HS18] Alan Hammond and Sourav Sarkar.  Modulus of continuity for polymer fluctuations and weight profiles in poissonian last passage percolation, 2018.
  * [Joh00a] Kurt Johansson.  Shape Fluctuations and Random Matrices.  Communications in Mathematical Physics, 209(2):437 – 476, 2000.
  * [Joh00b] Kurt Johansson.  Transversal fluctuations for increasing subsequences on the plane.  Probability Theory and Related Fields, 116(4):445 – 456, 2000.
  * [Liu22] Zhipeng Liu.  One-point distribution of the geodesic in directed last passage percolation, 2022.
  * [LR10] Michel Ledoux and Brian Rider.  Small deviations for beta ensembles.  Electron. J. Probab., 15:no. 41, 1319–1343, 2010.
  * [SS20] Timo Seppäläinen and Xiao Shen.  Coalescence estimates for the corner growth model with exponential weights.  Electronic Journal of Probability, 25(none):1 – 31, 2020.