[ude] organization=Fakultät für Physik, Universität Duisburg-
Essen,addressline=Lotharstraße 1, city=Duisburg, postcode=47048, state=NRW,
country=Germany

# Foreign exchange markets: price response and spread impact

Juan C. Henao-Londono 11footnotemark: 1<EMAIL_ADDRESS>Thomas
Guhr 22footnotemark: 2<EMAIL_ADDRESS>
###### Abstract

We carry out a detailed large-scale data analysis of price response functions
in the spot foreign exchange market for different years and different time
scales. Such response functions provide quantitative information on the
deviation from Markovian behavior. The price response functions show an
increase to a maximum followed by a slow decrease as the time lag grows, in
trade time scale and in physical time scale, for all analyzed years.
Furthermore, we use a price increment point (pip) bid-ask spread definition to
group different foreign exchange pairs and analyze the impact of the spread in
the price response functions. We find that large pip spreads have a stronger
impact on the response. This is similar to what has been found in stock
markets.

###### keywords:

Econophysics , Complex systems , Statistical physics , Price response function
, Spread impact , Foreign exchange market

##  1 Introduction

A major objective of data driven research on complex systems is the
identification of generic or universal statistical behavior. The tremendous
success of thermodynamics and statistical mechanics serves as an inspiration
when continuing this quest in complex systems beyond traditional physics.
Particularly interesting are large complex systems which consist of similar,
yet clearly distinguishable complex subsystems. Financial markets, for
example, have well defined subsystems as foreign exchange markets, stock
markets, bond markets, among others. The degree of universality found in one
particular subsystem can then be assessed if this type of universality is also
seen in another subsystem. If applicable, useful information on the impact of
specific system features on this universality may then be inferred.

In spite of the considerable interest, a thorough statistical analysis of the
microstructure in foreign exchange markets was hampered by limited access to
data. This changed, and nowadays such data analyses are possible down to the
level of ticks and over long time scales.

Here, we carry out such a study for finance, because a tremendous amount of
data is available [1]. Markets may be viewed as macroscopic complex systems
with an internal microscopic structure that is to a large extent accessible by
big data analysis [2]. Stock markets and foreign exchange markets are clearly
distinct, but share many common features. In previous analyses, we studied
response functions in stock markets to shed light on non-Markovian behavior.
Here, we extend that to the spot foreign exchange markets. To our surprise, we
did not find such an investigation in the literature. Hence, we believe that
this study is a rewarding effort. It helps to examine the behavior of the
functions applied to the foreign exchange market and it is suitable to compare
the similarities and differences to other markets.

The foreign exchange market has attracted a lot of attention in the last 20
years. Electronic trading has changed an opaque market to a fairly transparent
one with transaction costs that are a fraction of their former level. The
large amount of data that is now available to the public makes possible
different kinds of data analysis. Intense research is currently carried out in
different directions [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18].

McGroarty et al. [19] found that smaller volumes cause larger bid-ask spreads
for technical reasons related to the measurement, whereas Hau et al. [13, 20]
claim that larger bid-ask spreads caused smaller volumes due to the traders’
behavior.

Burnside et al. [11] found the spreads to be between two and four times larger
for emerging market currencies than for developed country currencies.
According to Huang and Masulis [14], bid-ask spreads increase when the foreign
exchange market volatility increases, and decrease when the competition
between the dealers increases. Ding and Hiltrop [9] showed that the Electronic
Broking Services (EBS) reduces spreads significantly, but dealers with
information advantage tend to quote relatively wider spreads. King [21]
analyzed the foreign exchange futures market and observed that the number of
transactions is negatively related with bid-ask spread, whereas volatility in
general is positively related. Serbinenko and Rachev [5] focus on the three
major market characteristics, namely efficiency, liquidity and volatility, and
found that the market is efficient in a weak form. Menkhoff and Schmeling [17]
used orders from the Russian interbank for Russian rouble/US dollar rate. They
analyzed the price impact in different regions of Russia, and found that
regions that are centers of political and financial decision making have high
permanent price impact.

Price response functions are a powerful tool to obtain dynamical information
because they measure price changes implied by execution of market orders.
Specifically, they measure how a buy or sell order at time $t$ influences on
average the price at a later time $t+\tau$. It was shown in different works
[22, 23, 24, 25, 26, 27, 28, 29, 30] that the price response functions
increase to a maximum and then slowly decrease as the time lag grows.

Little is known about price response functions or related quantities in the
foreign exchange markets [3, 16, 31]. Melvin and Melvin [16] simulate their
proposed model for different foreign exchange markets region to analyze the
impact of a one-standard-deviation shock using impulse response functions. The
general pattern of response was a fairly steep drop over the first couple of
days followed by a few days of gradual decline until the response is not
statistically different from zero. Mancini et al. [3] model the price impact
and return reversal to analyze liquidity. Their model predicts that more
liquid assets should exhibit narrower spreads and lower price impact.

To the best of our knowledge, no large-scale data analysis of response
functions for the spot foreign exchange market has been carried out. Response
functions are important observables as they give information on non-Markovian
behavior. It is the purpose of the present study to close this gap. Based on a
series of detailed empirical results obtained on trade by trade data, we show
that the price response functions in the foreign exchange markets behave
qualitatively similar as the ones in correlated stocks markets. We consider
different time scales, years and currency pairs to compute the price response
functions. Finally, we shed light on the spread impact in the response
functions for foreign exchange pairs. We use a pip bid-ask spread definition
to group different foreign exchange pairs and show that large pip spreads have
a stronger impact on the response. To facilitate the reproduction of our
results, the source code for the data analysis is available in Ref. [32].

The paper is organized as follows: in Sect. 2 we introduce the foreign
exchange market. In Sect. 3 we present our data set of spot foreign exchange
pairs and briefly describe the physical and trade time scale. We define the
time scale to be used in Sect. 4, and compute the price response functions for
foreign exchange pairs in Sect. 5. In Sect. 6 we show how the spread impact
the values of the response functions. Our conclusions follow in Sect. 7.

##  2 Foreign exchange market overview

In Sect. 2.1 we describe the basic characteristics of the foreign exchange
market. In Sect. 2.2 we establish the fundamental quantities used in the price
response definitions.

###  2.1 Foreign exchange market

The foreign exchange market is a 24-hour global decentralized or over-the-
counter (OTC) market for the trading of currencies closing only on the
weekends. The foreign exchange market is the most volatile, liquid and largest
of all financial markets [3, 4, 5, 6, 7, 33, 34, 35, 36], and it has a
paramount importance for the world economy. It affects employment, inflation,
international capital flows, among others [6]. The major participants trading
in this market include governments, central banks, global funds, retail
clients and corporations [35, 36]. Trading of currency in the foreign exchange
market involves the purchase and sale of two currencies at the same time [34,
35, 36]. The value of one of the currencies in that pair is relative to the
value of the other. The price one currency can be exchanged with another
currency is the foreign exchange rate. The foreign exchange market is a closed
system. As one value increases another value has to decrease. All foreign
exchange rates cannot appreciate, in contrast to the stock market [34, 36].

Depending on the country, the currencies can be “free float” or “fixed float”.
Free-floating currencies relative value is determined by free-market forces.
Some example of free-floating currencies include the U.S. dollar, Japanese yen
and Colombian peso. On the other hand, a fixed float is where a government
through the central bank set the currency’s relative value to other
currencies, usually by pegging it to some standard. Examples of fixed floating
currencies include the Chinese Yuan and the Indian Rupee [34]. In our case, we
only use free float currencies.

In the foreign exchange market, the trading day begins in Australia and Asia.
Then the markets in Europe open and finally the markets in America [6, 33, 35,
36]. As the market close time in New York overlaps the market open time in
Australia and Asia, the markets do not formally close during the week. Thus,
using the New York time as reference, the market opens on Sunday at 19h00 and
closes on Friday at 17h00. London, New York and Tokyo are the largest trading
centers of foreign exchange trading [37]

Currency markets are divided into spot market, forward, future, currency swaps
and currency options [35, 36, 37]. In our work we particularly focus on the
spot market, where as his name suggest, the trades are settled on the spot
[34, 36]. In a spot market, as the ccurency transactions are carried in the
OTC markets, information concerning open interest and volume is unavailable.
The transactions in this market represent up to the 40% of the total market
transactions in the foreign exchange market. The most traded currencies in the
spot market are the U.S. dollar, euro, Japanese yen, British pound and Swiss
franc [34].

In general, three categories of currency pairs are defined: majors, crosses,
and exotics. The “major” foreign exchange currency pairs are the most
frequently traded currencies that are paired with the U.S. dollar. The
“crosses” are those majors pairs paired between them and that exclude the U.S.
dollar. Finally, the “exotic” pairs usually consist of a major currency
alongside a thinly traded currency or an emerging market economy currency. The
majors are the most liquid pairs, in contrast with the exotics, who can be
much more volatile. In this work, we will refer as the “major currency pairs”
to the pairs of most traded currencies paired with the U.S. dollar, including
the so called commodity currencies: Canadian dollar, Australian dollar and New
Zealand dollar. The pairs and their corresponding symbol can be seen in Table
1.

Table 1: Analyzed currency pairs. Currency pair |  Symbol
---|---
euro/U.S dollar |  EUR/USD
British pound/U.S. dollar |  GBP/USD
Japanese yen/U.S. dollar |  JPY/USD
Australian dollar/U.S. dollar |  AUD/USD
U.S. dollar/Swiss franc |  USD/CHF
U.S. dollar/Canadian dollar |  USD/CAD
New Zealand dollar/U.S. dollar |  NZD/USD

The term pip (Price Increment Point) is commonly used in the foreign exchange
market instead of tick. The precise definition of a pip is a matter of
convention. Usually, it refers to the incremental value in the fifth non-zero
digit position from the left. It is not related to the position of the decimal
point. For example, one pip in the exchange rate USD/JPY of 124.21 would be
0.01, while one pip for EUR/USD of 1.1021 would be 0.0001 [6, 19, 33, 36, 38].

Compared with other markets like the stock market, there are some key
characteristics that differentiate the foreign exchange market. There are
fewer rules, there are no clearing houses and central bodies that oversee the
market. The investors will not have to pay fees or commissions as on another
markets. It is possible to trade at any time of day and regarding the risk and
reward, it is possible to get in and out whenever the investor want. In the
foreign exchange market, the bid-ask spread is the only transaction cost [35].

###  2.2 Key concepts

In spot foreign exchange markets, orders are executed at the best available
buy or sell price. Orders often fail to result in an immediate transaction,
and are stored in a queue called the limit order book [6, 33, 39, 40, 41, 42].
The order book is visible for all traders and its main purpose is to ensure
that all traders have the same information on what is offered on the market.
For a detailed description of the operation of the markets, we suggest to see
Ref. [28].

At any given time there is a best (lowest) offer to sell with price
$a\left(t\right)$, and a best (highest) bid to buy with price
$b\left(t\right)$ [25, 34, 42, 43, 44, 45]. The price gap between them is
called the spread $s\left(t\right)=a\left(t\right)-b\left(t\right)$ [7, 25,
26, 34, 43, 45, 46, 47, 48]. Spreads are significantly positively related to
price. Currencies with more liquidity tend to have lower spreads [22, 43, 49,
50]. In spot foreign exchange markets, the existing spread in any currency
will vary depending on the currency trader, the currency being traded and the
conditions in the market. Although the foreign exchange market is often cited
as the world’s largest financial market, this description fails to consider
the considerable differences in trading volume and liquidity across different
currency pairs [8, 35]. These differences can be directly seen in the spread.
The spread will tend to increase for currencies that do not generate a large
volume of trading [35]. Furthermore, the bid-ask spread is directly related
with the transaction costs to the dealer [7, 21, 35].

As we have the quotes prices in the data, we need to infer the trade price. We
consider a basic definition of the price given by [3, 12, 15]. The average of
the best ask and the best bid is the midpoint price, which is defined as [7,
25, 26, 28, 42, 45, 46, 48]

$m\left(t\right)=\frac{a\left(t\right)+b\left(t\right)}{2}.$ (1)

Price changes are typically characterized as returns. Using the midpoint price
$m\left(t\right)$ of a currency pair at time $t$, the return
$r\left(t,\tau\right)$, at time $t$ and time lag $\tau$ is simply the relative
variation of the price from $t$ to $t+\tau$ [25, 51, 52, 53, 54, 55],

$r\left(t,\tau\right)=\frac{m\left(t+\tau\right)-m\left(t\right)}{m\left(t\right)}.$
(2)

The distribution of returns is strongly non-Gaussian and its shape
continuously depends on the return period $\tau$. Small $\tau$ values have fat
tails return distributions [25]. The trade signs are defined for general cases
as

$\varepsilon\left(t\right)=\text{sign}\left(S\left(t\right)-m\left(t-\delta\right)\right),$
(3)

where $\delta$ is a positive time increment. Hence we have

$\varepsilon\left(t\right)=\left\\{\begin{array}[]{cc}+1,&\text{If
}S\left(t\right)\text{ is higher than the last }m\left(t\right)\\\
-1,&\text{If }S\left(t\right)\text{ is lower than the last
}m\left(t\right)\end{array}\right..$ (4)

Here, $\varepsilon(t)=+1$ indicates that the trade was triggered by a market
order to buy and a trade triggered by a market order to sell yields
$\varepsilon(t)=-1$ [25, 26, 56, 57, 58]. In our implementation of the price
response functions we need a more specific definition of trade signs. We give
a deeper explanation of trade signs depending on the time scale in Sect. 4.

The main objective of this work is to analyze the price response functions. In
general we define the price response functions in a foreign exchange market as

$R^{\left(\textrm{scale}\right)}_{i}\left(\tau\right)=\left\langle
r^{\left(\textrm{scale}\right)}_{i}\left(t-1,\tau\right)\varepsilon^{\left(\textrm{scale}\right)}_{i}\left(t\right)\right\rangle_{\textrm{average}},$
(5)

where the index $i$ corresponds to currency pairs in the market,
$r^{\left(\textrm{scale}\right)}_{i}$ is the return of the pair $i$ in a time
lag $\tau$ in the corresponding scale and
$\varepsilon^{\left(\textrm{scale}\right)}_{i}$ is the trade sign of the pair
$i$ in the corresponding scale. The superscript scale refers to the time scale
used, whether physical time scale ($\textrm{scale}=\textrm{p}$) or trade time
scale ($\textrm{scale}=\textrm{t}$). Finally, The subscript average refers to
the way to average the price response, whether relative to the physical time
scale ($\textrm{average}=P$) or relative to the trade time scale
($\textrm{average}=T$).

For correlated financial markets, the price response function increases to a
maximum and then slowly decreases. This result is observed empirically in
trade time scale and in physical time scale [28, 29].

##  3 Data set

The spot foreign exchange financial data was obtained from HistData.com. We
use a tick-by-tick database in generic ASCII format for different years and
currency pairs. This tick-by-tick data is sampled for each transaction. The
data comprises the date time stamp (YYYYMMDD HHMMSSNNN), the best bid and best
ask quotes prices in the Eastern Standard Time (EST) time zone. With both best
bid and best ask quotes it is easy to compute the pip bid-ask spread of the
data. No information about the size of each transaction is provided. Also, the
identity of the participants is not given. Furthermore, trading volumes in
spot foreign exchange market are not aggregated and the only volumes that are
possible to find are the Broker Specific Volumes. Therefore, the data provider
decided to remove the volume information from the delivered data.

Our aims is to compute the price response functions in two different time
scales: trade time scale and physical time scale. We give a complete
description on how the scales are defined in Sect. 4. Regarding the data for
this definitions, for the trade time scale we use the data as it is,
considering that it is sampled for each transaction. On the physical time
scale, for each exchange rate, we process the irregularly spaced raw data to
construct second-by-second price series, each containing 86,400 observations
per day. For every second, the midpoint of best bid and ask quotes are used to
construct one-second log-returns.

Another goal in this paper is to compare the price response functions in
different calendar years to see the differences and similarities along time.
To analyze the price response functions in Sect. 5, we select the seven major
currency pairs (see Table 1) in three different years: 2008, 2014 and 2019.

Additionally, we analyze the pip bid-ask spread impact in price response
functions (Sect. 6). We select 46 currency pairs in three different years
(2011, 2015 and 2019). The selected pairs are listed in A.

The selection of the calendar years to be analyzed was made considering the
availability of the data, the completeness of the time series and the option
to have a constant gap between the years.

In order to avoid overnight effects and any artifact due to the opening and
closing of the foreign exchange market, we systematically discard the first
ten and the last ten minutes of trading in a given week [26, 28, 30, 46, 56].
Therefore, we only consider trades of the same week from Sunday 19:10:00 to
Friday 16:50:00 New York local time. We will refer to this interval of time as
the “market time”.

##  4 Time scale

In Sect. 4.1 we describe the physical time scale and the trade time scale. In
Sect. 4.2 and Sect. 4.3 we define the trade and the physical time scales,
respectively.

###  4.1 Time definition

Due to the nature of the data, they are several options to define time for
analyzing data. In general, the time series are labeled in calendar time
(hours, minutes, seconds, milliseconds). Moreover, tick-by-tick data available
on financial markets all over the world is time stamped up to the millisecond,
but the order of magnitude of the guaranteed precision is much larger, usually
one second or a few hundreds of milliseconds [47, 51]. In several papers are
used different time definitions (calendar time, physical time, event time,
trade time, tick time) [51, 59, 60]. The spot foreign exchange market data
used in the analysis only has the quotes. In consequence, we have to infer the
trades during the market time. As we have tick-by-tick resolution, we can use
either trade time scale or physical time scale.

The trade time scale increases by one unit each time a transaction happens,
which in our case is every time the quotes change. The advantage of this count
is that limit orders far away in the order book do not increase the time by
one unit. The main outcome of trade time scale is its “smoothing” of data and
the aggregational normality [51].

The physical time scale is increased by one unit each time a second passes.
This means that computing the responses in this scale involves sampling [30,
59], which has to be done carefully when dealing for example with several
stocks with different liquidity. This sampling is made in the trade signs and
in the midpoint prices.

We use these two definitions of time scale to compute the price response
function. Between these two scales, there is not a judgment which is better or
worse. Their use directly depend on the application. Thus, our aim is to
present how the price response function behaves under these two different time
scales.

###  4.2 Trade time scale

As a first approximation, we use the trade sign classification in trade time
scale proposed in Ref. [30] and used in Refs. [28, 29, 61, 62] that reads

$\varepsilon^{\left(\textrm{t}\right)}\left(t,n\right)=\left\\{\begin{array}[]{cc}\text{sgn}\left(m\left(t,n\right)-m\left(t,n-1\right)\right),&\text{if
}\\\ m\left(t,n\right)\neq m\left(t,n-1\right)\\\
\varepsilon^{\left(\textrm{t}\right)}\left(t,n-1\right),&\text{otherwise}\end{array}\right..$
(6)

Here, $\varepsilon^{\left(\textrm{t}\right)}\left(t,n\right)=+1$ implies a
trade triggered by a market order to buy, and a value
$\varepsilon^{\left(\textrm{t}\right)}\left(t,n\right)=-1$ indicates a trade
triggered by a market order to sell.

In the second case of Eq. (6), if two consecutive trades with the same trading
direction do not exhaust all the available volume at the best quote, the
trades would have the same price, and they will thus have the same trade sign.

With this classification we obtain trade signs for every single trade in the
data set. According to Ref. [30], the average accuracy of the classification
is $85\%$ for the trade time scale.

###  4.3 Physical time scale

We use the trade sign definition in physical time scale proposed in Ref. [30]
and used in Refs. [29, 61], that depends on the classification in Eq. (6) and
reads

$\varepsilon^{\left(\textrm{p}\right)}\left(t\right)=\left\\{\begin{array}[]{cc}\text{sgn}\left(\sum_{n=1}^{N\left(t\right)}\varepsilon^{\left(\textrm{t}\right)}\left(t,n\right)\right),&\text{If
}N\left(t\right)>0\\\ 0,&\text{If }N\left(t\right)=0\end{array}\right.,$ (7)

where $N\left(t\right)$ is the number of trades in an interval of one second.
Here, $\varepsilon^{\left(\textrm{p}\right)}\left(t\right)=+1$ implies that
the majority of trades in the second $t$ are triggered by a market order to
buy, and a value $\varepsilon^{\left(\textrm{p}\right)}\left(t\right)=-1$
indicates a majority of sell market orders. In this definition, there are two
ways to obtain $\varepsilon^{\left(\textrm{p}\right)}\left(t\right)=0$. First,
there are no trades in a particular second and thus no trade sign. Second, the
sum of the trade signs in a given second amounts to zero, indicating an exact
balance of buy and sell market orders.

Market orders show opposite trade directions as compared to limit orders
executed simultaneously. An executed sell limit order corresponds to a buyer-
initiated market order. An executed buy limit order corresponds to a seller-
initiated market order. In this case we do not compare every single trade sign
in a second, but the net trade sign obtained for every second with the
definition, see Eq. (7). According to Ref. [30], this definition has an
average accuracy up to $82\%$ in the physical time scale.

##  5 Price response functions

In Sect. 5.1 we analyze the responses functions in trade time scale and in
Sect. 5.2 we analyze the responses functions in physical time scale.

###  5.1 Response functions on trade time scale

The price response function in trade time scale is defined as [28]

$R^{\left(\textrm{t}\right)}_{i}\left(\tau\right)=\left\langle
r^{\left(\textrm{t}\right)}_{i}\left(t-1,\tau\right)\varepsilon_{i}^{\left(\textrm{t}\right)}\left(t,n\right)\right\rangle_{T}.$
(8)

To compute the response functions on trade time scale, we use both, the trade
signs and the returns from the tick-by-tick original data during a week in
market time. Then, the response is averaged by the number of trades.

Figure 1: Price response functions
$R^{\left(\textrm{t}\right)}_{i}\left(\tau\right)$ versus time lag $\tau$ on a
logarithmic scale in trade time scale for the years 2008 (top), 2014 (middle)
and 2019 (bottom).

The results of Fig. 1 show the price response functions of the seven foreign
exchange major pairs used in the analysis (see Table 1) for three different
years. The results found for all the years are entirely in line with price
responses seen in other financial markets, particularly with correlated
financial markets. The response functions have an initial increasing trend to
a maximum, that flattens out and saturates at some level, and eventually
slowly decrease. This shape is explained by an initial increase caused by
autocorrelated transaction flow. The flattening out is due to the market
liquidity adapting to this flow and assuring diffusive prices [63]. For our
selected pairs, a time lag of $\tau=10^{3}$ trades is enough to see an
increase to a maximum followed by a decrease. Thus, the trend in the price
response functions is eventually reversed. The response signal is much more
noisier in the year 2008 for the first seconds in the time lag. This behavior
is because of the smaller amount of data of the corresponding year. In
general, more data was recorded in recent years than in past years. In the
three years analyzed, the more liquid currency pairs have a smaller response
in comparison with the non-liquid pairs. The strength of the response function
varies from one year to the other. In 2008 the strength of the signal was one
order of magnitude stronger than the response in 2014, but the signals in 2014
have approximately twice the strength of the signals of 2019. This behavior
can be explained by the fact that in recent times algorithm trading has been
used intensively. Thus, many more trades were carried out in the last years,
which means, the impact of each trade is reduced, and then the response
functions tend to decrease compared with previous years.

###  5.2 Response functions on physical time scale

One important detail to compute the price response function on physical time
scale is to define how the averaging of the function will be made, because the
response functions highly differ when we include or exclude
$\varepsilon^{\left(\textrm{p}\right)}_{j}\left(t\right)=0$ [30]. The price
responses including
$\varepsilon^{\left(\textrm{p}\right)}_{j}\left(t\right)=0$ are weaker than
the excluding ones due to the omission of direct influence of the lack of
trades. However, either including or excluding
$\varepsilon^{\left(\textrm{p}\right)}_{j}\left(t\right)=0$ does not change
the trend of price reversion versus the time lag, but it does affect the
response function strength [29]. For a deeper analysis of the influence of the
term $\varepsilon^{\left(\textrm{p}\right)}_{j}\left(t\right)=0$ in price
response functions, we suggest reviewing Refs. [29, 30]. We will only take
into account the price response functions excluding
$\varepsilon^{\textrm{p}}_{j}\left(t\right)=0$.

We define the price response functions on physical time scale, using the trade
signs and the returns sampled in seconds from the original data on physical
time scale. The price response function on physical time scale is defined as
[28]

$R^{\left(\textrm{p}\right)}_{i}\left(\tau\right)=\left\langle
r^{\left(\textrm{p}\right)}_{i}\left(t-1,\tau\right)\varepsilon_{i}^{\left(\textrm{p}\right)}\left(t\right)\right\rangle_{P}$
(9)

Figure 2: Price response functions
$R^{\left(\textrm{p}\right)}_{i}\left(\tau\right)$ excluding
$\varepsilon^{\left(\textrm{p}\right)}_{i}\left(t\right)=0$ versus time lag
$\tau$ on a logarithmic scale in physical time scale for the years 2008 (top),
2014 (middle) and 2019 (bottom).

The results shown in Fig. 2 are the price response functions on physical time
scale for three different years. The results show approximately the same
behavior observed in currency exchange pairs in trade time scale, and in
correlated financial markets, where we can see that an increase to a maximum
is followed by a decrease. Thus again, the trend in the price responses is
eventually reversed. An exception occurs in the year 2008, where the response
at short time lags seems to decrease, to then start to slightly increase, and
finally it decreases again.

The price response functions on physical time scale are smoother than the
responses on trade time scale. As we reduce from trade data all the returns
and trade signs in one second to one data point on physical time scale, and as
this sampling gives the same weight to every data point, the curves look
smoother.

Compared with the response functions on trade time scale, the strength of the
signal of the response functions on physical time scale are similar in
magnitude in the corresponding years. Thus, the strength of the signal in 2008
for trade time scale is similar to the strength of the signal in 2008 for
physical time scale, and so on. This behavior is different to the one
presented in correlated financial markets, where the results differ about a
factor of two depending on the time scale.

On physical time scale, we can see that the liquid pairs have a smaller price
response compared with non-liquid pairs. Therefore, the price response of a
foreign exchange pair with large activity is smaller to the small impact of
each trade. Also, the older the response, the stronger the signal. We consider
the same argument of algorithm trading to explain why the signals in recent
years are weaker than in older years.

##  6 Spread impact in price response functions

To analyze the spread impact in price response functions, we use 47 foreign
exchange pairs from three different years (A). As we showed in Sect. 2, due to
the difference in the position of the decimal points in the price between
foreign exchange pairs, to compare them we need to introduce a “scaling
factor” with the purpose of bringing the pip to the left of the decimal point.
For example, the scaling factor for the USD/JPY is $100$ and the one for the
EUR/USD is $10000$.

The pip bid-ask spread is defined as [19]

$s_{\textrm{pip}}=\left(a\left(t\right)-b\left(t\right)\right)\cdot\textrm{scaling
factor}.$ (10)

With the $s_{\textrm{pip}}$ we can group the foreign exchange pairs and check
how the average strength of the price response functions on trade time scale
and physical time scale behave. For each pair we compute the pip bid-ask
spread in every trade along the market time. Then we average the spread during
the trade weeks in the different years. With this value we group the foreign
exchange pairs.

Figure 3: Average price response functions
$R^{\left(t\right)}_{i}\left(\tau\right)$ versus time lag $\tau$ on a
logarithmic scale in trade time scale (Top) and
$R^{\left(p\right)}_{ii}\left(\tau\right)$ excluding
$\varepsilon^{\left(p\right)}_{i}\left(t\right)=0$ versus time lag $\tau$ on a
logarithmic scale in physical time scale (Bottom) for 47 foreign exchange
pairs divided in representative groups in three different years (2011, 2015
and 2019).

Depending on the year, we identify different numbers of groups according to
the pip spread $s_{\textrm{pip}}$. For the years 2011 and 2015, we use two
intervals to select the foreign exchange pairs groups ($s_{\textrm{pip}}<10$
and $10\leq s_{\textrm{pip}}$). In the year 2019 we use three intervals to
select the foreign exchange pairs groups ($s_{\textrm{pip}}<4$, $4\leq
s_{\textrm{pip}}<10$ and $10\leq s_{\textrm{pip}}$). The detailed information
of the foreign exchange pairs, spread and the groups can be seen in A. With
the groups of the stocks defined, we average the price response functions of
each group.

In Fig. 3 we show the average response functions for the corresponding groups
in three different years. From year to year the groups can vary depending on
the pip spread. The average price response function for the pairs with smaller
pip spreads (more liquid) have on average the weakest signal in the figure for
all the years and both time scales. On the other hand, the average price
responses for the pairs with larger pip spread (less liquid) have on average
the strongest signal for all the years in both time scales.

From Sect. 5 we expect the increase-maximum-decrease behavior. This behavior
can be seen in the figures of the year 2015 for both time scales and in the
figure of the year 2019 in trade time scale. In these figures the average
price response functions follow an increase, reach a maximum and then start to
slowly decrease. For the other figures, on average, the response functions
start to decrease from the beginning.

The response in trade time scale seems to be noisier, with large changes in
the first time steps. This aggregate noise can be related with the crosses and
exotics pairs, who tend to fluctuate more.

For the years 2015 and 2019 in physical time scale, the increase-maximum-
decrease behavior is not that well defined as in Sect. 5 or as in the average
response in trade time scale. However, some groups tend to behave in the
expected way. The groups that do not follow the trend, seem to have an
instantaneous high response that slowly decreases with time. This behavior is
mostly noticeable in the year 2019.

For the year 2011 in physical time scale, the increase-maximum-decrease shape
is not present. For both groups the response decreased almost immediately.

In the three plots, the foreign exchange market seems to have a global
influence over all the pairs in the corresponding years. A similar behavior
can be seen in correlated financial markets [28].

##  7 Conclusion

Price response functions provide quantitative information on the deviation
from Markovian behavior. They measure price changes resulting from execution
of market orders. We used these functions in big data analysis for spot
foreign exchange markets. Such a study was, to the best of our knowledge,
never done before.

We analyzed price response functions in spot foreign exchange markets for
different years and different time scales. We used trade time scale and
physical time scale to compute the price response functions for the seven
major foreign exchange pairs for three different years. These major pairs are
highly relevant in the dynamics of the market. The use of different time
scales and calendar years in the work had the intention to display the
different behaviors the price response function could take when the time
parameters differ.

The price response functions were analyzed according to the time scales. On
trade time scale, the signals were noisier. For both time scales we observe
that the signal for all the pairs increases to a maximum and then starts to
slowly decrease. However, for the year 2008 the shape of the signals is not as
well defined as in the other years. The increase-decrease behavior observed in
the spot foreign exchange market was also reported in correlated financial
markets [28, 29]. These results show that the price response functions
conserve their behavior in different years and in different markets. The shape
of the price response functions is qualitatively explained considering an
initial increase caused by the autocorrelated transaction flow. To assure
diffusive prices, price response flattens due to market liquidity adapting to
the flow in the initial increase.

On both scales, the more liquid pairs have a smaller price response function
compared with the non-liquid pairs. As the liquid pairs have more trades
during the market time, the impact of each trade is reduced. Comparing years
and scales, the price response signal is stronger in past than in recent
years. As algorithmic trading has gained great relevance, the quantity of
trades has grown in recent years, and in consequence, the impact in the
response has decreased.

Finally, we checked the pip spread impact in price response functions for
three different years. We used 46 foreign exchange pairs and grouped them
depending on the conditions of the corresponding year analyzed. We employ the
year average pip spread of every pair for each year. For all the year and time
scales, the price response function signals were stronger for the groups of
pairs with larger pip spreads and weaker for the group of pairs with smaller
spreads. For the average of the price response functions, it was only possible
to see the increase-maximum-decrease behavior in the year 2015 in both scales,
and in the year 2019 on trade time scale. Hence, the noise in the cross and
exotic pairs due to the lack of trading compared with the majors seems
stronger. A general average price response behavior for each year and time
scale was spotted for the groups, suggesting a market effect on the foreign
exchange pairs in each year.

Comparing the response functions in stock and spot currency exchange markets
from a more general viewpoint, we find a remarkable similarity. It triggers
the conclusion that the order book mechanism generates in a rather robust
fashion the observed universal features in these two similar, yet different
subsystems within the financial system.

##  8 Author contribution statement

TG proposed the research. JCHL developed the method of analysis. The idea to
analyze the spread impact was due to JCHL. JCHL carried out the analysis. Both
authors contributed equally to analyzing the results and writing the paper.

One of us (JCHL) acknowledges financial support from the German Academic
Exchange Service (DAAD) with the program “Research Grants - Doctoral
Programmes in Germany” (Funding programme 57381412).

We thank an unknown referee of an earlier version for helpful comments on the
shape of the response functions.

##  Appendix A Foreign exchange pairs used to analyze the spread impact

We analyze the spread impact in the price response functions for 47 foreign
exchange pairs in the foreign exchange market for the years 2011, 2015 and
2019\. In Table 2, we list the pairs in their corresponding pip spread groups.

Table 2: Foreign exchange pairs used in Sect. 6.

Symbol | Pair | Category | Scaling factor | $2011$ | $2015$ | $2019$
---|---|---|---|---|---|---
AUD/CAD | Australian dollar/Canadian dollar | Cross | 10000 | G1 | G1 | G1
AUD/CHF | Australian dollar/Swiss franc | Cross | 10000 | G1 | G1 | G1
AUD/JPY | Australian dollar/Japanese yen | Cross | 100 | G1 | G1 | G1
AUD/NZD | Australian dollar/New Zealand dollar | Cross | 10000 | G2 | G1 | G1
AUD/USD | Australian dollar/U. S. dollar | Major | 10000 | G1 | G1 | G1
CAD/CHF | Canadian dollar/Swiss franc | Cross | 10000 | G1 | G1 | G1
CAD/JPY | Canadian dollar/Japanese yen | Cross | 100 | G1 | G1 | G1
CHF/JPY | Swiss franc/Japanese yen | Cross | 100 | G1 | G1 | G1
EUR/AUD | euro/Australian dollar | Cross | 10000 | G2 | G1 | G1
EUR/CAD | euro/Canadian dollar | Cross | 10000 | G1 | G1 | G1
EUR/CHF | euro/Swiss franc | Cross | 10000 | G1 | G1 | G1
EUR/CZK | euro/Czech koruna | Exotic | 10000 | G2 | G2 | G3
EUR/GBP | euro/British pound | Cross | 10000 | G1 | G1 | G1
EUR/HUF | euro/Hungarian forint | Exotic | 100 | G2 | G2 | G3
EUR/JPY | euro/Japanese yen | Cross | 100 | G1 | G1 | G1
EUR/NOK | euro/Norwegian krone | Exotic | 10000 | G2 | G2 | G3
EUR/NZD | euro/New Zealand dollar | Cross | 10000 | G2 | G1 | G2
EUR/PLN | euro/Polish zloty | Exotic | 10000 | G2 | G2 | G3
EUR/SEK | euro/Swedish krona | Exotic | 10000 | G2 | G2 | G3
EUR/TRY | euro/Turkish lira | Exotic | 10000 | G2 | G2 | G3
EUR/USD | euro/U. S. dollar | Major | 10000 | G1 | G1 | G1
GBP/AUD | British pound/Australian dollar | Cross | 10000 | G1 | G1 | G2
GBP/CAD | British pound/Canadian dollar | Cross | 10000 | G2 | G1 | G2
GBP/CHF | British pound/Swiss franc | Cross | 10000 | G2 | G1 | G1
GBP/JPY | British pound/Japanese yen | Cross | 100 | G1 | G1 | G1
GBP/NZD | British pound/New Zealand dollar | Cross | 10000 | G2 | G1 | G2
GBP/USD | British pound/U. S. dollar | Major | 10000 | G1 | G1 | G1
NZD/CAD | New Zealand dollar/Canadian dollar | Cross | 10000 | G1 | G1 | G2
NZD/CHF | New Zealand dollar/Swiss franc | Cross | 10000 | G2 | G1 | G1
NZD/JPY | New Zealand dollar/Japanese yen | Cross | 100 | G1 | G1 | G1
NZD/USD | New Zealand dollar/U. S. dollar | Major | 10000 | G1 | G1 | G1
SGD/JPY | Singapore dollar/Japanese yen | Exotic | 100 | G1 | G1 | G1
USD/CAD | U. S. dollar/Canadian dollar | Major | 10000 | G1 | G1 | G1
USD/CHF | U. S. dollar/Swiss franc | Major | 10000 | G1 | G1 | G1
USD/CZK | U. S. dollar/Czech koruna | Exotic | 10000 | G2 | G2 | G3
USD/DKK | U. S. dollar/Danish krone | Exotic | 10000 | G1 | G1 | G2
USD/HKD | U. S. dollar/Hong Kong dollar | Exotic | 10000 | G1 | G1 | G2
USD/HUF | U. S. dollar/Hungarian forint | Exotic | 100 | G2 | G2 | G3
USD/JPY | U. S. dollar/Japanese yen | Major | 100 | G1 | G1 | G1
USD/MXN | U. S. dollar/Mexican peso | Exotic | 10000 | G2 | G2 | G3
USD/NOK | U. S. dollar/Norwegian krone | Exotic | 10000 | G2 | G2 | G3
USD/PLN | U. S. dollar/Polish zloty | Exotic | 10000 | G2 | G2 | G3
USD/SEK | U. S. dollar/Swedish krona | Exotic | 10000 | G2 | G2 | G3
USD/SGD | U. S. dollar/Singapore dollar | Exotic | 10000 | G1 | G1 | G1
USD/TRY | U. S. dollar/Turkish lira | Exotic | 10000 | G2 | G1 | G3
USD/ZAR | U. S. dollar/South African rand | Exotic | 10000 | G2 | G2 | G3

  * *

G1 = group 1, G2 = group 2 and G3 = group 3.

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0530IST

# Replicas for Random Matrices

Madhusudhan Raman  International Centre for Theoretical Sciences
Tata Institute of Fundamental Research
Shivakote, Hesaraghatta Hobli, Bengaluru 560 089, India Instituto de Física
Teórica, UNESP - Universidade Estadual Paulista, R. Dr. Bento Teobaldo Ferraz
271, São Paulo 01140-070, Brazil madhusudhan dot raman at unesp dot br

(Date: )

###### Abstract.

We discuss the use of the replica ansatz in computing free energies in random
matrix theory, and confirm a conjectured condition on analytic continuation in
the replica index at large-$N$.

###### Contents

  1. 1 Introduction
  2. 2 Exact Results
  3. 3 Replica Ansatz
  4. 4 Discussion

##  1\. Introduction

In this short note we show that at low temperatures, the free energy of a
Gaussian random matrix theory is specified by the left edge of the Wigner
semicircle distribution. We do this using the replica ansatz.

In this section we will situate this result in the context of contemporary
research into the connections between random matrix theory and two-dimensional
gravity.

### Gravitational Path Integrals

Recent work [Saad:2019lba] has established a striking correspondence between
the two-dimensional dilaton gravity [Jackiw:1984je, Teitelboim:1983ux] on one
hand, and a double-scaled matrix model with a Schwarzian density of states on
the other. This demonstration invites us to think of Jackiw-Teitelboim (JT)
gravity as holographically dual to an _ensemble_ of random Hamiltonians.

Concurrently, the significance of Euclidean replica wormholes to the black
hole information paradox — specifically, the observation that accounting for
wormhole saddles between disconnected boundaries allows one to derive a
unitary Page curve — was highlighted by [Penington:2019kki] and
[Almheiri:2019qdq]. This was similarly striking, as the inclusion of wormhole
saddles would _prima facie_ preclude the factorisability of the gravitational
path integral on disconnected boundaries. Once again, as discussed for example
by [Harlow:2018tqv], we find that the interpretation of a gravitational path
integral as computing an _ensemble_ average is an attractive, if curious,
possibility.

These developments were further clarified to us in a beautiful paper
[Engelhardt:2020qpv], which set out to understand how the gravitational path
integral is influenced by Euclidean wormhole solutions. Their argument is
worth recalling. Consider a gravitational path integral $P(B)$ anchored to a
boundary $B$:

(1.1) $P(B)=\int_{\partial M=B}\mathrm{D}g\,e^{-S}\ ,$

where we integrate over metrics $g$ on $M$ such that $\partial M=B$. For the
case of $m$ disconnected boundaries $B^{m}=B\cup\cdots\cup B$, we similarly
define $P(B^{m})$. Now, if Euclidean wormholes contribute to this partition
function, then we must have

(1.2) $P(B^{m})\neq P(B)^{m}\ .$

This is what one means when one says that the gravitational path integral
fails to factorise. This failure is naturally accommodated by the proposal
that the gravitational path integral computes an average over some ensemble of
theories, since it is well known that the average of products is not the same
as the product of averages.

This failure to factorise is seen, explicitly, in the difference between
“quenched” $(F_{\mathrm{q}})$ and “annealed” $(F_{\mathrm{a}})$ free energies
of these theories:

(1.3) $\displaystyle F_{\mathrm{q}}$ $\displaystyle\sim\left\langle\log
Z(\beta)\right\rangle\ ,$ $\displaystyle F_{\mathrm{a}}$
$\displaystyle\sim\log\left\langle Z(\beta)\right\rangle\ ,$

where $\left\langle\cdots\right\rangle$ denotes an ensemble average. Working
with $\widehat{\mathrm{CGHS}}$ and JT gravities, the authors of
[Engelhardt:2020qpv] observe that while at high temperatures the quenched and
annealed free energies agree, at low temperatures they begin to differ
considerably due to the increasingly dominant contributions of (wormhole)
connected correlators. This is also the regime where the annealed free energy
exhibits pathological behaviour: it is not monotonically decreasing, thereby
implying a negative thermodynamic entropy.

Interestingly, even after the contributions of replica wormholes are taken
into account at low temperatures, and although this inclusion goes some way
towards easing the tension signalled by non-monotonicity, the (quenched) free
energy continues to behave pathologically.

### Toy Models For Averaging

In light of these developments, attempts have been made to understand
averaging _in vitro_ , so to speak, in the relatively simple context of random
matrix theory. Notable among these and germane to our theme is
[Okuyama:2020mhl], which studies “small” (rather, finite-$N$) random matrices
using a combination of analytic and numerical methods, and finds that the
quenched free energy exhibits no signs of pathological behaviour, viz. it
decreases monotonically, ensuring a positive entropy at all temperatures. This
finite-$N$ eigenvalue model, the workhorse of [Okuyama:2020mhl], suggests that
at low temperatures, the free energy is equal to the smallest eigenvalue of
the matrix ensemble.

The questions we explore in this short note are: does this remain true at
large-$N$? And can the replica ansatz be used to demonstrate the same? After
reviewing the necessary background in Section 2, we answer both these
questions in the affirmative in Section 3.

### Random Energy Model and Rectangular Diagrams

We are motivated to pursue this question on the strength of a similarity
between this problem and an exactly solvable model of disordered systems
called the random energy model [Derrida:1981zz]. Correlators of $n$-point
functions of the partition function in both the random energy model and random
matrix theory, as we will describe in more detail in subsequent sections, can
be written as a sum of $p(n)$ terms, where $p(n)$ is defined as

(1.4) $\prod_{k=1}^{\infty}\frac{1}{1-x^{k}}=\sum_{n=0}^{\infty}p(n)x^{n}\ ,$

and counts the number of partitions of $n$. Each of these $p(n)$ terms is
associated canonically to a Young diagram corresponding to a partition of $n$.
In the random energy model, at high temperatures the totally disconnected
partition $n=1+\cdots+1$ dominates, and the replica ansatz for computing the
quenched free energy

(1.5) $\left\langle\log Z\right\rangle=\lim_{n\rightarrow 0}\frac{\left\langle
Z^{n}\right\rangle-1}{n}\ ,$

works swimmingly, since $\left\langle Z^{n}\right\rangle\simeq\left\langle
Z\right\rangle^{n}$. At low temperatures, where connected correlators begin to
compete for dominance, the replica trick naively fails to return sensible
results.

It turns out that the way to extract physically sensible results using the
replica ansatz is to assume that some rectangular Young diagram is dominant at
low temperatures, and solve for its width by requiring that the corresponding
free energy is extremised. In the language of Parisi’s replica symmetry
breaking (RSB) scheme (see [Mezard:1987] for a thorough discussion and a
useful collection of reprints) this is essentially the “$1$-step RSB.” We will
adopt this strategy when studying the matrix model as well.

Our new results are: (i) a much simpler closed-form expression for the
generating function of $n$-point connected correlators in eq. 2.12, and (ii) a
proof that this $n$-point functions at large-$N$ and low temperatures
satisfies the following condition on analytic continuation:

(1.6) $\lim_{\beta\rightarrow\infty}\left\langle
Z(\beta)^{n}\right\rangle=e^{-n\beta E_{0}}\ .$

In the above formula, $E_{0}=-2$, the left edge of the Wigner semicircle
distribution. This condition was first proposed by [Okuyama:2020mhl] and we
prove this using the replica ansatz.

### Acknowledgments

The author is grateful to Sujay Ashok, Diptarka Das, Shouvik Datta, Oliver
Janssen, and Dileep Jatkar for discussions, and to Rukmini Dey and Suvrat Raju
for arranging a short-term visiting position at the International Centre for
Theoretical Sciences.

##  2\. Exact Results

As we discussed in the previous section, it is our goal in this paper to
explore the use of the replica ansatz in computing free energies in random
matrix theory. Concretely, we consider a system governed by a random
Hamiltonian $H$ drawn from a Gaussian unitary ensemble (GUE). Each instance of
this ensemble is associated to a partition function

(2.1) $Z(\beta)=\operatorname{Tr}e^{-\beta H}\ ,$

where $H$ is a Hermitian $N\times N$ matrix and $\beta=T^{-1}$ is the inverse
temperature. Quantities of interest in this theory are averaged over the GUE
like so:

(2.2) $\left\langle
f(H)\right\rangle=\frac{1}{\mathsf{Z}}\int\text{d}H\,e^{-\frac{N}{2}\operatorname{Tr}H^{2}}f(H)\
,$

where $\mathsf{Z}$ is defined such that $\left\langle 1\right\rangle=1$. In
particular, the correlation function we are interested in is the “quenched”
free energy $\left\langle\log Z(\beta)\right\rangle$, which according to the
replica ansatz is given by

(2.3) $\langle\log Z(\beta)\rangle=\lim_{n\rightarrow 0}\frac{\left\langle
Z(\beta)^{n}\right\rangle-1}{n}\ .$

For this reason, the objects of central interest to us in this note will be
$\left\langle Z(\beta)^{n}\right\rangle$, which we will occasionally call
$n$-point functions without qualification.

Happily, the correlation functions $\left\langle Z(\beta)^{n}\right\rangle$ in
random matrix theory are well-known from the study of $\tfrac{1}{2}$-BPS
Wilson loops in four-dimensional $\mathcal{N}=4$ super Yang-Mills theory, as
was highlighted in [delCampo:2017bzr, Okuyama:2018yep]. Formally, the results
of these computations are related by the replacement

(2.4) $\sqrt{\lambda}\leftrightarrow 2\beta\ ,$

where $\lambda$ is the ’t Hooft coupling of the four-dimensional gauge theory.
In light of this correspondence, we will frequently use results from the gauge
theory and the matrix model interchangeably.

It will be useful for us to observe that for any integer $n$, the correlation
functions $\left\langle Z(\beta)^{n}\right\rangle$ may be decomposed into sums
of connected components — identified as
$\left\langle\,\cdot\,\right\rangle_{c}$ — and each contribution to this
decomposition is labelled by a partition of the integer $n$, which is
canonically associated to a Young diagram

(2.5) $Y=\left[1^{k_{1}}\cdots j^{k_{j}}\cdots n^{k_{n}}\right]\ ,$

which, for all $j\in\\{1,\cdots,n\\}$, has $k_{j}$ rows of $j$ boxes. Of
course, we will require that $|Y|$, the total number of boxes, is $n$:

(2.6) $|Y|=\sum_{j}jk_{j}=n\ .$

With this in place, the correlation function is decomposed as

(2.7) $\left\langle Z(\beta)^{n}\right\rangle=\sum_{|Y|=n}Z_{Y}(\beta)\ ,$

where each contribution labelled by a partition of $n$ (alternatively, an
$n$-box Young diagram) is given by

(2.8)
$Z_{Y}(\beta)=n!\,\prod_{j=1}^{n}\frac{1}{k_{j}!}\left[\frac{\left\langle
Z(\beta)^{j}\right\rangle_{c}}{j!}\right]^{k_{j}}\ .$

The $n$-point function thus receives contributions from all partitions of $n$.
As the temperature is varied, the various partitions of $n$ jockey for
dominance within the sum, and it is conceivable that there will exist some
temperatures at which dominance is traded. A simpler and more explicit version
of this phenomenon is presented in [Engelhardt:2020qpv], where in their
discussion of the $\widehat{\mathrm{CGHS}}$ model, they show that at lower
temperatures connected correlators dominate the partition sum. We will briefly
discuss this theory in the final section.

The $1$-point function $\left\langle Z(\beta)\right\rangle$ in random matrix
theory was computed exactly at finite $N$ [Drukker:2000rr] and it is given in
terms of associated Laguerre polynomials as

(2.9) $\langle
Z(\beta)\rangle=e^{\frac{\beta^{2}}{2N}}L_{N-1}^{1}\left(-\frac{\beta^{2}}{N}\right)\
.$

For future convenience, let us define

(2.10) $A_{k}=\left\langle Z(k\beta)\right\rangle\ .$

Higher-point correlators of the partition function can also be computed in
terms of these $A_{k}$, as was demonstrated in [Okuyama:2018aij] through the
following contour integral formula:

(2.11) $\left\langle
Z(\beta)^{n}\right\rangle_{c}=\oint\prod_{i=1}^{n}\frac{\text{d}z_{i}}{2\pi\mathrm{i}\,z_{i}^{2}}\operatorname{Tr}\log\left[\sum_{m=0}^{n}\sum_{i_{1}<\cdots<i_{m}}z_{i_{1}}\cdots
z_{i_{m}}A_{m}\right]\ .$

We now propose a much simpler formula that captures the same content as the
above equation:

(2.12) $\left\langle
e^{tZ(\beta)}\right\rangle_{c}=1+\log\left[1+\sum_{k=1}^{\infty}A_{k}\frac{t^{k}}{k!}\right]\
,$

Observe that the l.h.s. of eq. 2.12 is in fact the generating function of
connected correlators of the partition function. The above expression allows
us to relate connected higher-point correlators of the partition function to
the $A_{k}$, in addition to sidestepping the difficulty of performing a multi-
dimensional contour integral. In particular, by expanding each side in a
Taylor series around $t=0$, we recover the same content as eq. 2.11. For
example, for low $k$ we find by simple Taylor expansions

$\displaystyle\left\langle Z(\beta)^{3}\right\rangle_{c}$
$\displaystyle=A_{3}-3A_{1}A_{2}+2A_{1}^{3}\ ,$ $\displaystyle\left\langle
Z(\beta)^{4}\right\rangle_{c}$
$\displaystyle=A_{4}-4A_{1}A_{3}-3A_{2}^{2}+12A_{1}^{2}A_{2}-6A_{1}^{4}\ ,$
$\displaystyle\left\langle Z(\beta)^{5}\right\rangle_{c}$
$\displaystyle=A_{5}-5A_{1}A_{4}-10A_{2}A_{3}+20A_{1}^{2}A_{3}+30A_{1}A_{2}^{2}-60A_{1}^{3}A_{2}+24A_{1}^{5}\
,$

and it is easily verified that these expressions match the results of the
contour integral representation.

That eq. 2.12 is correct is easy to see once we recognise that it is simply an
“inversion” of the standard relation between a correlation function and its
connected part:

(2.13) $\left\langle
e^{tZ(\beta)}\right\rangle=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\left\langle
Z(\beta)^{n}\right\rangle=\exp\left[\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\left\langle
Z(\beta)^{j}\right\rangle_{c}\right]\ .$

The above relation tells us that the correlation function $\left\langle
Z(\beta)^{n}\right\rangle$ is given by the $n$-th Bell polynomial
$B_{n}(x_{1},\cdots,x_{n})$ in the connected correlation functions

(2.14) $x_{k}=\left\langle Z(\beta)^{k}\right\rangle_{c}\ .$

Essentially, where the $\left\langle Z(\beta)^{n}\right\rangle$ are like
moments of a distribution, the connected correlation functions $\left\langle
Z(\beta)^{n}\right\rangle_{c}$ are like cumulants of a distribution.111See
[McCullagh:2009] for more details.

It will be useful for us to recall also the large-$N$ limits of eq. 2.9. It
was first observed in [Erickson:2000af] that

(2.15) $\langle Z(\beta)\rangle=\frac{N}{\beta}I_{1}(2\beta)\ ,$

where $I_{\nu}(z)$ is a modified Bessel function. Since we have succeeded in
expressing all connected $n$-point correlators in terms of the $A_{k}$ using
eq. 2.10 and eq. 2.12, this result is sufficient for our purposes.

##  3\. Replica Ansatz

Our interest is in computing the quenched free energy using the replica
ansatz:

(3.1) $\langle\log Z(\beta)\rangle=\lim_{n\rightarrow 0}\frac{\left\langle
Z(\beta)^{n}\right\rangle-1}{n}\ .$

The trouble with implementing the replica ansatz, as we have discussed, arises
at low temperatures. Before discussing this, we quickly recall some results at
high temperature.

### High Temperatures

Let us first consider the behaviour of $\left\langle
Z(\beta)^{n}\right\rangle$ at high temperatures. In this regime, the dominant
contribution is the totally disconnected correlator corresponding to the
partition $Y=\left[1^{n}\right]$. Thus, at high temperatures we have

(3.2) $\left\langle Z(\beta)^{n}\right\rangle\simeq\left\langle
Z(\beta)\right\rangle^{n}\ ,$

and the replica ansatz eq. 3.1 finds that

(3.3) $\left\langle\log Z(\beta)\right\rangle\simeq\log\left\langle
Z(\beta)\right\rangle\ .$

That is, at high temperatures the quenched and annealed free energies agree
and the replica ansatz returns an answer that is physically reasonable; as
[Okuyama:2020mhl] notes and as one can observe from eq. 2.15, in the the limit
$\beta\rightarrow 0$ we have

(3.4) $\lim_{\beta\rightarrow 0}Z(\beta)=N\ ,$

and so the quenched free energy computed using eq. 3.1 approaches $\log N$,
the maximal entropy of the system.

### Low Temperatures

At low temperatures, however, the application of the replica ansatz frustrated
by well-known ambiguities involving analytic continuation. For example, it was
shown through the numerical studies of [Okuyama:2019xvg] that at low
temperatures the dominant contribution to the $n$-point function comes from
the fully connected correlator labelled by $Y=\left[n^{1}\right]$, and this in
turn implies the following simplification:

(3.5) $\left\langle Z(\beta)^{n}\right\rangle\simeq A_{n}=\left\langle
Z(n\beta)\right\rangle\ .$

Unfortunately, when the above simplification is plugged into the replica
ansatz in eq. 3.1, the limit is ill-defined and in this way the replica ansatz
appears to fail. The temperature at which quenched and annealed quantities
begin to differ — a signature of what is referred to as replica symmetry
breaking — is typically denoted $T_{\mathrm{RSB}}$.

It will be important in subsequent sections to note that the demonstration of
connected correlator dominance at low temperatures was carried out for integer
$n$. This is interesting in its own right, but for the purposes of the replica
ansatz is inconclusive since the replica ansatz necessitates a departure from
integer $n$. In other words: the dominance of $Y=\left[n^{1}\right]$ for
integer $n$ in no way guarantees its dominance as $n$ is analytically
continued away from the integers.

The goal of the subsequent sections will be to argue that the replica ansatz
can in fact return physically reasonable answers at low temperatures. For
this, we must first chart the landscape of contributions to the free energy.

### Parametrising The Landscape

In eq. 2.12, we have seen how to express connected correlation functions in
terms of sums of products of the functions $A_{k}$. Schematically, each
contribution to an $n$-point function is, schematically, a product of $p$
factors of the form

(3.6) $A_{k_{1}}\cdots A_{k_{p}}\quad\text{such
that}\quad\sum_{i=1}^{p}k_{i}=n\ .$

The replica ansatz requires that we analytically continue from
$n\in\mathbb{Z}_{+}$ to $n\in\mathbb{R}_{\geq 0}$; crucially, we will retain
the above constraint on the $k_{i}$, i.e. that they sum to $n$.

An important observation of [Okuyama:2019xvg] is that near the the temperature
at which replica symmetry breaking occurs, all partitions of $n$ contribute in
roughly equal measure to any computation of the free energy. There is, thus,
at $T_{\mathrm{RSB}}$, a landscape of free energies which contribute in
roughly equal magnitude to the partition sum, and there is no clear dominance
of one partition over another.

This picture of the landscape of free energies is productive because at fixed
integer $n$, we can explicitly see an exchange of dominance from totally
connected to totally disconnected correlation functions as the temperature is
ramped up. At fixed $\beta$, however, varying $p$ essentially interpolates
between different partitions.

In light of this, we suppose that within the replica ansatz, it is _a priori_
unclear which partition dominates at low temperatures. (This is to be
contrasted with the case of integer $n$, where we know that the partition
$\left[n^{1}\right]$ dominates.) We should then attempt to parametrise the
landscape of contributions to the free energy and extremise it with respect to
these parameters.

The simplest such parametrisation is to suppose that a “rectangular” partition

(3.7) $\underbrace{\left(p,\,\cdots,p\right)}_{\frac{n}{p}\,\mathrm{times}}$

is dominant. This proposal takes a leaf out of earlier work on the random
energy model [Derrida:1981zz] and was more recently suggested by
[Okuyama:2020mhl]. Observe that $p=1$ corresponds to the totally disconnected
correlator (dominant at high temperatures), $p=n$ corresponds to the totally
connected correlator (dominant at low temperatures when $n$ is an integer),
and intermediate values of $p$ correspond to other allowed terms. It is clear
that $p$ parametrises the shape of the Young diagram, and such a term
contributes the following to the $n$-point function:

(3.8) $\left(A_{p}\right)^{n/p}\ .$

For some $p=p_{\star}$ this quantity will be extremal — this contribution,
when inserted into the replica ansatz, ought to correspond to the correct free
energy.

### Low Temperatures Revisited

Let us try and compute the quenched free energy at large $\beta$ (low
temperatures) using the technique we have just discussed. We start with

(3.9)
$\left(A_{p}\right)^{n/p}\simeq\left[\frac{N}{(p\beta)^{3/2}}e^{2p\beta}\right]^{n/p}\
,$

where we have ignored numerical factors that appear in the asymptotic
expansion for the modified Bessel functions since they will not matter.
Extremising this with respect to $p$, we find that

(3.10) $p_{\star}=e\left(\frac{N^{2/3}}{\beta}\right)\ ,$

which when we plug back into eq. 3.9 gives us

(3.11)
$\left(A_{p_{\star}}\right)^{n/p_{\star}}=\exp\left\\{n\beta\left(2-\frac{3}{2eN^{2/3}}\right)\right\\}\
.$

Now, using the standard thermodynamic prescription

(3.12) $\displaystyle F$ $\displaystyle=-\beta^{-1}\langle\log Z\rangle\ ,$
$\displaystyle=-2+\frac{3}{2eN^{2/3}}\ ,$ $\displaystyle\simeq-2\ ,$

where in the last line we have dropped terms that are negligible in the limit
$N\rightarrow\infty$.

This is in fact consistent with expectations first elaborated in
[Okuyama:2020mhl]: it is natural to expect that for finite $N$ and
$\beta\rightarrow\infty$ (low temperatures) only the smallest eigenvalue of
$H$ will contribute to

(3.13) $F=-\lim_{\beta\rightarrow\infty}\beta^{-1}\left\langle\log
Z(\beta)\right\rangle=-\lim_{\beta\rightarrow\infty}\beta^{-1}\left\langle\log\sum_{i=1}^{N}e^{-\beta
E_{i}}\right\rangle=E^{(N)}_{\mathrm{min.}}\ ,$

and further, that at large-$N$, $E^{(N)}_{\mathrm{min.}}$ approaches the left
edge of the Wigner semi-circle distribution, which in our matrix integral
conventions is $E^{(\infty)}_{\mathrm{min.}}=E_{0}=-2$, so

(3.14) $\lim_{\beta\rightarrow\infty}F=E_{0}\ .$

This computation may be interpreted as confirmation at large-$N$ of a
condition on analytic continuation in the replica index $n$

(3.15) $\lim_{\beta\rightarrow\infty}\left\langle
Z(\beta)^{n}\right\rangle=e^{-n\beta E_{0}}\ ,$

originally proposed in [Okuyama:2020mhl]. While the original condition was
explicitly tested at finite $N$ using a combination of analytic and numerical
methods, the above can be seen as a large-$N$ version of the same
demonstration that successfully employs the replica ansatz.

##  4\. Discussion

Before we conclude, it is interesting to consider the case of
$\widehat{\mathrm{CGHS}}$ gravity and ask if a similar strategy would work
there too. We recall some basic facts about this dilaton gravity, and refer
the reader to [Godet:2021cdl] for more details.

The original CGHS model is a dilaton gravity theory in two dimensions
[Callan:1992rs]. Solutions to this theory are flat, and have a fixed
temperature determined by the cosmological constant and the boundary value of
the dilaton. To get solutions with different values of temperatures, following
[Afshar:2019axx] we use the action

(4.1)
$I_{\widehat{\mathrm{CGHS}}}=-\frac{\kappa}{2}\int\text{d}^{2}x\sqrt{g}\,\left(\Phi
R+2\Lambda-2\Lambda\varepsilon^{\mu\nu}\partial_{\mu}A_{\nu}\right)\ ,$

and note that the above action reproduces the dynamics of the original CGHS
model, except this time $\Lambda$ can take any constant value, thereby
allowing for solutions for varying temperatures. A judicious choice of
boundary conditions selects geometries with asymptotic boundaries that are
thermal circles, and since dilaton path integral constrains these geometries
to be locally flat, they can only be discs or cylinders. The full
gravitational path integral (with an appropriate boundary action and a
topological term that controls the topological expansion) on a manifold with
$n$ boundaries can be computed just as in the case of JT gravity.

The connected correlators in this theory were computed in [Godet:2020xpk] and
are given by

(4.2) $\left\langle
Z(\beta)\right\rangle=\frac{2\pi}{\beta^{2}}\quad,\quad\left\langle
Z(\beta)^{2}\right\rangle_{c}=\frac{2\pi^{2}}{\beta}\quad,\quad\left\langle
Z(\beta)^{n\geq 3}\right\rangle_{c}=0\ ,$

since there are no flat connected surfaces with $n\geq 3$ connected
boundaries. That is, $Z(\beta)$ can be thought of as a Gaussian-distributed
random variable with non-zero mean and variance, both specified by the
temperature as above. For such a theory, the quantity $F_{\mathrm{q}}$ is ill-
defined, since $Z(\beta)$ takes values on the entire real line. For this
reason, we _define_ the quenched free energy of $\widehat{\mathrm{CGHS}}$
gravity, following [Godet:2020xpk, Engelhardt:2020qpv], to be

(4.3) $F_{\mathrm{q}}=-\beta^{-1}\left\langle\log|Z(\beta)|\right\rangle\ .$

That $Z(\beta)$ may be thought of as a Gaussian-distributed random variable is
interesting in light of the wormhole network picture of [Okuyama:2019xvg],
where all $n$-point functions are decomposed in terms of Young diagrams /
partitions of $n$, each corresponding to a product of connected parts. The
constraint eq. 4.2 tells us that $(n\geq 3)$-point correlators vanish
identically, which means only _restricted_ partitions are permitted. For
example, while the (unrestricted) partitions of $n=4$ are

$\left\\{\quad\tiny\yng(1,1,1,1)\quad,\quad\tiny\yng(2,1,1)\quad,\quad\tiny\yng(2,2)\quad,\quad\tiny\yng(3,1)\quad,\quad\tiny\yng(4)\quad\right\\}\
,$

we see that the set of (restricted) partitions that contribute to the
$4$-point multi-boundary correlators in $\widehat{\mathrm{CGHS}}$ gravity are

$\left\\{\quad\tiny\yng(1,1,1,1)\quad,\quad\tiny\yng(2,1,1)\quad,\quad\tiny\yng(2,2)\quad\right\\}\
.$

In general, for any $n$, Young diagrams with rows having more than two columns
are disallowed. We conclude from this that an $n$-point function can be
written as

(4.4) $\left\langle Z(\beta)^{n}\right\rangle=\left\langle
Z(\beta)\right\rangle^{n}\sum_{k=0}^{n}T_{n,k}\,\frac{\left\langle
Z(\beta)^{2}\right\rangle_{c}}{\left\langle Z(\beta)\right\rangle^{2}}\ ,$

where

(4.5) $T_{n,k}=\frac{n!}{k!\,(n-2k)!\,2^{k}}\ ,$

are Bessel numbers, which tell us how many ways there are to partition $n$
objects into $k$ sets, each of size one or two.222For more details, see
[Cheon:2013].

This brings us to an important point: the restricted nature of the partitions
in $\widehat{\mathrm{CGHS}}$ gravity means our method of implementing the
replica ansatz, which involved a variational problem that solved for the width
of dominant rectangular Young diagrams, is unlikely to find purchase in this
context.

Although our work has dealt solely with the replica ansatz, we would be remiss
if we did not mention a new, exact relation quantifying the difference between
quenched and annealed free energies first reported in [Okuyama:2021pkf] and
explicated elegantly in [Johnson:2021rsh]. This relation, which essentially
derives from an integral representation of the natural logarithm:

(4.6) $\log
x=\int_{0}^{\infty}\frac{e^{-u}-e^{-ux}}{u}\mathrm{\leavevmode\nobreak\ d}u\
,$

reads

(4.7) $\langle\log Z\rangle=\log\langle
Z\rangle-\int_{0}^{\infty}\frac{\mathrm{d}u}{u}\left[\left\langle
e^{-Zu}\right\rangle-e^{-\left\langle Z\right\rangle u}\right]\ .$

It has been suggested that the term quantifying the difference between
quenched and annealed free energies — which, as we will see, cleverly encodes
the contributions of connected correlators — corresponds to a superposition of
spacetime D-branes [Marolf:2020xie]. Further, subsequent work
[Alishahiha:2020jko, Janssen:2021mek, Johnson:2021zuo] explicitly finds that
the quenched free energy of JT gravity and its deformations is well-behaved,
i.e. is monotonically decreasing.

We can, for example, still use eq. 4.7 to compute the quenched free energy of
$\widehat{\mathrm{CGHS}}$ gravity as defined above exactly. This computation
is simplified by the fact that only connected correlators factor into the term
quantifying the difference between quenched and annealed free energies, and as
we have seen, almost all of them vanish. A simple calculation, which requires
analytic continuation from $\beta<0$ followed by picking out the real part,
fully analogous to the prescription implied by eq. 4.3, finds

(4.8) $F_{\mathrm{q}}=F_{\mathrm{a}}+\frac{T}{2}\left[\gamma+\log
4T^{3}-2T^{3}\times{}_{2}F_{2}\left(1,1;\tfrac{3}{2},2;-T^{3}\right)\right]\
.$

This confirms a result of [Godet:2021cdl].

## References