# Pressure tuning the Jahn-Teller transition temperature in NaNiO2

Liam A. V. Nagle-Cocco<EMAIL_ADDRESS>Cavendish Laboratory, University of
Cambridge, JJ Thomson Avenue, Cambridge, CB3 0HE, United Kingdom.  Craig L.
Bull  ISIS Neutron and Muon Facility, Rutherford Appleton Laboratory, Didcot,
OX11 0QX, United Kingdom.  School of Chemistry, University of Edinburgh, David
Brewster Road, Edinburgh, EH9 3FJ, United Kingdom.  Christopher J. Ridley
ISIS Neutron and Muon Facility, Rutherford Appleton Laboratory, Didcot, OX11
0QX, United Kingdom.  Siân E. Dutton<EMAIL_ADDRESS>Cavendish Laboratory,
University of Cambridge, JJ Thomson Avenue, Cambridge, CB3 0HE, United
Kingdom.

###### Abstract

Abstract: NaNiO2 is a layered material consisting of alternating layers of
NaO6 and Jahn-Teller-active NiO6 edge-sharing octahedra. At ambient pressure
it undergoes a broad phase transition from a monoclinic to rhombohedral
structure between $\sim$465 K and $\sim$495 K, associated with the loss of
long-range orbital ordering. In this work, we present the results of a neutron
powder diffraction study on powdered NaNiO2 as a function of pressure and
temperature from ambient pressure to $\sim$5 GPa between 290 K and 490 K. The
290 K and 460 K isothermal compressions remained in the monoclinic phase up to
the maximum pressures studied, whereas the 490 K isotherm was mixed-phase
throughout. The unit-cell volume was fitted to a 2nd-order Birch-Murnaghan
equation of state, with $B=113(1)$ GPa. We observe at 490 K that the fraction
of Jahn-Teller-distorted phase increases with increasing pressure, from
67.8(6)% at 0.71(2) GPa to 80.2(9)% at 4.20(6) GPa. Using this observation, in
conjunction with neutron diffraction measurements at 490 K from 5.46(9) GPa to
0.342(13) GPa, we show that the Jahn-Teller transition temperature increases
with pressure. Our results are used to present a structural pressure-
temperature phase diagram for NaNiO2. To our knowledge, this is the first
diffraction study of the effect of pressure on the Jahn-Teller transition
temperature in materials with edge-sharing Jahn-Teller-distorted octahedra,
and the first variable-pressure study on a JT-active edge-sharing nickelate.

##  1 Introduction

Many transition metal oxides exhibit a Jahn-Teller (JT) distortion due to
degeneracy in the 3$d$ orbitals, manifesting as an elongation or compression
of the MO6 (M=transition metal ion) octahedra, generally with associated
orbital ordering. Previous studies on the effect of pressure on materials
containing JT-active ions have found that pressure can entirely suppress the
JT distortion and orbital ordering 1, 2. It has also been observed that
application of pressure reduces the magnitude of distortion in $M$O6 octahedra
3, 4, 5, 6, 7.

One well-studied material under pressure is LaMnO3 (with JT active d4 Mn3+
ions) 3, 8, 2. At ambient pressure it adopts the perovskite structure with
corner-sharing MnO6 octahedra. An ordered JT distortion results in an
orthorhombic symmetry. At $T>\sim 750$ K the JT distortion is suppressed, and
there is an increase in symmetry first to a cubic phase with octahedral
tilting, and then at higher temperatures to a rhombohedral phase 9. The
temperature-driven suppression of the JT distortion coincides with a marked
increase in electronic conductivity 10. On application of pressure at room
temperature $P<8$ GPa, the JT distortion is decreased through reduction of the
long Mn-O bond lengths 3. At $\sim$11 GPa a rhombohedral phase with no JT
distortion co-exists with the distorted orthorhombic phase 2, becoming single-
phase at $\sim$12 GPa. Similarly, the manganese (III) quadruple perovskite
LaMn7O12, exhibits a complete suppression of the JT distortion at $\sim$34 GPa
1.

There are several interesting studies on JT-distorted compounds with edge-
sharing octahedra. Here we describe three different examples classes, all
containing JT-active $d^{4}$ Mn3+. Mn3O4, a spinel containing both Mn3+ and
Mn2+, has been found to exhibit different pressure-dependence of JT-distorted
octahedra depending on morphology; for instance, in single-crystal Mn3O4 the
JT disortion survives to 60 GPa 11, whereas there are observed transitions to
JT-free phases at much lower pressures in powdered 12 and nanorod 13 Mn3O4.
ZnMnO2, also with spinel-type structure but with Zn2+ on the Mn2+ of Mn3O4,
has been studied to very high pressure ($\sim 52$ GPa) 14, with a transition
reported at $\sim 23$ GPa which has been alternately described as a transition
from JT-elongation to a slight JT-compression 14 or a spin-crossover
transition resulting in an insulator$\rightarrow$metal transition 15. CuMnO2,
with delafossite structure, has also had the pressure-dependence of its JT
distortion studied 16. It exhibits a higher compressibility in the long Mn-O
bond than the short Mn-O bond similar to LaMnO3 3 and other materials 4, 5 up
to around around $\sim 10$ GPa; above this pressure there is an isostructural
phase transition associated with a collapse in the interlayer (c-axis) and an
increase in the volume of the Mn3+O6 JT-distorted octahedra.

Nickelates containing JT-active $d^{7}$ Ni3+ are far less studied than the
manganates under pressure. This may be partly because many materials
containing $d^{7}$ Ni3+ octahedra do not exhibit a co-operative JT distortion,
where the JT distortion is long-range ordered. NdNiO3, which has been subject
to a variable-pressure structural study 17, is not considered to contain a JT
distortion 18, 19, 20, as is the case for most Ni-containing perovskites 21.
Similarly, AgNiO2 is widely accepted to not contain any kind of JT distortion
22, 23. LiNiO2 is an interesting case as it does not display long-range
magnetic or orbital ordering, likely due to Li/Ni site mixing; some
experimental results have been interpreted as evidence for a non-cooperative
JT distortion 24, 25, although this is debated 26, 27, 28. Similarly various
nickel-containing perovskites 29 are subject to discussion regarding whether
there exists any kind of JT distortion.

NaNiO2 is a layered $d^{7}$ nickelate. The presence of the JT distortion in
NaNiO2 is not subject to debate 30, 31, 28, even amongst proponents of
alternative theories for degeneracy-breaking in LiNiO2 28. NaNiO2 is therefore
an ideal choice for studying the effect of pressure on the JT distortion in a
material which is both a nickelate and has edge-sharing octahedra. The room-
temperature phase of NaNiO2 is a semiconductor, based on its black colour and
by analogy with LiNiO2 32, but we do not know of any measurement of the
conductivity properties of the high-temperature phase. NaNiO2 is of interest
because of its magnetic ground state, consisting at ambient pressure of intra-
layer ferromagnetism and inter-layer anti-ferromagnetism 33, 34, 35. It has
also been studied in recent years because $A$NiO2 ($A=\mathrm{alkali\ metal}$)
is the template compound for Ni-rich alkali metal-transition metal oxides
within the field of batteries 36, 37.

NaNiO2 has an ordered JT distortion at room temperature due to degeneracy in
$e_{g}$ orbitals in low-spin Ni3+. It exhibits a first-order phase transition
between 465 K and 495 K to an undistorted phase. The crystal structures are
shown in Figure 1. The monoclinic ($C2/m$) JT-distorted phase consists of
alternating layers of edge-sharing NiO6 and NaO6 octahedra. Both the NiO6 and
NaO6 octahedra exhibit angular and bond-length distortions from geometrically
regular octahedra. Ni, Na, and O ions occupy the 2a(0,0,0),
2d(0,$\frac{1}{2}$,$\frac{1}{2}$), and 4i($x$,0,$z$) Wyckoff sites
respectively. The rhombohedral ($R\bar{3}m$) phase consists of the same
arrangement of alternating NiO6 and NaO6 layers of edge-sharing octahedra,
with octahedra bound within layers by O4 tetrahedra. In this phase, Ni, Na,
and O ions occupy the 3b(0,0,$\frac{1}{2}$), 3a(0,0,0), and 6c(0,0,$z$)
Wyckoff sites respectively. The unit cell remains centrosymmetric, with the
change in symmetry due solely to the repression of the JT distortion and the
resulting non-variable $r_{\mathrm{\textit{M}-O}}$ bond lengths.

Figure 1: (a) and (b) show the monoclinic, Jahn-Teller distorted NaNiO2 phase
along the $b$-axis and $c$-axis respectively. (c) and (d) show the
rhombohedral, JT-inactive NaNiO2 phase along the $b$ and $c$ axes
respectively. Ni3+ cations are shown in green, O2- anions in red, and Na+
cations in cyan. Na+ ions and octahedra are hidden in (b) and (d). The solid
black quadrilaterals denote the unit cell. The black arrows represent the
directions of principal axes of compression projected into the (a,c)
$ac$-plane and (b,d) $ab$-plane.

In this work, we present a structural study of NaNiO2 as a function of
temperature between 290 K and 500 K, and pressure up to 5.46(9) GPa. We
demonstrate using the 490 K isotherm that the JT transition temperature
increases between 2 GPa and 4.2 GPa, increasing more rapidly with pressure at
higher pressures, while the degree of distortion decreases over this pressure
range.

##  2 Methods

Sample preparation and characterisation. Samples were prepared by solid state
synthesis. Na2O2 (Alfa Aesar; 95%) and NiO (Alfa Aesar; 99.995%) were mixed
and pelletised in a 1.05:1 molar ratio of Na:Ni, with excess Na to account for
Na-loss during heating. Sample was heated to 973 K for 70 hrs in a tube
furnace under constant flow of O2. To prevent reaction with moisture, the
sample was stored and handled in an inert Ar-atmosphere. X-Ray Diffraction
(XRD) data were obtained using a Bruker D8 Discover powder (Cu
K$\alpha_{1,2}$, $\lambda=1.541$ Å) diffractometer. A Mira3 TESCAN Scanning
Electron Microscopy was used to obtain SEM images of the morphology of NaNiO2,
with an accelerating electron voltage of 3 kV (for SEM images, see SI).

Ambient-pressure neutron diffraction. Ambient-pressure neutron diffraction was
performed using the NOMAD instrument 38 at the Spallation Neutron Source, Oak
Ridge National Laboratory, USA. NaNiO2 was sealed in a glass ampoule for the
measurements. Heating was performed using a furnace. The sample was measured
during heating at 293 K, 450 K, 500 K, and after cooling at 316 K.

Variable-pressure neutron diffraction. Variable-temperature and -pressure
neutron diffraction studies were performed at the PEARL instrument 39
($2\theta=90^{\circ}$), ISIS Neutron and Muon Source, UK, using a V3 Paris-
Edinburgh press. The sample was measured between 0.107(8) GPa and 4.24(5) GPa
at 290 K, 0.130(10) GPa and 5.29(8) GPa at 460 K, and 0.254(17) GPa and
4.20(6) GPa at 490 K. NaNiO2 was packed into a encapsulated null scattering
TiZr gasket which was loaded in a zirconia-toughened alumina toroidal profile
anvil, with a lead pellet for pressure calibration 40. Anhydrous deuterated
methanol:ethanol (4:1 by volume) was used as a pressure transmitting medium
for the ambient temperature isothermal compression experiment. Preliminary
measurements indicated that NaNiO2 reacted with the methanol:ethanol solution
at higher temperatures [SI Figure S2], and so a 1:1 ratio by volume of
FC77:FC84 fluorinert (purchased from 3M) was used for the 460 K and 490 K
isotherms. The data were processed and corrected using Mantid 41.

Diffraction Analysis. Diffraction data were analysed using the software
package topas 5 42, utilising Pawley fitting 43 and Rietveld refinement 44.
For NaNiO2, preliminary analysis of NOMAD data indicated Na occupancy was 1
within error, hence site occupancy of all sites during all further refinement
was fixed at 1. Thermal Beq parameters were allowed to refine but constrained
to be positive and not exceed a value of 5 Å2. All atomic positions were
refined within symmetry constraints. The background was fitted by a Chebyschev
polynomial (order 6 for PEARL data, order 11 for NOMAD data, order 19 for XRD
data). For XRD data, a TCHZ peak-shape was used 45. Peak-shapes used for
neutron data are discussed in SI Section II. For PEARL only the 90∘ detection
bank was used, but for NOMAD a combined refinement was performed using banks
2-5 ($2\theta=31^{\circ},67^{\circ},122^{\circ},154^{\circ}$ respectively).

##  3 Results

###  3.1 Ambient-pressure structural properties.

Powder X-ray diffraction of the as-synthesised NaNiO2 indicated the formation
of a phase-pure product. SEM on the material indicates the sample is
polycrystalline with particulates between 0.2 $\micro$m and 5 $\micro$m in
diameter [SI Figure S13]. Rietveld refinement using the reported monoclinic
$C2/m$ space group [SI Figure S1; Table S1] yielded lattice parameters
consistent with prior reports 30, 46.

The reported monoclinic$\Rightarrow$rhombohedral phase transition in NaNiO2
was investigated using neutron powder diffraction at ambient pressure on the
NOMAD instrument. Rietveld refinement [Figure 2] shows the phase transition
occurs between 450 K and 500 K, and is reversible on cooling. The lattice
parameters [SI Table S2] all exhibit positive thermal expansion, and are
consistent with previous measurements 30, 31.

Figure 2: Rietveld refinements for the ambient-pressure, variable-temperature
neutron diffraction measurements of NaNiO2 on bank 5 of NOMAD
($2\theta=154^{\circ}$) at 293 K (top) and 500 K (bottom). Black dots:
measured data; red line: calculated diffraction pattern from Rietveld
refinement; grey line: $\mathrm{Y}_{\mathrm{obs}}-\mathrm{Y}_{\mathrm{calc}}$.
Green and orange tick marks show expected reflections for the monoclinic and
rhombohedral phase respectively.

In the monoclinic structure the NiO6 octahedra exhibit a cooperative JT
distortion with 2 longer Ni-O bonds, whereas in the high-temperature
rhombohedral phase all six Ni-O bonds are equivalent [Figure 1]. The degree of
bond length distortion within individual NaO6 and NiO6 octahedra can be
evaluated using a number of distortion metrics, calculated using topas 5 42.
Here we consider the effective coordination number 47 and the bond length
distortion indices 48, which measure distortion in octahedra by quantifying
the difference from the average value of the distances between the central
cation and the coordinated oxygen anions. The general form of effective
coordination, $\mathrm{ECoN}$, and bond length distortion index, $D$, is given
in the SI. The equations as applicable to monoclinic NaNiO2 are:

$\mathrm{ECoN}=4\exp{\left[1-\left(\frac{l_{\mathrm{short}}}{l^{\prime}_{\mathrm{av}}}\right)^{6}\right]}+2\exp{\left[1-\left(\frac{l_{\mathrm{long}}}{l^{\prime}_{\mathrm{av}}}\right)^{6}\right]},$
(1)

where $l^{\prime}_{\mathrm{av}}$ is a modified average bond length defined in
SI, and;

$D=\frac{1}{3}\frac{l_{\mathrm{long}}-l_{\mathrm{av}}}{l_{\mathrm{av}}}+\frac{2}{3}\frac{l_{\mathrm{av}}-l_{\mathrm{short}}}{l_{\mathrm{av}}}$
(2)

where $l_{\mathrm{av}}$ is the average bond length, and $l_{\mathrm{long}}$
and $l_{\mathrm{short}}$ are the long and short bonds respectively.

In the rhombohedral structure the effective coordination number and bond
length distortion index of both the NiO6 and NaO6 octahedra are constrained by
symmetry to values of 6 and 0 respectively. In the monoclinic structure,
departure from these values indicates bond length disproportionation and is
primarily attributable to the JT distortion. These changes are significantly
larger for the JT-active NiO6 octahedra than the NaO6 octahedra. Throughout
the measurement, the bond length distortion index
$\\{\mathrm{effective~{}coordination}\\}$ of NaO6 octahedra remains very near
its high-symmetry value of 0$\\{6\\}$, for example at 293 K the value of bond
length distortion index $\\{\mathrm{effective~{}coordination}\\}$ in NaO6
octahedra is 0.00581(11)$\\{5.99232(19)\\}$, compared with
0.05463(14)$\\{5.309(3)\\}$ in the NiO6 octahedra. This is indicative of much
greater distortion in bond lengths for NiO6 octahedra, consistent with the JT
distortion. The values of bond length distortion index are on the same order
of magnitude as recent studies on JT-distorted Mn3+O6-containing compounds 49,
16.

Inconsistency in bond length is not the only distortion of the octahedra from
regular octahedra. A regular octahedron would have bond angles
$\theta_{\mathrm{O-}M\mathrm{-O}}=90^{\circ}$ for nearest-neighbour O anions.
However, in both the JT-active monoclinic phase and the JT-inactive
rhombohedral phase there is variance from this ideal bond angle. Non-nearest-
neighbour oxygen anions are constrained to have 180∘ bond angles via the
central cation, and so the 12 bond angles in an octahedron are each paired
with another O-$M$-O bond, with the paired bond angles sharing one oxygen in
common and with their non-shared oxygen anions occurring along a straight line
through the central cation (for a visual representation, see SI Figure S14).
We define these bond angles as
$\theta_{\mathrm{O-}M\mathrm{-O}}=90^{\circ}\pm\Delta$, where the two angles
in a pair have opposite sign preceding the $\Delta$. $\Delta$ can also be
thought of as a measure of the extent of angular distortion. In the
rhombohedral structure, there is only one value of $\Delta$ for each type of
octahedron, with half the O-$M$-O bond angles being $90^{\circ}+\Delta$ and
the other half being $90^{\circ}-\Delta$. In the monoclinic unit cell where
octahedra have two long $M$-O ($M$=Na,Ni) bonds and four short $M$-O bonds,
there are four nearest-neighbour bond angles between short and short bonds and
eight nearest-neighbour bond angles between short and long bonds. We therefore
must define two values of $\Delta$ for the bond angles in the monoclinic
phase, $\Delta_{\mathrm{short-short}}$ and $\Delta_{\mathrm{long-short}}$,
respectively. Table 1 shows these values of $\Delta$ at each temperature. It
is clear that NaO6 octahedra exhibit far higher bond angle distortion than
NiO6 octahedra, in contrast to the bond length distortion which is greater for
NiO6 octahedra. This is not unexpected, given that crystal field effects will
result in much greater stability for open-shell $d^{7}$ Ni3+ in an octahedral
configuration, minimising bond angle variance, whereas this won’t be a factor
for closed-shell Na+ cations.

Phase | T (K) | NiO6 (∘) | NaO6 (∘)
---|---|---|---
$\Delta^{\mathrm{Ni}}_{\mathrm{short-short}}$ | $\Delta^{\mathrm{Ni}}_{\mathrm{long-short}}$ | $\Delta^{\mathrm{Na}}_{\mathrm{short-short}}$ | $\Delta^{\mathrm{Na}}_{\mathrm{long-short}}$
$C2/m$ | 293 (-) | 6.134(17) | 5.456(19) | 14.494(12) | 9.708(14)
$C2/m$ | 450 ($\uparrow$) | 6.163(19) | 5.50(2) | 14.564(14) | 9.844(16)
$R\bar{3}m$ | 500 ($\uparrow$) | 6.135(15) | 11.777(13)
$C2/m$ | 316 ($\downarrow$) | 6.121(18) | 5.46(2) | 14.505(13) | 9.731(15)
Table 1: This table shows the value of $\Delta$ for bond angles O-$M$-O
($M$=Na, Ni) which take the value $90^{\circ}\pm\Delta$, as a function of
temperature. For definitions, check the main text. The arrow next to the
temperature indicates whether the data were collected on warming or cooling of
the sample.

###  3.2 Variable-pressure neutron diffraction.

The effect of pressure on the JT distortion in NaNiO2 was explored at 290 K,
460 K, and 490 K, with an example Rietveld refinement shown in Figure 3 and
data shown in Figures 4-8. Over the entire pressure and temperature range
studied NaNiO2 could be described using the previously-reported ambient
pressure crystal structures. Diffraction data also included contributions from
alumina and zirconia in the sample environment and the lead used to determined
the applied pressure - these are also included in the structural refinements.
In addition, at higher temperatures (460 K and 490 K) and pressures additional
peaks attributed to crystallisation of the fluorinert pressure media [SI
Figure S3] are observed in the measurements.

Figure 3: Rietveld refinements for the variable-pressure neutron diffraction
data of NaNiO2. Top: a representative plot with monoclinic NaNiO2 only;
bottom: a representative plot with both monoclinic and rhombohedral NaNiO2.
Black dots: measured data; red line: calculated diffraction pattern from
Rietveld refinement; grey line:
$\mathrm{Y}_{\mathrm{obs}}-\mathrm{Y}_{\mathrm{calc}}$. Unfitted peaks are
marked with an asterisk and arise from crystalline fluorinert [SI Figure S3].
Figure 4: Fractional contraction of lattice parameters obtained by Rietveld
refinement 44 as a function of temperature and pressure for the monoclinic
$C2/m$ and rhombohedral $R\bar{3}m$ phases of NaNiO2. Error bars are shown for
all plots; where they are not visible it is because the error is smaller than
the data point. Lines are a guide for the eye.

Rietveld analysis, Figure 3, shows that NaNiO2 remained in the monoclinic
phase at 290 K (up to 4.24(5) GPa) and 460 K (up to 5.29(8) GPa). However, the
measurements at 490 K capture NaNiO2 midway through its transition from JT-
distorted $C2/m$ monoclinic to JT-inactive $R\bar{3}m$ rhombohedral, and
throughout this isotherm the NaNiO2 is mixed-phase.

The lattice parameters [Figure 4] show the expected variation with temperature
and pressure. The rhombohedral and monoclinic phases have similar
compressibility, and in both NaNiO2 is considerably more compressible in the
inter-layer direction ($c$-axis) than the intra-layer ($ab$-) plane. With
reference to Figure 1, we note that compression within the plane results in
changes to the highly ionic Na+-Na+ interactions and the less ionic, but still
repulsive Ni3+-Ni3+ interactions, whereas compression in the inter-layer
direction will compress the Ni3+-O2- and Na+-O2- bonds which are softer due to
the nearest-neighbour interaction lacking a Coulomb repulsive force. This
higher compressibility in the inter-layer direction is consistent with that
seen in another material with alternating layers of edge-sharing octahedra,
the honeycomb iridate Na2IrO3 50, 51.

Within the plane, in monoclinic NaNiO2, the $b$-axis is less compressible than
the $a$-axis. A reason for this might be that Na+-Na+ and Ni3+-Ni3+
interactions are parallel to the direction of compression for the $b$-axis,
maximising the increase in Coulomb repulsion with decreasing lattice parameter
due to compression, whereas there are no Na+-Na+/Ni3+-Ni3+ interactions with
components only along the $a$-axis. Another contribution may be that the
Na+-Na+ and Ni3+-Ni3+ ionic distances parallel to the $b$ axis are
considerably shorter than the distances which can be projected onto the
$a$-axis ($\sim$2.85 Å and $\sim$3.02 Å respectively at 290 K and 0.107(8)
GPa).

Structure | Temperature |  $V_{0}$ ($\AA^{3}$) |  $B_{0}$ (GPa)
---|---|---|---
$R\bar{3}m$ | 490 K | 119.83(2) | 113(1)
$C2/m$ | 490 K | 79.900(16) | 110(1)
| 460 K | 79.798(9) | 113.5(6)
| 290 K | 79.258(7) | 119.6(5)
Table 2: Parameters determined from the 2nd-order Birch-Murnaghan Equation of
State, obtained using PASCal 52 [SI Section V, Table S12].

PASCal 52 was used to obtain the bulk modulus for each isotherm, using a 2nd-
order Birch-Murnaghan equation of state 53. A plot of the unit cell volume
obtained by Rietveld refinement, as a function of pressure, with a fit of this
equation of state, is shown in Figure 5 and tabulated in Table 2. For the
monoclinic phase, $\frac{dV_{0}}{dT}>0$ which is consistent with a structure
with positive thermal expansion. $B$ decreases with increasing temperature,
meaning that compressibility increases with temperature. At 290 K, $B$ is
119.6(5) GPa. This is comparable with a similar JT-distorted material with
edge-sharing octahedra, CuMnO2, which has bulk modulus 116(2) GPa 16. It is,
however, substantially less than the reported bulk modulus for ZnMn2O4 of
197(5) GPa 14, and although there are several different reported values for
Mn3O4 depending on phase and morphology 11, 12, 13, all are higher than what
we report for NaNiO2. LaMnO3 is not entirely comparable owing to the LaO12
units and corner-sharing octahedra, but for reference it has a reported bulk
modulus of 108(2) GPa 3.

Figure 5: Variation in unit cell volume per formula unit for monoclinic $C2/m$
phase. Solid data points show experimental derived values and the solid line
the determined 2nd-order Birch-Murnaghan equation of state. Full lattice
parameters are found in SI Tables S3-5. Where error bars are not visible it is
because they are smaller than the data points.

The directions of the principal axes of compression are determined using
PASCal 52 [SI Figure S13]. These are the axes in which compression occurs
linearly with pressure, and do not necessarily align with the crystallographic
axes in crystalline materials. The principal axis directions projected onto
the $ab$-plane do not change between the monoclinic and rhombohedral phases.
However, the inter-layer direction is a principal axis for the rhombohedral
phase, but not for the monoclinic phase where two principal axes are at an
angle to the inter-layer direction [Fig. 1]. Interestingly, the axis of JT
elongation does not correspond to any of the principal axes [SI Table S11].
There is some temperature dependence in the principal axis directions, likely
owing to variation in the lattice parameters with temperature [SI Tables.
S3-S5]. The compressibilities of NaNiO2 in each of the principal axes is
consistent with variation in lattice parameters [SI Table S11].

We now consider pressure-dependence of the bond length distortion index and
effective coordination [SI Figure S8; Tables S8-10]. As in the ambient
pressure measurements the bond length distortions are significantly larger in
the NiO6 compared to the NaO6 octahedra, with the most significant variation
being the increase in effective coordination (5.387(10) at 0.107(8) GPa to
5.504(13) at 4.24(5) GPa at 290 K) and decrease in bond length distortion
index (from 0.0512(5) to 0.0458(7) in the same pressure range at 290 K) of the
NiO6 octahedra in the monoclinic phase on application of pressure. NaO6
octahedra exhibit far smaller changes in bond length distortion index and
effective coordination, with the overall behaviour not seeming to exhibit a
consistent change with pressure; effective coordination remains between 5.98
and 5.99 throughout the 290 K isotherm.

The differing behaviour of bond length distortion index between NiO6 and NaO6
octahedra is likely attributable to the fact that NiO6 is JT-active and NaO6
is not, and suggests that pressure is decreasing the magnitude of JT
distortion. We investigate this by considering the direct manifestation of the
JT effect in NaNiO2. The Ni-O bond lengths of both the monoclinic and
rhombohedral phases as a function of pressure are shown in Figure 6. The short
Ni-O bonds are less sensitive to the effect of pressure than the long Ni-O
bonds, indicating that the difference between long and short Ni-O bond lengths
is decreasing with pressure. We also observe that the average monoclinic bond
length is consistently larger than the rhombohedral bond length at 490 K [SI
Figure S9]. In the NaO6 octahedra, SI Figure S5, there is an approximately
linear variation of the Na-O bond lengths with pressure. We conclude that
anisotropy of the Ni-O bond compression is a consequence of the JT distortion
in NiO6 octahedra.

Figure 6: Ni-O bond lengths, as a function of pressure, and associated error
of monoclinic NaNiO2 at 290 K, 460 K, and 490 K, with the JT-inactive
rhombohedral phase bond lengths shown for 490 K. Lines are a guide for the
eye.

The observed decrease in difference between long and short Ni-O bonds with
pressure is also reported for other materials containing a JT distortion, such
as LaMnO3 3, KCuF3 4, and CuAs2O4 5. This is equivalent to the observed
tendency with pressure of NiO6 octahedra bond length distortion index and
effective coordination towards their symmetry-constrained values of 0 and 6,
respectively. It indicates that the symmetry of JT-distorted octahedra
increases with application of pressure in monoclinic NaNiO2, consistent with
prior reports 3, 4, 5, 6, 7.

A previous study on LaMnO3 attempted to extrapolate a linear fit to the
pressure-dependence of JT-distorted bond length, and estimated a critical
pressure of $\sim$18 GPa 3. Such an extrapolation could be performed for
NaNiO2 yielding a critical pressure of $\sim$50 GPa, converging at a Ni-O bond
length of 1.85 Å at 290 K. However, this value is unlikely to be
representative of the true critical pressure of the JT distortion in NaNiO2. A
later study on LaMnO3 found that the JT distortion was suppressed at a lower
pressure of around $\sim$12 GPa, suggesting such extrapolation does not yield
accurate predictions. Further, studies of other JT-distorted materials such as
[(CH3)2NH2][Cu(HCOO)3] 6 and CuMnO2 16 have found that this pressure-
dependence of JT-disproportionated bond length exists only up to a certain
pressure, beyond which there is a change in behaviour which renders such
extrapolation of low-pressure behaviour meaningless.

We earlier defined the bond angles $\theta^{\mathrm{M}}_{\mathrm{short-
short}}$ and $\theta^{\mathrm{M}}_{\mathrm{long-short}}$ ($M$=Na,Ni) for
monoclinic NaNiO2, and the associated $\Delta$ values which reduce the number
of parameters needed to describe the behaviour. We plot these $\Delta$ values
in Figure 7 for the 290 K isotherm. These plots show that throughout the
studied pressure range, the degree of angular distortion is far greater for
NaO6 than NiO6, as was the case at ambient pressure [Table 1]. We can also see
that with application of pressure, $\Delta$ is decreasing in value; this
indicates increasing symmetry towards the 90∘ degree bond angle for a perfect
octahedron, analogous to the increasing symmetry with pressure we see with
bond length distortion index.

Figure 7: Values of $\Delta$ for NiO6 and NaO6 octahedra as a function of
pressure at 290 K, representing the magnitude of angular distortion as
nearest-neighbour bond angles take the value $90^{\circ}\pm\Delta$. The two
different $\Delta$ values in monoclinic NaNiO2 are between two short bonds
(red) and between a short and long bond (blue), where bonds are short or long
due to the JT distortion. Lines are a guide for the eye.

The pressure dependence of the NaO6 and NiO6 octahedral volume in NaNiO2 [SI
Figure S7] show that the changes in volume displays different pressure-
dependence for NiO6 octahedra and NaO6 octahedra, as compared with the unit
cell. NaO6 octahedra have higher relative compressibility than the entire unit
cell, and NiO6 octahedra are much more resistant to compression. It has been
shown that for perovskites with $A$O12 and $B$O6 polyhedra the parameters
$M_{A}$ and $M_{B}$ can be used to predict the relative compressibility of the
polyhedra via $\beta_{B}/\beta_{A}=M_{A}/M_{B}$, in which
$\beta_{i}=-\frac{1}{R_{i}}\frac{dR_{i}}{dP}$ is the bond compressibility,
$R_{i}$ is the distance between the central cation and the ith O anion, and
$M_{i}$ is a bond-valence parameter defined in SI 54. We apply this model to
NaNiO2 and find that $M_{\mathrm{Ni}}>M_{\mathrm{Na}}$ throughout the 290 K
isotherm [SI Figure S12]. Accounting for the different values of $R_{i}$, this
indicates that $\frac{dR_{\mathrm{Na-O}}}{dP}>\frac{dR_{\mathrm{Ni-O}}}{dP}$,
which is consistent with our observation that NaO6 octahedra are more
compressible than NiO6 octahedra. This may be due to differences in electronic
configuration for closed-shell Na+ and open-shell Ni3+, or Na+ being a much
larger ion than Ni3+.

We now consider a related model proposed by Angel et al., again in the context
of perovskites 55, whereby a transition temperature $T_{\mathrm{c}}$
associated with an octahedral phase transition will exhibit
$dT_{\mathrm{c}}/dP<0$ if octahedra are more compressible than the extra-
framework cation sites (analogous to the NaO6 octahedra in NaNiO2), and
$dT_{\mathrm{c}}/dP>0$ if octahedra are less compressible. Our structural
analysis shows the enhanced compressibility of NaO6 octahedra when compared to
NiO6, and so this model predicts the observed increase in $T_{\mathrm{JT}}$
with pressure. It is worth noting that there are more degrees of freedom in
the layered NaNiO2 structure so the relationships between distortions in NiO6
and NaO6 may not be so strongly coupled as in the perovskites. However, the
basic hypothesis of the model in Angel et al. appears to be applicable to
NaNiO2.

Along the 490 K isotherm, both monoclinic and rhombohedral NaNiO2 were
observed in coexistence. The fraction of NaNiO2 in the low-temperature, JT-
distorted monoclinic phase is shown in Figure 8. The fraction remains
approximately stagnant to $\sim$2 GPa, beyond which it consistently increases
with increasing pressure. In the range where it is increasing, the monoclinic
fraction at 490 K increases from 67.8(6)% at 0.71(2) GPa to 80.2(9)% at
4.20(6) GPa. This indicates that $T_{\mathrm{JT}}$ increases with increasing
pressure beyond $\sim$2 GPa, consistent with our prediction based on
octahedral compressibility.

Figure 8: The fraction of NaNiO2 which is in the monoclinic phase at 490 K, as
a function of pressure.

To explore the P-T dependence of the transition, the sample was heated at
5.29(8) GPa from 460 K to 490 K after measuring the variable-pressure 460 K
isotherm. At ambient pressure, this would result in a mixed
monoclinic/rhombohedral phase. However, at the resulting high pressure of
5.46(9) GPa, we did not observe emergence of any rhombohedral peaks in the
diffraction pattern. Subsequent reduction in pressure to 0.342(13) GPa at the
same temperature, 490 K, did yield emergence of rhombohedral peaks [Figure
S4], further supporting our interpretation that $T_{\mathrm{JT}}$ is
increasing with pressure.

##  4 Discussion

The results of our $P$-T study on NaNiO2 are summarised in a phase diagram,
Figure 9. To our knowledge, this is the first study on the effect of pressure
on the JT transition temperature in a material containing edge-sharing MO6
octahedra, and the first variable-pressure study on a JT-active edge-sharing
nickelate. Comparison between the results of this study and previous works
must therefore rely on the work done on non-nickelate materials.

Figure 9: Tentative phase diagram showing the structure of NaNiO2 as a
function of pressure and temperature. Triangles denote diffraction
measurements and point left/right if $P$ was decreasing/increasing or up/down
if $T$ was increasing or decreasing. The precise boundaries of the three
regions are estimates based on available data, with the results in Refs. 46,
31 used to estimate the broadness of the transition.

Like the perovskite materials LaMnO3 3, 8 and KCuF3 4, NaNiO2 exhibits far
greater compressibility in the JT-elongated O-Ni-O axis than the JT-compressed
O-Ni-O axes, with the JT distortion in both NaNiO2 and the previously-
discussed perovskites decreasing in magnitude with pressure. The consistent
behaviour with other JT-active materials is also clear evidence that the
charge disproportionation model proposed for LiNiO2 26, 27, 28 and some
Ni3+-containing perovskites 21 is not applicable to NaNiO2.

A novel behaviour we observe in NaNiO2 is that $T_{\mathrm{JT}}$ increases
with application of pressure NaNiO2. For comparison, in LaMnO3 the JT
distortion is suppressed at $\sim$12 GPa, indicating that $T_{\mathrm{JT}}$ is
reduced to room temperature from $\sim$750 K by 12 GPa 2. This mechanism seems
unlikely in NaNiO2 due to the increasing $T_{\mathrm{JT}}$ with pressure,
although we cannot exclude the possibility that a reversal above our maximum
measured pressure may result in a decrease in $T_{\mathrm{JT}}$. Additionally,
there is a trend observed in this, and other 3, 4, 5, 6, 7, works, that the
magnitude of distortion due to the JT effect decreases with pressure. This
could be interpreted as meaning that there is some pressure where the
distortion is entirely suppressed and the NiO6 octahedra achieve a bond length
distortion index of zero, consistent with absence of an ordered JT distortion.
However, this is at odds with recent reports 6, 16 which show that at some
pressure the long and short bonds in JT-distorted octahedra eventually
stabilise at different lengths. It is therefore not clear how exactly the JT
distortion is suppressed in NaNiO2 with high pressure, and further
investigation is needed to elucidate this.

We should once again note that the conductivity behaviour of the high-
temperature phase of NaNiO2 also remains unexplored. There is significant
reduction in resistivity with JT suppression in LaMnO3, 10 and there is a
possibility for similar behaviour in high-temperature rhombohedral NaNiO2.
Density Functional Theory calculations on rhombohedral, JT-free LiNiO2 (which
is isostructural with the high-temperature phase of NaNiO2) have suggested
metallic behaviour 56. On a similar note, a broad first-order transition
between two structures with a group-subgroup relationship in SrCrO3 57
featured coexistence of electronic phases. If the high-temperature phase of
NaNiO2 were indeed metallic, this metallic behaviour could explain why
$dT_{\mathrm{JT}}/dP>0$ in this material, as application of pressure may
result in narrowing of Ni(3d)-O(2p) bands, pushing the metal-to-insulator
phase transition to higher and higher temperatures, and electronic phase
coexistence could provide an explanation for the very broad nature of the
transition.

##  5 Conclusion

The key finding of this study is that in NaNiO2, $T_{\mathrm{JT}}$ increases
slightly with application of pressure while JT-distorted NiO6 octahedra become
more symmetric, as demonstrated by the pressure-dependence of two distortion
metrics (effective coordination and bond length distortion index). While the
latter is a well-documented property of JT-distorted materials, the former is
in contrast to the JT distortion in LaMnO3 2. NaNiO2 is more resistant to
pressure than other similar materials, having a higher bulk modulus
($B=119.6(5)$ GPa at 290 K) than similar perovskites 3, 4, Prussian Blue
analogues 7, and layered honeycomb structures 50, although its bulk modulus is
very similar to JT-distorted edge-sharing CuMnO2 16, and is less than Mn3O4
11, 12, 13, NiO 58, and ZnMn2O4 14. NaNiO2 also displays a much smaller
magnitude of $\frac{dT_{\mathrm{JT}}}{dP}$ than LaMnO3, with LaMnO3 shifting
$T_{\mathrm{JT}}$ from $\sim$750 K at ambient pressure to room temperature in
12 GPa 2 compared with a very small shift from $\sim$480 K at ambient pressure
in NaNiO2.

Further variable-pressure diffraction measurements, at several temperatures
and to higher pressures, are needed to fully understand the process of
suppressing the JT distortion in NaNiO2. Variable-pressure Raman spectroscopy
measurements on NaNiO2 could also be useful and may help identify phase
transitions at higher pressures.

Additionally, future investigations are needed to investigate whether other
JT-distorted materials exhibit a $dT_{\mathrm{JT}}/dP>0$ pressure dependence,
for example a study building on previous work on CuMnO2 16 by measuring at
multiple isotherms.

Data availability: The variable-pressure neutron diffraction data from the
PEARL instrument at ISIS is available at doi:10.5286/ISIS.E.RB2000219 59. All
other data can be found at doi:10.17863/CAM.81605 60.

## Acknowledgements

This work was supported by the Faraday Institution grant number FIRG017. LNC
acknowledges a scholarship EP/R513180/1 to pursue doctoral research from the
UK Engineering and Physical Sciences Research Council (EPSRC). Experiments at
the ISIS Neutron and Muon Source were supported by a beamtime allocation
RB2000219 from the Science and Technology Facilities Council of the United
Kingdom. A portion of this research used resources at the Spallation Neutron
Source, a DOE Office of Science User Facility operated by the Oak Ridge
National Laboratory of the United States of America, with data collection
performed by Joerg C Neuefeind and Jue Liu (ORNL). Figure 1 was generated
using CrystalMaker®: a crystal and molecular structures program from
CrystalMaker Software Ltd, Oxford, UK. Heather Greer assisted with the SEM
images in Supplementary Information. We also thank others whose ideas and
comments were useful: Joshua D Bocarsly, Farheen N Sayed, Andrew G Seel,
Siddharth S Saxena, Euan N Bassey, Nicola D Kelly, Venkateswarlu Daramalla,
Chloe S Coates, Camilla Tacconis, and Debasis Nayak.

## Supplementary Information

The Supporting Information is available free of charge at ————-. Additional
neutron diffraction data and refinement details. SEM images. Tabulated
diffraction data and distortion parameters. Transformation matrices.

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# The coverage ratio of the frog model on complete graphs

Gustavo O. de Carvalho  and  Fábio P. Machado  Institute of Mathematics and
Statistics
University of São Paulo
Rua do Matão 1010, CEP 05508-090, São Paulo, SP, Brazil -
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract.

The frog model is a system of interacting random walks. Initially, there is
one particle at each vertex of a connected graph $\mathcal{G}$. All particles
are inactive at time zero, except for the one which is placed at the root of
$\mathcal{G}$, which is active. At each instant of time, each active particle
may die with probability $1-p$. Once an active particle survives, it jumps on
one of its nearest vertices, chosen with uniform probability, performing a
discrete time simple symmetric random walk (SRW) on $\mathcal{G}$. Up to the
time it dies, it activates all inactive particles it hits along its way. From
the moment they are activated on, every such particle starts to walk,
performing exactly the same dynamics, independent of everything else. In this
paper, we take $\mathcal{G}$ as the $n-$complete graph ($\mathcal{K}_{n}$, a
finite graph with each pair of vertices linked by an edge). We study the limit
in $n$ of the coverage ratio, that is, the proportion of visited vertices by
some active particle up to the end of the process, after all active particles
have died.

###### Key words and phrases:

complete graph, coverage ratio, frog model, random walks system.

###### 2010 Mathematics Subject Classification:

60K35, 05C81

Research supported by Capes (88887.676435/2022-00), CNPq (303699/2018-3 and
132598/2020-5), FAPESP (17/10555-0)

##  1\. Introduction

The frog model is a system of interacting random walks on a given rooted
connected graph, finite or infinite. Initially, the graph contains some
configuration (maybe one per vertex) of inactive particles on its vertices.
The particles at the root of the graph start out active and perform
independent simple nearest-neighbor discrete time random walks. Whenever a
vertex with inactive particles is first visited, all the particles at the
vertex wake up and begin their own independent random walks, waking up
inactive particles as they visit them. The lifetime of a particle, in case it
is active, can be finite or infinite. This model is often associated with the
dynamics of the spread of a rumor or the spread of a virus in a connected
population. A formal definition of the frog model can be found in [AMP02a].

Part of the literature on the frog model is focused on the case where active
particles have infinite lifetime and $\mathcal{G}$ is an infinite connected
graph. In this setup, a goal is to study recurrence/transience, that is,
conditions for the root of the graph to be visited infinitely often, as in
[Pop01, HJJ17, KZ17]. Another approach on infinite graphs is to study the
limit shape of the set of visited vertices, also known as shape theorem, as
found in [AMP02b]. On finite graphs and particles with infinite lifetime, one
can study the first time where each vertex of the graph is visited at least
once (the coverage time) as in [Her18, HJJ19]. Finite lifetime on finite
graphs are considered in [BFHM20, Her18].

In this paper, we are interested in studying the limit coverage ratio for the
frog model on $\mathcal{K}_{n}$, the $n-$complete graph, when the lifetime of
an active particle follows a geometric random variable with parameter $1-p$.
That is a consequence of the fact that at each instant of time, indenpendently
of everything else, each active particle may die with probability $1-p$ or
survive with probability $p$. We know from [ALMM06] that, starting from a one
particle per vertex configuration on $\mathcal{K}_{n}$, there exists a phase
transition property, which means the existence of a non-trivial critical
parameter $p_{c}$ such that the coverage ratio converges to zero in
distribution for $p<p_{c}$, while this convergence does not occur for
$p>p_{c}$. See [LE19] for some interesting result with this setup. It is also
shown in [ALMM06] that $p_{c}=1/2$. Our main theorem extend the previous
result, giving more details about the number of visited vertices, and proving
also, for $p>p_{c}=1/2$, the exact limit in $n$ of the probability that the
coverage ratio is strictly larger than zero.

Let us start with a basic definition

######  Definition 1.1.

Let $V_{\infty}=V_{\infty}(\mathcal{K}_{n})$ be the number of vertices of
$\mathcal{K}_{n}$, visited at least once, up to the time there are no more
active particles.

Our main result is the following.

######  Theorem 1.2.

Consider the frog model on $\mathcal{K}_{n}$, starting from the configuration
with one particle per vertex. Then

(i) For $p\leq 1/2$, for every function $f:\mathbb{N}\rightarrow\mathbb{R}$
such that $\lim_{n\to\infty}f(n)=+\infty$,

$\lim_{n\to\infty}P(V_{\infty}\leq f(n))=1.$

(ii) For $p>1/2$, there are constants $c>0$ e $c^{\prime}\in(0,1)$ such that

$\lim_{n\to\infty}P(V_{\infty}\leq c\log n)=\frac{1-p}{p},$
$\lim_{n\to\infty}P(V_{\infty}\geq c^{\prime}n)=\frac{2p-1}{p}.$

Observe that the function $f(n)$ in part (i) of Theorem 1.2 can be replaced by
$c\log n$, for any $c>0$. Still, part $(i)$ of Theorem 1.2 is valid for any
sequence of connected graphs. The proof we provide in this paper works also in
this case.

The ideas and techniques used in the proof of the Theorem 1.2 are inspired by
the study of the existence of a giant component on Erdös–Rényi graph (see
[Gri18, sec. 11.2] for details). Some basic concepts of branching processes
(see [GS01, sec. 5.4]) are also considered.

The intuition behind Theorem 1.2 comes from the fact that if one considers a
stage when there is a small number of visited vertices on a large graph, what
one sees is that the next surviving particle will, most likely, visit an
unvisited vertex, activating a new particle. Therefore, in the beginning of
the process, the number of vertices visited is near to the number of
individuals in a branching process that generates 2 offspring with probability
$p$ and 0 offspring with probability $1-p$. In this setup, the extinction
probability equals to $1$ for $p\leq 1/2$ and to $\frac{1-p}{p}$ for $p>1/2$.
Besides that, for $p>1/2$, the branching process mentioned above grows
infinitely with probability $\frac{2p-1}{p}$, an event that is in some sense
analogous to $V_{\infty}$ being on the order of $n$.

The paper is organized as follows. In section 2 we describe an auxiliary
process aiming to simplify the computations needed. In section 3 one finds the
proofs of Theorem 1.2. The sub-critical phase ($p\leq 1/2$) is proved in
subsection 3.1, for any sequence of connected graphs, while the supercritical
phase ($p>1/2$) is proved in subsection 3.2.

##  2\. The auxiliary process

Let us define an auxiliary process starting from a small modification on the
frog model. Let us start from the $n+1$-complete graph, $\mathcal{K}_{n+1}$,
picking one of its vertices and defining it as its root. We denote the set of
vertices of $\mathcal{K}_{n+1}$ by $\mathcal{V}(\mathcal{K}_{n+1})$. At time
zero, there is one active particle at the root and one inactive particle at
each other vertices of $\mathcal{K}_{n+1}$. All particles present at time zero
are declared original. This is done because at some point, extra particles are
considered.

The auxiliary process has the following characteristics

  * •

It is taken in rounds, so that only one particle acts (dying or moving) in
each round. The particle which is being considered each time, survives with
probability $p$ and, if so, it jumps on one of its nearest vertices, chosen
with uniform probability.

  * •

Let $R$ be the time that the last original active particle dies. It is the
time that the original frog model dies out. At this very time, every remaining
inactive particles become extra inactive particles. Besides that, a brand new
extra active particle is placed at the root of $\mathcal{K}_{n+1}$, so it acts
in round $R+1$.

  * •

From now on every time the last extra active particle dies, another extra
active particle is placed at the root of $\mathcal{K}_{n+1}$.

Observe that there are only extra particles from the round $R+1$ on. Observe
also that $R<\infty$ with probability 1, as the number of original particles
activated up to a given time is at most
$|\mathcal{V}(\mathcal{K}_{n+1})|=n+1<\infty$ and the lifetime of any active
particle is given by a geometric random variable of parameter $1-p$. The
lifetime is finite with probability 1, since we are restricted to $p<1$.

Even though $R<\infty$, the auxiliary process as a whole is infinite, as it
always renews itself when we consider an extra particle at the origin. This
rule makes it possible to analyze the behavior of the particle (original or
extra) that performs the $k$th round for any $k\in\mathbb{N}$, without the
need to check whether the original process has finished or not.

Now, for $k\in\mathbb{N}$ we define the random variable $X_{k}$ which depends
on what happens with the particle that acts in round $k$ during the auxiliary
process:

  * •

$X_{k}=0$ if it dies.

  * •

$X_{k}=1$ if it survives and jumps to a vertex which has been visited before
round $k$.

  * •

$X_{k}=2$ if it survives and jumps to a vertex which has never been visited
before round $k$.

Therefore, $X_{k}$ can be interpreted as the number of descendants (in the
branching process sense) of the particle that acts in the $k$th round. Observe
that the sequence $X_{1},X_{2},...$ is neither independent nor identically
distributed.

From the number of descendants in each round, we define the number of
potentially active particles at the end of the $k$-th round as

$A^{\prime}_{0}:=1;\hskip
14.22636ptA^{\prime}_{k}:=1+\sum_{j=1}^{k}(X_{j}-1),\hskip 2.84526ptk\geq 1.$
(2.1)

The term potential comes from the fact that the previous expression also
considers descendants of extra particles. We can disregard this by taking

$R=\inf\\{k:A^{\prime}_{k}=0\\}$ (2.2)

and setting the number of active particles at the end of the $k$th round as

$A_{k}:=A^{\prime}_{k}{\mathbbm{1}}_{(k<R)},\hskip 5.69054ptk\in\mathbb{N}.$
(2.3)

Similarly, we define the number of potentially visited vertices by the end of
the $k$th round as

$V^{\prime}_{0}:=1;\hskip
14.22636ptV^{\prime}_{k}:=1+\sum_{j=1}^{k}{\mathbbm{1}}_{(X_{j}=2)},\hskip
2.84526ptk\in\mathbb{N}$ (2.4)

and the number of vertices visited at the end of the $k$th round as

$V_{k}:=V^{\prime}_{k}{\mathbbm{1}}_{(k<R)}+V^{\prime}_{R}{\mathbbm{1}}_{(k\geq
R)},\hskip 2.84526ptk\in\mathbb{N}.$ (2.5)

We define the total number of vertices visited during the whole process by

$V_{\infty}:=\lim_{k\to\infty}V_{k}=V_{R}.$ (2.6)

Note that $V_{\infty}=V_{R}$ is not affected by the extra particles, as they
are triggered only after $R$. The extra particles are useful for computational
purposes, as they make it possible to keep track of the number of potentially
active particles and potentially visited vertices without having to worry
about whether all the original particles have died or not.

It is important to note that $V_{\infty}$ also describes the total number of
vertices visited in the original frog model. Therefore, the study of the frog
model will be done from the auxiliary process, using $V_{\infty}$.

Note that the auxiliary process and its definitions could be done for any
conected graph. In some cases one could have $R=\infty$ with positive
probability. The auxiliary process fits well the complete graph due to the
fact that some computations can be easily done, as

$P(X_{k}=x|V^{\prime}_{k-1}=v)=\begin{cases}1-p&\text{if $x=0$,}\\\
\frac{p(v-1)}{n}&\text{if $x=1$,}\\\ \frac{p(n-v+1)}{n}&\text{if
$x=2$.}\end{cases}$ (2.7)

##  3\. Proofs

We start off with a basic probability result involving Chernoff bounds that
will be useful later on.

######  Lemma 3.1 ([MU05], Theorem 4.5).

For $X$ a random variable with binomial distribution, $X\sim B(n,p)$, it holds
that

$P(X\leq E(X)-t)\leq\exp(\frac{-t^{2}}{2E(X)}),\hskip 8.53581ptt>0.$

###  3.1. The sub-critical phase: $p\leq 1/2$

###### Proof of Theorem 1.2 part (i).

Consider $\\{\mathcal{G}_{n}\\}_{n\in\mathbb{N}}$ a sequence of connected
graphs. Let us define a process such that every time there is an active
particle jumping to a neighboring vertex, it chooses one that has never been
visited before. Our aim is to define a process whose number of visited
vertices at any given time dominates the same quantity computed for the
auxiliary process defined in Section 2.

Let us define a sequence of random variables
$\\{X_{j}^{+}\\}_{j\in\mathbb{N}}$

$X_{j}^{+}=\begin{cases}0&\text{if $X_{j}=0$,}\\\ 2&\text{if
$(X_{j}=1)\cup(X_{j}=2)$.}\end{cases}$ (3.1)

Note that $X^{+}_{1},X^{+}_{2},...$ are independent and identically
distributed, as they rely only on the survival event of particles. Consider
also $X^{+}$ a random variable with the same distribution as $X^{+}_{1}$. It
holds that

$E(X^{+})=2P(X^{+}=2)=2p\leq 1.$ (3.2)

Next, we consider a branching process where the number of children of each
individual is distributed as $X^{+}$. As done for the auxiliary process, we
keep track of this branching process by the number of descendants of each
individual per turn.

Analogously to definitions (2.1), (2.2), (2.3), (2.4), (2.5) and (2.6), denote
the number of potentially alive individuals at the end of the $k$th round of
the branching process by

$A^{\prime+}_{k}:=1+\sum_{j=1}^{k}(X_{j}^{+}-1),$ (3.3)

and the number of rounds until the extinction of this population by

$R^{+}:=\inf\\{k\in\mathbb{N}:A^{\prime+}_{k}=0\\}.$ (3.4)

The total number of individuals in this branching process can be written as

$V_{\infty}^{+}:=1+\sum_{j=1}^{R^{+}}{\mathbbm{1}}_{(X_{j}^{+}=2)}.$

Note that (3.2) tells us that the branching process with offspring
distribution $X^{+}$ becomes extinct, and therefore its total number of
individuals is finite, with probability 1. The extinction probability does not
change if we count the descendants of one individual at a time each round. As,
by hypothesis, $\lim_{n\to\infty}f(n)=+\infty$, it follows that:

$\lim_{n\to\infty}P(V_{\infty}^{+}\leq f(n))=1.$

From (3.1), we have that for every $k\in\mathbb{N}$ it holds that
$X^{+}_{k}\geq X_{k}$, which by its turn implies that $A^{\prime+}_{k}\geq
A^{\prime}_{k}$ and so $R^{+}\geq R$. It is also true that $V_{\infty}^{+}\geq
V_{\infty}$. So, we conclude that

$\lim_{n\to\infty}P(V_{\infty}\leq f(n))=1.$

$\square$

###  3.2. The super-critical phase: $p>1/2$

###### Proof of Theorem 1.2 part (ii).

As the case $p=1$ is trivial, we focus on $p\in(1/2,1)$, splitting the proof
in the three following statements

  * (a)

For $k_{-}=2\frac{4p}{(1+2p)[\frac{2p-1}{8p+4}]^{2}}\log n$ and
$k_{+}=(1-\frac{2}{1+2p})n$, it holds that

${\lim_{n\to\infty}P(R\in[k_{-},k_{+}])=0}.$

  * (b)

$\lim_{n\to\infty}P(V_{\infty}\leq c\log n|R<k_{-})=1$ and
${\lim_{n\to\infty}P(V_{\infty}\geq c^{\prime}n|R>k_{+})=1}$.

  * (c)

$\lim_{n\to\infty}P(R<k_{-})=\frac{1-p}{p}$.

Note that with these 3 items, the proof comes to an end. Next, we prove (a),
(b) and (c).

Proof of (a). First we define

$k_{-}=k_{-}(n):=2\frac{4p}{(1+2p)[\frac{2p-1}{8p+4}]^{2}}\log n;$

and

$k_{+}=k_{+}(n):=(1-\frac{2}{1+2p})n.$

See that $0<1-\frac{2}{1+2p}<1$ when $p>1/2$. For
$k\in\mathbb{N}\cap[k_{-},k_{+}]$, we may consider independent random
variables $Y_{1},...,Y_{k}$ such that for $j\in\\{1,...,k\\}$

$P(Y_{j}=x)=\begin{cases}1-p&\text{if $x=0$,}\\\ \frac{pk_{+}}{n}&\text{if
$x=1$,}\\\ \frac{p(n-k_{+})}{n}&\text{if $x=2$.}\end{cases}$ (3.5)

For any $j\in\\{1,...,k\\}$ it is true that $V^{\prime}_{j-1}-1\leq j\leq
k\leq k_{+}$. Note that $Y_{j}$ can be interpreted as the number of
descendants of the particle participating in the $k$th round, when we maximize
the number of visited vertices (compare with the equations (3.5) and (2.7)).
So, by coupling, we can assume that $\sum_{j=1}^{k}X_{j}$ is always greater
than $\sum_{j=1}^{k}Y_{j}$. Let us denote this fact by
$\sum_{j=1}^{k}X_{j}\succeq\sum_{j=1}^{k}Y_{j}.$

Next, let us define define $a_{k}:=(\frac{2p}{1+2p}-\frac{1}{2})k$. Note that
$0<\frac{2p}{1+2p}-\frac{1}{2}<1$ for $p>1/2$. By using the Chernoff bounds
from Lemma 3.1, together with the fact that

$\sum_{j=1}^{k}{\mathbbm{1}}_{(Y_{j}=2)}\sim Bin(k,\frac{p(n-k_{+})}{n})$

and (2.1), we have that

$\displaystyle P($ $\displaystyle A^{\prime}_{k}\leq a_{k}+1)$
$\displaystyle=P(\sum_{j=1}^{k}X_{j}\leq k+a_{k})$ $\displaystyle\leq
P(\sum_{j=1}^{k}Y_{j}\leq k+a_{k})$ $\displaystyle\leq
P(\sum_{j=1}^{k}{\mathbbm{1}}_{(Y_{j}=2)}\leq\frac{k+a_{k}}{2})$
$\displaystyle=P(\sum_{j=1}^{k}{\mathbbm{1}}_{(Y_{j}=2)}\leq
kp-\frac{pkk_{+}}{n}-k(p-\frac{1}{2})+\frac{pkk_{+}}{n}+\frac{a_{k}}{2})$
(3.6) $\displaystyle=P(\sum_{j=1}^{k}{\mathbbm{1}}_{(Y_{j}=2)}\leq
E(\sum_{j=1}^{k}{\mathbbm{1}}_{(Y_{j}=2)})-k[\frac{2p-1}{8p+4}])$
$\displaystyle\leq\exp(-\frac{k^{2}[\frac{2p-1}{8p+4}]^{2}}{2kp(1-\frac{k_{+}}{n})})$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{\leq}}\exp(-\frac{k_{-}[\frac{2p-1}{8p+4}]^{2}(1+2p)}{4p})$
$\displaystyle=\exp(-2\log n)=o(n^{-1}),$

where for two sequences of functions $f_{1},f_{2},...$ and $g_{1},g_{2},...$,
we write $f_{n}=o(g_{n})$ if $\lim_{n\to\infty}\frac{f_{n}}{g_{n}}=0$. The
Chernoff bounds can be applied as $\frac{2p-1}{8p+4}>0$ for $p>1/2$. Observe
that we can exchange $k$ by $k_{-}$ in (*) as $k_{-}\leq k$ and $\exp(-x)$ is
decreasing in $x$.

Now, consider the set

$A:=\bigcup_{k\in\mathbb{N}\cap[k_{-},k_{+}]}(A^{\prime}_{k}\leq a_{k}+1)$

In words, it means that for some ${k\in\mathbb{N}\cup[k_{-},k_{+}]}$ there is
at most $a_{k}+1$ potentially active particles. By the subaditivity of the
probability measure and (3.2), we have

$P(A)\leq\sum_{k\in\mathbb{N}\cap[k_{-},k_{+}]}P(A^{\prime}_{k}\leq
a_{k}+1)\leq k_{+}o(n^{-1})=o(1).$

Then

$P(A^{c})=P(\cap_{k\in\mathbb{N}\cap[k_{-},k_{+}]}(A^{\prime}_{k}>a_{k}+1))\stackrel{{\scriptstyle
n\to\infty}}{{\rightarrow}}1.$

As $R:=\inf\\{k:A^{\prime}_{k}=0\\},$

$\lim_{n\to\infty}P(R\in[k_{-},k_{+}])=0.$

Proof of (b). From the definition of $R,$ we know that $X_{j}=0$ for some
$j\in\\{1,2,..,R\\}$. Therefore, for $R<k_{-}$, from (2.4), (2.5) and (2.6),
we know that

$V_{\infty}=V_{R}=V^{\prime}_{R}=1+\sum_{j=1}^{R}{\mathbbm{1}}_{(X_{j}=2)}\leq
1+R-1<k_{-}.$

So, we conclude that

$\lim_{n\to\infty}P(V_{\infty}\leq k_{-}|R<k_{-})=1.$

Furthermore, for $\lfloor k_{+}\rfloor=\max\\{x\in\mathbb{Z}:x\leq k_{+}\\}$,
we have that the probability of having
$\frac{k_{+}+a_{k_{+}}}{2}=n(\frac{12p^{2}-4p-1}{4(1+2p)^{2}})$ or fewer
potentially visited vertices equals to

$\displaystyle P(V^{\prime}_{\lfloor
k_{+}\rfloor}\leq\frac{k_{+}+a_{k_{+}}}{2})$
$\displaystyle=P(\sum_{k=1}^{\lfloor
k_{+}\rfloor}{\mathbbm{1}}_{(X_{k}=2)}+1\leq\frac{k_{+}+a_{k_{+}}}{2})$ (3.7)
$\displaystyle=P(\sum_{k=1}^{\lfloor
k_{+}\rfloor}{\mathbbm{1}}_{(X_{k}=2)}\leq\frac{(k_{+}-1)+(a_{k_{+}}-1)}{2})$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{\leq}}P(\sum_{k=1}^{\lfloor
k_{+}\rfloor}{\mathbbm{1}}_{(X_{k}=2)}\leq\frac{\lfloor
k_{+}\rfloor+a_{\lfloor k_{+}\rfloor}}{2})$ $\displaystyle\leq
P(\sum_{k=1}^{\lfloor k_{+}\rfloor}{\mathbbm{1}}_{(Y_{k}=2)}\leq\frac{\lfloor
k_{+}\rfloor+a_{\lfloor k_{+}\rfloor}}{2})=o(1).$

The inequality (*) holds as $\lfloor k_{+}\rfloor\geq k_{+}-1$ and that,
$a_{k}=ck$, for some constant $c\in(0,1)$. Therefore $a_{\lfloor
k_{+}\rfloor}=c\lfloor k_{+}\rfloor\geq c(k_{+}-1)=a_{k_{+}}-c\geq
a_{k_{+}}-1$.

Moreover, see that (3.7) is $o(1)$ because $\lfloor
k_{+}\rfloor\in\mathbb{N}\cap[k_{-},k_{+}]$ and that term shows up in (3.2).

Therefore

$P(V^{\prime}_{\lfloor
k_{+}\rfloor}\leq\frac{k_{+}+a_{k_{+}}}{2}|R>k_{+})P(R>k_{+})+P(V^{\prime}_{\lfloor
k_{+}\rfloor}\leq\frac{k_{+}+a_{k_{+}}}{2}|R\leq k_{+})P(R\leq
k_{+})\stackrel{{\scriptstyle n\to\infty}}{{\rightarrow}}0.$

So, if $\lim_{n\to\infty}P(R>k_{+})>0$ (see part c), it holds that
$\lim_{n\to\infty}P(V^{\prime}_{\lfloor
k_{+}\rfloor}>\frac{k_{+}+a_{k_{+}}}{2}|R>k_{+})=1$. In addition, by (2.4),
(2.5) and (2.6) when $R>k_{+}\geq\lfloor k_{+}\rfloor$, it is true that
$V_{\infty}=V_{R}\geq V_{\lfloor k_{+}\rfloor}=V^{\prime}_{\lfloor
k_{+}\rfloor}$.

From this, we conclude that

$\lim_{n\to\infty}P(V_{\infty}\geq\frac{k_{+}+a_{k_{+}}}{2}|R>k_{+})=1.$

Remember that $k_{+}=c_{1}n$ for a constant $c_{1}\in(0,1)$ and that
$a_{k_{+}}=c_{2}k_{+}=c_{1}c_{2}n$ for a constant $c_{2}\in(0,1)$, then
$\frac{k_{+}+a_{k_{+}}}{2}=c^{\prime}n$ for some $c^{\prime}\in(0,1)$, thus
concluding the proof of (b).

Proof of (c).

Let us return to the random variable $X^{+}$ presented in sub-section 3.1.

$P(X^{+}=x)=\begin{cases}1-p,&\text{if $x=0$,}\\\ p,&\text{if $x=2$.}\\\
\end{cases}$ (3.8)

Again, we consider a branching process with offspring distribution given by
$X^{+}$, where the number of descendants of the individuals is evaluated in
turns, and return to definitions (3.3), (3.4), rewritten below.

$A^{\prime+}_{k}:=1+\sum_{j=1}^{k}(X_{j}^{+}-1)$

representing the number of potentially active particles of this branching
process at the end of the $k$th round and

$R^{+}:=\inf\\{k\in\mathbb{N}:A^{\prime+}_{k}=0\\}$

representing the instant where the branching process ends. It is now necessary
to define $R^{+}:=+\infty$ if
$\\{k\in\mathbb{N}:A^{\prime+}_{k}=0\\}=\emptyset$, since it can happen, with
positive probability in the case $p>1/2$, of having a never ending process.
Furthermore, we define

$A^{+}_{k}:=A^{\prime+}_{k}\mathbb{I}(k<R^{+})$ (3.9)

representing the number of effectively active particles of this branching
process at the end of the $k$th round.

Let us define the event for which the branching process with offspring
distribution $X^{+}$ extinguishes, but after the round $k_{-}$ by

$B^{+}_{k_{-}}:=\\{k_{-}<R^{+}<+\infty\\}.$

We define one last branching process, with the same rounds scheme, with
offspring distribution $X^{-}$, where

$P(X^{-}=x)=\begin{cases}1-p&\text{if $x=0$,}\\\ \frac{pk_{-}}{n}&\text{if
$x=1$,}\\\ \frac{p(n-k_{-})}{n}&\text{if $x=2$.}\end{cases}$ (3.10)

Similarly, consider $X^{-}_{1},X^{-}_{2},...$ as identically distributed
independent random variables with the same distribution of $X^{-}$ and define

$A^{\prime-}_{k}:=1+\sum_{j=1}^{k}(X_{j}^{-}-1)$

as the number of potentially alive individuals at the end of the $k$th round.

Furthermore, define

$R^{-}:=\inf\\{k\in\mathbb{N}:A^{\prime-}_{k}=0\\}$

as the moment when the branching process with offspring given by $X^{-}$ ends.
Consider that $R^{-}:=+\infty$ if
$\\{k\in\mathbb{N}:A^{\prime-}_{k}=0\\}=\emptyset$.

We have that $k\leq k_{-}\Rightarrow
V^{\prime}_{k-1}=1+\sum_{j=1}^{k-1}{\mathbbm{1}}_{(X_{j}=2)}\leq k_{-}$.
Therefore, comparing (3.8) and (3.10) with (2.7), we conclude that
$\sum_{j=1}^{k}X^{-}_{j}\preceq\sum_{j=1}^{k}X_{j}\preceq\sum_{j=1}^{k}X^{+}_{j}$,
and so $A^{\prime-}_{k}\preceq A^{\prime}_{k}\preceq A^{\prime+}_{k}$ for all
$k\in\mathbb{N}\cap[1,k_{-}]$. We then see that it is possible to couple the
three process so that $R^{+}\leq k_{-}\Rightarrow R\leq k_{-}$ and also $R\leq
k_{-}\Rightarrow R^{-}\leq k_{-}$.

So, it is true that

$P(R^{+}<\infty)-P(B^{+}_{k_{-}})=P(R^{+}\leq k_{-})\leq P(R\leq k_{-})\leq
P(R^{-}\leq k_{-})\leq P(R^{-}<\infty).$ (3.11)

By properties of branching processes, $P(R^{+}<\infty)$ is the smallest
positive solution of

$P(R^{+}<\infty)=1-p+P(R^{+}<\infty)^{2}p.$

Solving the quadratic equation above, we realize that

$P(R^{+}<\infty)=\frac{1-p}{p}.$

It also holds that $P(R^{-}<\infty)$ is the smallest positive solution of

$P(R^{-}<\infty)=1-p+P(R^{-}<\infty)\frac{pk_{-}}{n}+P(R^{-}<\infty)^{2}\frac{p(n-k_{-})}{n}.$

Solving the quadratic equation above, we see that

$P(R^{-}<\infty)=\frac{n(1-p)}{pn-pk_{-}}\stackrel{{\scriptstyle
n\to\infty}}{{\rightarrow}}\frac{1-p}{p}.$

Denote $\lceil k_{-}\rceil=\min\\{x\in\mathbb{Z}:x\geq k_{-}\\}$. From (3.2),
it holds that $\lim_{n\to\infty}P(A^{\prime}_{\lceil k_{-}\rceil}>a_{\lceil
k_{-}\rceil}+1)=1$ and therefore $\lim_{n\to\infty}P(A^{\prime+}_{\lceil
k_{-}\rceil}>a_{\lceil k_{-}\rceil}+1)=1$. Then, if
$\lim_{n\to\infty}P(R^{+}>k_{-})>0$ (which is true because $E(X^{+})>1$), we
have that $\lim_{n\to\infty}P(A^{\prime+}_{\lceil k_{-}\rceil}>a_{\lceil
k_{-}\rceil}+1|R^{+}>k_{-})=1$. Besides that, when $R^{+}>k_{-}$, it holds
that $R^{+}\geq{\lceil k_{-}\rceil}$ and from (3.9), we have that
$A^{+}_{\lceil k_{-}\rceil}=A^{\prime+}_{\lceil k_{-}\rceil}$. Then, we
conclude

$\displaystyle P(B_{k_{-}}^{+})$ $\displaystyle=P(R^{+}>k_{-},R^{+}<\infty)$
$\displaystyle=P(R^{+}>k_{-},R^{+}<\infty,A^{+}_{\lceil k_{-}\rceil}>a_{\lceil
k_{-}\rceil}+1)+P(R^{+}>k_{-},R^{+}<\infty,A^{+}_{\lceil k_{-}\rceil}\leq
a_{\lceil k_{-}\rceil}+1)$
$\displaystyle\leq\frac{P(R^{+}>k_{-},R^{+}<\infty,A^{+}_{\lceil
k_{-}\rceil}>a_{\lceil k_{-}\rceil}+1)}{P(R^{+}>k_{-},A^{+}_{\lceil
k_{-}\rceil}>a_{\lceil k_{-}\rceil}+1)}+\frac{P(R^{+}>k_{-},A^{+}_{\lceil
k_{-}\rceil}\leq a_{\lceil k_{-}\rceil}+1)}{P(R^{+}>k_{-})}$
$\displaystyle=P(R^{+}<\infty,|R^{+}>k_{-},A^{+}_{\lceil
k_{-}\rceil}>a_{\lceil k_{-}\rceil}+1)+P(A^{+}_{\lceil k_{-}\rceil}\leq
a_{\lceil k_{-}\rceil}+1|R^{+}>k_{-})$
$\displaystyle\leq(\frac{1-p}{p})^{a_{\lceil k_{-}\rceil}+1}+P(A^{+}_{\lceil
k_{-}\rceil}\leq a_{\lceil k_{-}\rceil}+1|R^{+}>k_{-})\stackrel{{\scriptstyle
n\to\infty}}{{\rightarrow}}0$

where the last inequality is given by the extinction probability of at least
$a_{\lceil k_{-}\rceil}+1$ independent branching processes, each with
extinction probability $\frac{1-p}{p}$.

Applying to (3.11) all limits found, we conclude that

$\lim_{n\to\infty}P(R<k_{-})=\frac{1-p}{p}.$

$\square$

## References

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With decomposition (54), the mean-field equation (1) for $\rho$ can be
rewritten in terms of equations for $\rho_{\text{low}},\rho_{\text{high}},$
and $y$ as follows:

$\displaystyle\partial_{t}\rho_{\text{low}}(p,t)$
$\displaystyle=-s(1-2\lambda)\big{(}p-\overline{p}_{\text{low},t}\big{)}\rho_{\text{low}}(p,t)\
,$ (56) $\displaystyle\partial_{t}\rho_{\text{high}}(p,t)$
$\displaystyle=-s(1-2\lambda)\big{(}p-\overline{p}_{\text{high},t}\big{)}\rho_{\text{high}}(p,t)+2\lambda\frac{1-s\overline{p}_{t}}{1-y(t)}\big{(}\delta(p-R(\overline{p}_{t}))-\rho_{\text{high}}(p,t)\big{)}\
,$ (57) $\displaystyle\partial_{t}y(t)$
$\displaystyle=y(t)\big{(}-2\lambda(1-s\overline{p}_{\text{low},t})+s(1-y(t))(\overline{p}_{\text{high},t}-\overline{p}_{\text{low},t})\big{)}\
.$ (58)

We note that the decomposition (54) of $\rho$ with equations (56-58) is not
unique, but this choice of decomposition enables the characterization of
heterogeneous stationary distributions and, thus, phenotypic heterogeneity.

The temporal evolution equation (56) for $\rho_{\text{low}}$ has the form of
the continuous replicator equation (see equation (1) with $\lambda=0$) with
renormalized selection strength $s(1-2\lambda)$. Following the analysis that
resulted in equation (32), the solution for $\rho_{\text{low}}$ is given by:

$\displaystyle\rho_{\text{low}}(p,t)=\rho_{\text{low},0}(p)e^{-s(1-2\lambda)tp}/\mathcal{L}[\rho_{\text{low},0}](s(1-2\lambda)t)\
,\text{ with}\quad\rho_{\text{low},0}(p)=\rho_{\text{low}}(p,t=0)\ ,$ (59)

if $\lambda\leq 1/2$. As shown in the main text, the condition $\lambda\leq
1/2$ is consistent with the condition for the upper threshold of the response
probability $\lambda\leq s/2<1/2$, above which heterogeneous stationary
distributions cannot occur. For the mean $\overline{p}_{\text{low},t}$, one
obtains:

$\displaystyle\overline{p}_{\text{low},t}=-\partial_{v}\ln{\mathcal{L}[\rho_{\text{low},0}}](v)|_{v=s(1-2\lambda)t}\
.$ (60)

In other words, $\rho_{\text{low}}$ approaches a stationary
$\delta$-distribution:

$\displaystyle\rho_{\text{low}}(p,t\to\infty)$
$\displaystyle=\rho_{\text{low},\infty}(p)=\delta(p-p_{\text{low}})\ ,$ (61)
$\displaystyle\text{ with}\quad p_{\text{low}}$
$\displaystyle=\overline{p}_{\text{low},\infty}=\min(\mathrm{supp}(\rho_{\text{low},0}))=\min(\mathrm{supp}(\rho_{0}))\
.$

The temporal evolution equation (57) for $\rho_{\text{high}}$ has a similar
form as the original mean-field equation (1): it involves the sense-and-
response term with prefactor $2\lambda$, and the replicator term with
prefactor $1-2\lambda$. The sense-and-response term, however, couples to the
full production distribution $\rho$ through the argument $R(\overline{p}_{t})$
in the $\delta$-function and the prefactor $(1-s\overline{p}_{t})/(1-y(t))$,
whereas the replicator term does not couple to $\rho_{\text{low}}$ or $y$.
Equation (57) is most suitably analyzed in the space of moment and cumulant
generating functions with:

$\displaystyle
M_{\text{high}}(u,t)\mathrel{\mathop{:}}=\int_{0}^{1}\mathrm{d}{p}\
e^{up}\rho_{\text{high}}(p,t)\ ,\text{ and}\quad
C_{\text{high}}(u,t)\mathrel{\mathop{:}}=\mathrm{ln}\left(M_{\text{high}}(u,t)\right)\
,\quad u\in(-\infty,\infty)\ .$ (62)

The moments and cumulants of $\rho_{\text{high}}$ are obtained as
$M_{\text{high},k}(t)\mathrel{\mathop{:}}=\partial_{u}^{k}M_{\text{high}}(u,t)|_{u=0}$
and
$C_{\text{high},k}(t)\mathrel{\mathop{:}}=\partial_{u}^{k}C_{\text{high}}(u,t)|_{u=0}$
for $k\geq 1$. With this notation, it is
$\overline{p}_{\text{high},t}=M_{\text{high},1}(t)=C_{\text{high},1}(t)$. By
applying these transformations to the temporal evolution equation (57) of
$\rho_{\text{high}}$, one obtains:

$\displaystyle\partial_{t}M_{\text{high}}(u,t)$
$\displaystyle=-(1-2\lambda)s\left(\partial_{u}M_{\text{high}}(u,t)-M_{\text{high},1}(t)M_{\text{high}}(u,t)\right)$
$\displaystyle\quad+2\lambda\frac{1-s\overline{p}_{t}}{1-y(t)}\left(e^{uR(\overline{p}_{t})}-M_{\text{high}}(u,t)\right)\
,$ (63) $\displaystyle\partial_{t}C_{\text{high}}(u,t)$
$\displaystyle=-(1-2\lambda)s\left(\partial_{u}C_{\text{high}}(u,t)-C_{\text{high},1}(t)\right)$
$\displaystyle\quad+2\lambda\frac{1-s\overline{p}_{t}}{1-y(t)}\left(e^{uR(\overline{p}_{t})}e^{-C_{\text{high}}(u,t)}-1\right)\
,$ (64)

in which the coupling of $\rho_{\text{high}}$ to $\rho_{\text{low}}$ and $y$
is apparent explicitly through the occurrence of the factor $1-y(t)$ and
implicitly through the occurrence of
$\overline{p}_{t}=y(t)\overline{p}_{\text{low},t}+(1-y(t))\overline{p}_{\text{high},t}$.
The corresponding equations of motion for the first three cumulants are, thus,
obtained as:

$\displaystyle\partial_{t}C_{\text{high},1}(t)$
$\displaystyle=-(1-2\lambda)sC_{\text{high},2}(t)+2\lambda\frac{1-s\overline{p}_{t}}{1-y(t)}\left(R(\overline{p}_{t})-C_{\text{high},1}(t)\right)\
,$ (65) $\displaystyle\partial_{t}C_{\text{high},2}(t)$
$\displaystyle=-(1-2\lambda)sC_{\text{high},3}(t)+2\lambda\frac{1-s\overline{p}_{t}}{1-y(t)}\left(-C_{\text{high},2}(t)+(R(\overline{p}_{t})-C_{\text{high},1}(t))^{2}\right)\
,$ $\displaystyle\partial_{t}C_{\text{high},3}(t)$
$\displaystyle=-(1-2\lambda)sC_{\text{high},4}(t)$
$\displaystyle\quad+2\lambda\frac{1-s\overline{p}_{t}}{1-y(t)}\left(-C_{\text{high},3}(t)-3(R(\overline{p}_{t})-C_{\text{high},1}(t))C_{\text{high},2}(t)+(R(\overline{p}_{t})-C_{\text{high},1}(t))^{3}\right)\
.$

At stationarity, it is $\partial_{t}y(t)=0$ and $y(t)\equiv y_{\infty}$ with
(see equation (58); recall also that
$\overline{p}_{\infty}=y_{\infty}p_{\text{low}}+(1-y_{\infty})p_{\text{high}}$):

$\displaystyle
2\lambda(1-sp_{\text{low}})=s(1-y_{\infty})(p_{\text{high}}-p_{\text{low}})\
,\text{ or
equivalently}\quad(1-2\lambda)(1-sp_{\text{low}})=1-s\overline{p}_{\infty}\ .$
(66)

Thus, assuming that a stationary value $0<y_{\infty}<1$ exists, it fulfils the
self-consistency relation:

$\displaystyle
y_{\infty}=1-\frac{2\lambda}{s}\frac{1-sp_{\text{low}}}{p_{\text{high}}-p_{\text{low}}}=\frac{p_{\text{high}}-\overline{p}_{\infty}}{p_{\text{high}}-p_{\text{low}}}\
.$ (67)

Note that we denoted $y_{\infty}$ simply as $y$ in the main text.

If $0<y_{\infty}<1$ exists, it follows that the stationary solution for
$\rho_{\text{high}}$ can be obtained via equation (63) in terms of the
stationary moment generating function
$M_{\text{high},\infty}(u)=M_{\text{high}}(u,t\to\infty)$ with:

$\displaystyle\partial_{u}M_{\text{high},\infty}(u)-p_{\text{low}}M_{\text{high},\infty}(u)=(p_{\text{high}}-p_{\text{low}})e^{uR(\overline{p}_{\infty})}\
,\text{ and}\quad
p_{\text{high}}=\partial_{u}M_{\text{high},\infty}(u)|_{u=0}\ ,$ (68)

where the relation between $p_{\text{low}},p_{\text{high}},$ and $y_{\infty}$
in equation (66) was exploited and the definition
$p_{\text{high}}=\overline{p}_{\text{high},\infty}$ translates into the
boundary condition. In total, one obtains
$M_{\text{high},\infty}(u)=e^{up_{\text{high}}}$ with the self-consistency
relation $p_{\text{high}}=R(\overline{p}_{\infty})$. In other words,
$\rho_{\text{high}}$ approaches a stationary $\delta$-distribution:

$\displaystyle\rho_{\text{high}}(p,t\to\infty)$
$\displaystyle=\rho_{\text{high},\infty}(p)=\delta(p-p_{\text{high}})\ ,$ (69)
$\displaystyle\text{ with}\quad p_{\text{high}}$
$\displaystyle=\overline{p}_{\text{high},\infty}=R(\overline{p}_{\infty})=R(2\lambda/s+(1-2\lambda)p_{\text{low}})\
.$

For $p_{\text{low}}=\min(\mathrm{supp}(\rho_{0}))=0$, one recovers from
equations (61, 67, 69) the heterogeneous stationary distribution
(LABEL:eq:adaptation:quasistationary:solution) that was given in the main
text.

For Fig. 2(F) of the main text, equations (58), (60), and (65) were
numerically integrated with $C_{\text{high},i}(t)=0$ for $i\geq 3$ and for all
$t$, and initial conditions $y_{0}=0.99$,
$\rho_{\text{low},0}\sim\mathrm{Uniform}(0,1)$, and
$\rho_{\text{high},0}\sim\delta(\cdot-R(0.5))$. The choice of initial
conditions, however, is not important for the asymptotic behavior, see
Supplementary Fig. 1.

###  13.10 Linear stability analysis of heterogeneous stationary distributions

Here, we supplement the statements from the main text on the stability of
heterogeneous stationary distributions
(LABEL:eq:adaptation:quasistationary:solution) in the linear approximation
around stationarity. For the sake of simplicity and feasibility, we carry out
the stability analysis in the space of cumulants. To this end, we define the
vector:

$\displaystyle\mathbf{c}(t)=(y(t),C_{\text{low},1}(t),C_{\text{high},1}(t),C_{\text{low},2}(t),C_{\text{high},2}(t),\dots)=(c_{0}(t),c_{1}(t),c_{2}(t),\dots)\
,$ (70)

which is at stationarity:

$\displaystyle\mathbf{c}(t\to\infty)=\mathbf{c}_{\infty}=(y,C_{\text{low},1},C_{\text{high},1},C_{\text{low},2},C_{\text{high},2},\dots)=(y,p_{\text{low}},p_{\text{high}},0,0,\dots)=(c_{0},c_{1},c_{2},\dots)\
.$ (71)

The cumulants of $\rho_{\text{low}}$ are obtained in the same way as for
$\rho_{\text{high}}$, that is as
$C_{\text{low},k}(t)\mathrel{\mathop{:}}=\partial_{u}^{k}C_{\text{low}}(u,t)|_{u=0}$
for $k\geq 1$ from
$M_{\text{low}}(u,t)\mathrel{\mathop{:}}=\int_{0}^{1}\mathrm{d}{p}\
e^{up}\rho_{\text{low}}(p,t)$ and
$C_{\text{low}}(u,t)\mathrel{\mathop{:}}=\mathrm{ln}\left(M_{\text{low}}(u,t)\right)$
for $u\in(-\infty,\infty)$. With this notation, the equations of motion for
$y(t)$ in equation (58) and the cumulants of $\rho_{\text{low}}$ and
$\rho_{\text{high}}$, respectively, are cast into the compact form:

$\displaystyle\partial_{t}c_{i}(t)=F_{i}(\mathbf{c}(t))\ ,\text{ for }i\geq 0\
.$ (72)

Upon introducing the distance $\Delta\mathbf{c}$ to the stationary vector
$\mathbf{c}_{\infty}$, that is
$\Delta\mathbf{c}=\mathbf{c}-\mathbf{c}_{\infty}$, one obtains the temporal
behavior of $\Delta\mathbf{c}$ as follows:

$\displaystyle\partial_{t}\Delta c_{i}(t)$
$\displaystyle=F_{i}(\mathbf{c}_{\infty}+\Delta\mathbf{c}(t))=\sum_{j=0}^{\infty}J_{ij}(\mathbf{c}_{\infty})\Delta
c_{j}(t)+\mathcal{O}(\lVert\Delta\mathbf{c}\rVert^{2})\ ,\text{ for }i\geq 0\
,$ (73)

and with Jacobian $J_{ij}(\mathbf{c}_{\infty})=\frac{\partial
F_{i}(\mathbf{c})}{\partial c_{j}}|_{\mathbf{c}=\mathbf{c}_{\infty}}$, whose
entries are obtained after some algebra as:

$\displaystyle J_{00}$ $\displaystyle=-sy(p_{\text{high}}-p_{\text{low}})\ ,$
(74) $\displaystyle J_{01}$
$\displaystyle=-sy(1-y)\frac{1-sp_{\text{high}}}{1-sp_{\text{low}}}\ ,$ (75)
$\displaystyle J_{02}$ $\displaystyle=sy(1-y)\ ,$ (76) $\displaystyle J_{10}$
$\displaystyle=0\ ,$ (77) $\displaystyle J_{11}$ $\displaystyle=0\ ,$ (78)
$\displaystyle J_{12}$ $\displaystyle=0\ ,$ (79) $\displaystyle J_{20}$
$\displaystyle=-s(1-2\lambda)(p_{\text{high}}-p_{\text{low}})^{2}R^{\prime}(\overline{p}_{\infty})\
,$ (80) $\displaystyle J_{21}$
$\displaystyle=s(1-2\lambda)y(p_{\text{high}}-p_{\text{low}})R^{\prime}(\overline{p}_{\infty})\
,$ (81) $\displaystyle J_{22}$
$\displaystyle=s(1-2\lambda)(p_{\text{high}}-p_{\text{low}})((1-y)R^{\prime}(\overline{p}_{\infty})-1)\
,$ (82)

and,

$\displaystyle J_{i,i+2}$ $\displaystyle=-s(1-2\lambda)\ ,\text{ for }i\geq 1\
,$ (83) $\displaystyle J_{2i,2i}$
$\displaystyle=-s(1-y)(p_{\text{high}}-p_{\text{low}})\ ,\text{ for }i\geq 2\
,$ (84) $\displaystyle J_{i,j}$ $\displaystyle=0\ ,\text{ otherwise }.$ (85)

The eigenvalues of the matrix $J$ determine the stability of the heterogeneous
stationary distribution up to linear order in perturbations at the level of
cumulants around stationarity. Its eigenvalues are given by:

  * •

the two eigenvalues $\gamma_{1,2}$ of the $2\times 2$ matrix,

$\displaystyle\tilde{J}$
$\displaystyle=\left(\begin{array}[]{cc}J_{00}&J_{02}\\\
J_{20}&J_{22}\end{array}\right)$ (88)
$\displaystyle=\left(\begin{array}[]{cc}-sy(p_{\text{high}}-p_{\text{low}})&sy(1-y)\\\
-s(1-2\lambda)(p_{\text{high}}-p_{\text{low}})^{2}R^{\prime}(\overline{p}_{\infty})&s(1-2\lambda)(p_{\text{high}}-p_{\text{low}})((1-y)R^{\prime}(\overline{p}_{\infty})-1)\end{array}\right)\
,$ (91)

  * •

one eigenvalue 0,

  * •

and infinitely many pairs of eigenvalues with values 0 and
$-s(1-y)(p_{\text{high}}-p_{\text{low}})<0$ (because
$p_{\text{high}}-p_{\text{low}}>0$ and $1-y>0$ for the considered bimodal
distributions).

For simplicity of the discussion, we assume
$p_{\text{low}}=\min(\mathrm{supp}(\rho_{0}))=0$ in the following, and also
introduce the parameter $\beta=2\lambda/s$ as in the main text. The two
eigenvalues $\gamma_{1,2}$ of $\tilde{J}$ are given by:

$\displaystyle\gamma_{1,2}$
$\displaystyle=\frac{1}{2}\mathrm{Tr}(\tilde{J})\pm\left(\frac{1}{4}\mathrm{Tr}(\tilde{J})^{2}-\mathrm{Det}(\tilde{J})\right)^{1/2}\
,$ (92) $\displaystyle\text{with}\quad\mathrm{Tr}(\tilde{J})$
$\displaystyle=s(1-2\lambda)(\beta
R^{\prime}(\beta)-R(\beta))-s(R(\beta)-\beta)\ ,$ (93)
$\displaystyle\text{and}\quad\mathrm{Det}(\tilde{J})$
$\displaystyle=s^{2}(1-2\lambda)R(\beta)(R(\beta)-\beta))\ .$ (94)

Linear stability for small $\lambda$.
Under the assumptions $R(0)=0$ and $1<R^{\prime}(0)<\infty$, one checks that
for $0<\lambda\ll 1$ the eigenvalues of the Jacobian $\tilde{J}$ in equation
(92) are given by:

$\displaystyle\gamma_{1,2}=-\lambda(R^{\prime}(0)-1)+\mathcal{O}(\lambda^{3})\pm
i\lambda\big{(}(R^{\prime}(0)-1)(3R^{\prime}(0)+1)+\mathcal{O}(\lambda)\big{)}^{1/2}\
,$ (95)

and, thus, $\mathrm{Re}(\gamma_{1,2})<0$ as $\lambda\searrow 0$.

Therefore, for small response probabilities, the heterogeneous stationary
distribution (equation (LABEL:eq:adaptation:quasistationary:solution) of the
main text) is stable up to linear order in perturbations at the level of
cumulants around stationarity (here shown under the assumptions
$p_{\text{low}}=\min(\mathrm{supp}(\rho_{0}))=0$, $R(0)=0$, and
$1<R^{\prime}(0)<\infty$).

Linear stability for the specific response function
$R(\beta)=\beta+\kappa\cdot\sin(\pi\beta)$.
Upon choosing the response function $R(\beta)=\beta+\kappa\cdot\sin(\pi\beta)$
with $\beta\in[0,1]$ (that is $\lambda\in[0,s/2]$) and with
$\kappa\in[0,1/\pi]$, one checks that all eigenvalues of the Jacobian
$\tilde{J}$ in equation (92) have negative real part.

Therefore, for the special choice of the response function that up-regulates
the cellular autoinducer production for all sensed average productions in the
population, all heterogeneous stationary distributions (equation
(LABEL:eq:adaptation:quasistationary:solution) of the main text) for choices
of the parameters $\lambda\in[0,s/2]$ and $\kappa\in[0,1/\pi]$ are stable up
to linear order in perturbations at the level of cumulants around
stationarity.