We survey some recent rigorous results regarding the dynamics of spin glasses. We first survey recent results with Gheissari and Ben Arous regarding spectral gaps for these models. In particular, we briefly present the extension of the large deviations based approach of Ben Arous and Jagannath [Commun. Math. Phys. **361**, 1–52 (2018)] to the setting of spherical spin glasses, unifying the treatment of the Ising and spherical models. We then turn to the new *bounding flows* method introduced by Ben Arous *et al.* [“Bounding flows for spherical spin glass dynamics,” e-print arXiv:1808.00929] regarding the nonactivated dynamics. We end with a report on progress on activated dynamics.

## I. INTRODUCTION

Let *H* be a smooth, random function on the sphere $SN={x\u2208RN:||x||22=N}$. Suppose that we run Langevin dynamics corresponding to this Hamiltonian and let *X*_{t} denote the corresponding process. Consider the following naive questions:

How well does the dynamics,

*X*_{t}, optimize*H*? That is, how low does*H*(*X*_{t}) reach on observable time scales?How long does it take to produce a sample from the Gibbs distribution at inverse temperature

*β*> 0, namely,

Specifically, for which *β* would sampling take exponential time from a *worst case* initialization?

Here, *dx* is the uniform measure on $SN$, which we call the *spherical setting*. Alternatively, one can take *dx* to be the counting measure on {−1,1}^{N}, which we call the *Ising setting.* We review here some recent progress toward a rigorous understanding of some of these questions.

At this point, we should be a bit more specific to have a hope of answering these questions from a mathematical perspective. So let us consider the following assumptions on *H*:

*Assumptions.*

*H*(*x*) is a stochastic process on $SN$ which satisfies the following properties:

*H*is Gaussian,*H*is centered, $EH=0$, andthe law of

*H*is isotropic and well-defined for all*N*≥ 1, i.e.,

*ξ*(

*t*), where “·” denotes the Euclidean inner product.

^{1}states that the most general class of such

*ξ*are of the form $\xi (t)=\u2211p\u22651ap2tp$ for some sequence (

*a*

_{p}) with

*ξ*(1 +

*ϵ*) <

*∞*for some

*ϵ*> 0. Put differently, for

*p*≥ 1, let

*H*must be of the form

*a*

_{p}) as above. Throughout the following, we refer to

*ξ*as the

*model.*

A truly remarkable observation is that these Gaussian processes play a central role in the statistical physics of disordered media. To see this, consider the following spin systems. Take as configuration space either the discrete hypercube, Ω_{N} = {−1, +1}^{N}, or the hypersphere, $\Omega N=SN$. On Ω_{N}, we have a Hamiltonian *H*. When *H* = *H*_{p}, this corresponds to Derrida’s *p*-spin model, with the case *p* = 2, of course, being the classical Sherrington–Kirkpatrick (SK) model on either the hypercube or the sphere. Thus, the above questions naturally connect to questions regarding the dynamics of mean field spin glasses.

The dynamics of spin glasses are expected to have a rich phenomenology. Traditionally, the study of these dynamics has split in o three regimes at low temperatures:

the time scale to equilibrium;

the

*activated dynamics*: time scales that are exponentially long but much shorter than the time to equilibrium; andthe

*nonactivated dynamics*: time scales that are order 1 in*N*.

*bounding flows*approach for the nonactivated dynamics. We end with a review of the activated dynamics.

Notably absent from this review is the expansive but nonrigorous literature from the theoretical physics community in this direction. For an in-depth review of the predictions surrounding spin glasses and, more broadly, glassy systems, we refer the reader to the surveys in Refs. 3–5.

## II. WARM-UP

In order to understand the essential issues at play, consider the following warm-up problem. Suppose that we want to optimize the function *H*_{p}(*x*) constrained to the sphere $SN$, that is,

At first glance, this is a fairly tame sounding problem of elementary analysis. How hard could this be? Let us roll up our sleeves and think a little about the problem.

When *p* = 1, this problem is, of course, not so hard: we are optimizing a random linear form, *H*_{1}(*x*) ∼ (*g*, *x*), and this has two critical points almost surely. When *p* = 2, if we symmetrize the polynomial, we are optimizing the Rayleigh quotient of a real symmetric matrix, and since the matrix is Gaussian, it has almost surely (a.s.) nondegenerate spectrum, in particular, optimization turns into the question of finding the 2*N* critical points corresponding to the eigenvectors. But what about the case *p* ≥ 3?

It turns out that for *p* ≥ 3, *H*_{p} has *exponentially* many critical points eventually almost surely! In fact, it was shown by Auffinger, Ben Arous, and Černy^{6} and Subag^{7} that there are exponentially many critical points of every index (note that by Gaussianity, all critical points are nondegenerate a.s.). It is expected (and shown in some cases, see below) that this complexity plays a vital role in the evolution of the dynamics.

## III. TIME TO EQUILIBRIUM

Let us begin by considering the time to equilibrium. As *X*_{t} is a Markov process, we can define the corresponding infinitesimal generator, *L*. Let *λ*_{1} denote the first nontrivial eigenvalue of *L*. It is a classical fact that *λ*_{1} measures the time to equilibrium via the elementary inequality

where *P*_{t} = *e*^{tL} is the heat semigroup.

The literature surrounding the spectral gap of spin glass dynamics has been much sparser than those surrounding the nonactivated and activated regimes. The behavior of this gap was understood in great detail in the case of local and global Glauber-type dynamics of the Random Energy Model in the early work of Fontes, Isopi, Kohakayawa, and Picco.^{8} An exponential lower bond for the mixing time was obtained in the work of Mathieu^{9} for general models; furthermore, bounds on the correlation time for a single spin were studied by Montanari and Semerjian^{10} for Glauber dynamics of diluted *p*-spin models.

More recently, there has been quite a bit of progress on general reversible dynamics of *p*-spin models. At high temperatures, there has been work on proving order 1 logarithmic Sobolev constants (which, in particular, imply order 1 lower bounds on *λ*_{1}). In Ref. 11, it was shown that in the case of spherical mixed *p*-spin models, at very high temperature, one obtains an order 1 logarithmic Sobolev inequality. (We remark here that in the case of the spherical 2-spin model, this argument should be sharp.) On the other hand, for the Ising spin SK model (*p* = 2), Bauerschmidt and Bodineau showed^{12} that the log Sobolev constant is order 1 by an elegant supersymmetric argument.

The bulk of our review concerns the development of a characterization of the slow mixing regime for general *p*-spin models. This program was started in the work of Gheissari and Jagannath in Ref. 11, where it was shown that slow mixing occurs at very low temperatures for spherical *p*-spin models using the recent geometric study of Subag.^{13} More recently, Ben Arous and Jagannath^{2} proved that any local, reversible dynamics for a general class of mean field spin glasses are exponentially slow in the entire low temperature, or replica symmetry breaking (RSB), phase for a broad class of Ising and spherical models. More precisely, we give sufficient conditions for the spectral gap of these dynamics to be exponentially small. In the case of Ising spin models, we provided a far wider criterion that holds in a broader range of temperatures. In particular, we show that the dynamical glass transition (we define this momentarily) occurs at a strictly higher temperature than the static transition in the case of convex models with no *p* = 2 component, e.g., *p*-spin models with *p* ≥ 3. While the latter results were originally stated in Ref. 2 in the Ising spin setting, they can easily be extended to the spherical setting due to the recent work of Ko on constrained free energy bounds for spherical models^{14} by only minor modifications. Let us now describe these broader results for spherical models.

### A. Free energy barriers in spherical spin glasses

For the rest of this section, let us fix a model *ξ* that is convex and suppose that we are working in the spherical setting. We will also add an external field term so that the Hamiltonian we will consider is of the form

where *h* ≥ 0 is the strength of the external field and *H*_{0} is a Hamiltonian with model *ξ*.

We are interested in the time to equilibrium for the Langevin dynamics for this Hamiltonian, namely, the heat flow on $SN$ with infinitesimal generator

where *β* > 0 is the inverse temperature, *g* is the usual metric, ∇ is the covariant derivative, and Δ is the spherical Laplacian. Our aim is to prove exponentially small upper bounds on *λ*_{1}, the first nontrivial eigenvalue of *L*.

The key issue is to show that at sufficiently low temperatures the model admits what is called a *free energy barrier* for the overlap distribution. To see this, we begin by observing a quenched large deviation principle for the overlap distribution. Let $QN$ denote the *quenched overlap distribution*,

(If *x*^{1}, *x*^{2} are iid drawn from $\pi N,\beta \u22972$, the normalized inner product *x*^{1} · *x*^{2}/*N* is called *the overlap*, hence the name overlap distribution.) One then has the following theorem:

*For any ξ convex, β >* 0, *and h ≥* 0, *the sequence* ${QN}$ *admits a large deviation principle with good rate function I from* (1) *and rate N almost surely.*

In the spherical setting, this large deviation principle is a consequence of a recent result of Ko,^{14} where he proved the sharpness of the Guerra-Talagrand bounds for spherical models, following the work of Panchenko^{15} in the Ising spin setting. The proof of this result is identical to that of Theorem 1.13 from Ref. 2 except applying the lower bound of Ko instead of that of Panchenko from Ref. 15 and so it is omitted.

^{16}

*Definition 1.*

We say that there is a *free energy barrier for the overlap* or simply **FEB** holds, if $H>0$.^{17}

Notice that $H$ is positive if and only if the large deviation rate function is nonmonotone to the left or right of a zero. (As *I* is a rate function, it must have at least one zero.) We invite the reader here to compare the notion of free energy barriers to that of free energy wells from Ref. 18. The former relates to mixing times and the latter relates to hitting times, and the quantities defined are evidently complementary.

With this definition in hand, one may then show the following theorem.

*We have that*

*almost surely. In particular, if ξ is convex and*

*FEB**holds, this is strictly negative.*

The proof of this result is again identical to the Proof of Theorem 1.16 from Ref. 2, except applying Theorem 1 in place of Theorem 1.13 from Ref. 2 and Theorem 3.3 from Ref. 2 in place of Theorem 3.2 from Ref. 2.

Let us pause for a moment to understand the core idea behind this inequality. The goal is to bound the spectral gap for the Langevin dynamics which is reversible with respect to *π*_{N}. The idea here is to understand the dynamical transition via *replicated dynamics*—the evolution of two independently evolving diffusions in the same environment. If there is a free energy barrier for the overlap, then the replicated dynamics must cross this barrier in the free energy landscape in order to reach equilibrium and this transit will take an amount of time that is exponential in the height of this barrier. The corresponding result for the original dynamics then follows by applying a standard tensorization argument. For a rigorous treatment of free energy barriers that is unified between the discrete and manifold state spaces setting, as well as the use of replicated dynamics, see Ref. 2.

The question is then how one checks this condition. To understand the task at hand, let us take a moment to define *I*. We begin by defining the *Crisanti-Sommers functional*.

^{*}lower semicontinuous and strictly convex so that the existence and uniqueness of its minimizer are guaranteed. Here and in the following, we refer to the minimizer,

*μ*

_{β,h}, as the

*Parisi measure*.

*q*∈ [−1, 1]. Let $Qt:[0,1]\u2192P2$ be a continuous, weakly differentiable,

^{19}nondecreasing path

^{20}in $P2$ with boundary data

_{q}.

*ξ*(

*Q*) coordinatewise, then, by Lemma 4.2 from Ref. 2,

*ξ*(

*Q*) is positive semidefinite for any

*Q*that is positive semidefinite. For

*b*

_{1},

*b*

_{2}≥ 0 and $\lambda \u2208R$, let

*b*

_{i}and

*λ*are such that

*A*is positive definite. For $\nu \u2208M1([0,1])$, consider the matrix integral

**=**

*h**h*(1, 1). One then has the following equation:

*b*

_{i}and

*λ*such that

*A*

_{1}is positive definite. Evidently, the task of controlling the shape of

*I*is a major challenge.

To side step this issue, we introduce a simple set of analytical conditions that imply the desired inequality. The first condition is that of replica symmetry breaking.

*Definition 2.*

We say that **RSB** holds if the Parisi measure, *μ*_{β,h}, has support of cardinality at least 2, Card(supp(*μ*_{β,h})) ≥ 2.

The condition **RSB** is the defining property of the *static glass phase.* In particular, note that we have the following theorem:

*For any ξ convex, β >*0,

*and h ≥*0,

*we have that*

*replicon eigenvalue*. Fix a model

*ξ*and let

*μ*be its Parisi measure, for

*q*∈ supp(

*μ*

_{β,h}), we denote the replicon eigenvalue by

*Definition 3.*

We say that **PREV** holds if for some *q* ∈ supp (*μ*_{β,h}), we have that Λ_{R}(*q*, *μ*_{β,h}) > 0.

One can then show the following theorem:

*If* *PREV**and* *RSB**hold, then* *FEB**holds.*^{21}

### B. Static vs dynamical transitions

When studying glassy systems, one often finds that the notion of *glassiness* is defined in the literature both in terms of “static” type phenomena—namely, complexity of a landscape—and dynamical properties, namely, dynamical slow down. Various competing notions of static and dynamical transitions have been introduced in the physics literature and the mathematics literature over the years. A natural question regards the relation between these transitions. Here, we review recent results regarding the relation between the “static” spin glass transition, the static replica symmetry breaking transition, and a dynamical transition. In particular, we focus on a dynamical transition which is naturally connected to static quantities in other systems, namely, the transition in the change of scaling of the spectral gap which has natural connections to the static quantities.

Let us first recall the definition of these two transitions. Let *μ*_{β,h} denote the Parisi measure corresponding to the model *ξ*, inverse temperature *β* > 0, and external field *h* ≥ 0. Consider now the following two temperatures:

The first critical temperature, $\beta s\u22121$, corresponds to the static replica symmetry breaking transition. That is, as soon as *β* > *β*_{s}, we say that the replica symmetry has been broken and that the system is in the spin glass phase. The second critical temperature, $\beta d\u22121$, corresponds to the onset of exponentially slow *L*^{2} mixing.

As a consequence of the preceding results, one can show that there is a *strict* inequality between these two temperatures at zero external field.

*If h =* 0, $\xi 0\u2032\u2032(0)=0$, *and ξ is convex, then β*_{s} *> β*_{d}*.*

Put simply, Theorem 5 states that there is a regime of temperature *β*_{d} < *β* < *β*_{s} in which the system is in the “trivial” or high temperature phase from a static perspective, but is still in a glassy phase from the perspective of the dynamics. More precisely, the proof shows that if we define $\beta FEB\u22121$ to be the temperature below which there is a free energy barrier for the overlap, then *β*_{FEB} < *β*_{s}. Thus, even though the overlap distribution is (heuristically) trivial (since the Parisi measure is trivial), there is still a free energy barrier as the rate function is nonmonotone. It remains an exciting question to determine under what (if any) conditions *β*_{d} is also the critical temperature for the existence of a free energy barrier.

The Proof of Theorems 3–5 are minor modifications of the results from Ref. 2. As the proof of these results in the spherical setting do not appear elsewhere in the literature, sketches of these modifications are presented in the Appendix.

## IV. NONACTIVATED DYNAMICS

The dynamics of spin glasses are also expected to be very interesting in the so-called *nonactivated* regime. This is the regime of times that are order 1 in *N*, i.e., one first takes the limit in *N* before studying the dynamics.

The short-time dynamics of the SK model in a double-well potential as well as with Ising spins were also studied in this regime by Ben Arous and Guionnet^{22,23} and Grunwald,^{24} respectively. Here, one finds that an appropriate scaling limit of the dynamics is given by a non-Markovian process that is characterized by a self-consistent equation.

The most common approach in the physics literature to studying the nonactivated dynamics of spin glasses goes via the *Crisanti–Horner–Sommers–Cugliandolo–Kurchan* (CHSCK) equations (sometimes simply called the Cugliandolo–Kurchan equations) which were introduced in Refs. 25 and 26 for spherical spin glass models. Here, the key idea is to look at observables at two different times, namely, the *correlation*—the normalized inner product of the dynamics with itself at times *t* and *s*—and the *response*—the time derivative (in *t*) of the inner product between the driving Brownian motion at time *t* and the dynamics at time *s*. In the mathematics literature, this approach was placed on a rigorous footing by the seminal works of Ben Arous, Dembo, and Guionnet^{27,28} for a “soft” relaxation of the spherical constraint, which in the case *p* = 2 led to a proof of the aging phenomenon. At high temperature, the problem was studied in the limit in which the relaxation goes to zero.^{29}

At this time, however, it seems challenging to use either of these approaches to answer the following questions:

Do the dynamics reach a certain energy level and remain below there in finite time?

Do the dynamics get trapped by critical regions?

When started near a critical point how does the dynamics escape?

How do the answers to these questions change as one varies the initialization?

Motivated by these questions, Ben Arous, Gheissari, and Jagannath introduced^{30} an elementary new approach to answering these questions by studying one-time observables, namely,

where *X*_{t} is Langevin dynamics on the sphere. It was found in Ref. 30 that the weak limit points of these dynamics satisfy a system of autonomous differential inequalities whose solutions are called the *bounding flows*:

*For every p, β, T, and all initial data, the law of U*

_{N}

*(t) from (*

*2*

*) is tight in*$M1(C([0,T]2))$

*. Moreover, every limit point, U(t) = (u(t),*$v$

*(t)), is C*

^{1}

*and satisfies*

*where, for some*Λ

_{p}

*< ∞,*

As a consequence of this system of differential inequalities, one can obtain answers to the preceding questions. Indeed, one can develop an approximate phase diagram for these dynamics. One obtains the following answers to the first two questions:

There is a constant

*T*_{0}such that for all initializations, the energy must reach and remain below a certain energy level which has an explicit formula.This absorbing region is uniformly bounded away from critical points. In particular, one obtains an explicit, uniform lower bound on the gradient in this region.

Suppose that one starts the dynamics in the neighborhood of a critical point which has negative energy and is a saddle point of diverging index. It is not hard to show that for such a point, the empirical spectrum of the Hessian straddles zero so that there are order *N* positive, negative, and “nearly flat” eigenvalues. How does the dynamics evolve if, working in the local coordinates of the Hessian, the initial data are entirely supported in the nearly flat part of the spectrum? What about a part of the spectrum that is uniformly negative? Naively, one might expect that at very low temperatures, *β* ≫ 1 but still order 1, the dynamics should instantaneously slide down the saddle. It turns out, however, that this is not the case. Instead the dynamics will *climb* the saddle by an *extensive amount* in order 1 time, regardless of where it is initialized within a certain ball around the critical point (this ball being of extensive size as well).

*u*, $v$ > 0, consider the set

*Fix β >*0

*and p >*2

*. For every η >*0

*and δ < pηβ*

^{−1}

*, there exists c*

_{1}

*, c*

_{2}

*, ρ >*0

*such that*$P\u2212$

*almost surely,*

## V. ACTIVATED DYNAMICS

So far we have reviewed the longest time scales—the time to equilibrium—and the shortest time scales, time scales that are order 1. At low temperature, the intermediate regime—time scales that are exponentially large but much shorter than the time to equilibrium–have received a tremendous amount of attention particularly in the mathematics literature. It is in this regime in which one expects a deep connection to trapped models and, in particular, one expects aging to occur.

Following the seminal works of Bouchaud^{31} and Bouchaud and Dean^{32} in the physics literature, the question of aging in the activated regime was studied for *random hopping time* dynamics for the Random Energy Model (REM) by Ben Arous, Bovier, and Gayrard.^{33–35} This was later extended to the Ising *p*-spin model in several works^{36–38} again for random hopping time dynamics. A general approach to proving aging for trapped models was introduced by Ben Arous and Černy.^{39} A spectral approach to aging in some models was studied by Bovier and Faggionato.^{40} There is a beautiful and deep theory of aging, and it is beyond the scope of this review to give a detailed account of this theory, its connections to fractional kinetics, stable subordinators, etc. Instead, we refer the reader to the early survey in Ref. 41.

Finally, we note here that understanding aging for Metropolis or Glauber dynamics for Ising spin models (or Langevin dynamics for spherical models) remains a major challenge except in the case of the Random Energy Model where this has been achieved in the recent remarkable works of Černy and Wassmer^{42} and Gayard.^{43} These results have since led to a revisiting of these questions in the physics literature as well.^{44}

## ACKNOWLEDGMENTS

A.J. is thankful to the organizers of the ICMP 2018 session on equilibrium statistical mechanics for the opportunity to present some of the work discussed here. A.J. also thanks his collaborators G. Ben Arous and R. Gheissari with whom these results were developed. This work was supported by an NSF mathematical sciences postdoctoral research fellowship, NSF Grant No. OISE-1604232.

### APPENDIX: PROOF OF RESULTS FROM SEC. III A

In this section, we collect the proof of the results from the previous sections that did not already appear in the literature. These are, by and large, minor modifications of arguments already appearing in Ref. 2, so we only present here the key changes to be made to those proofs.

*Proof of Theorem 3.*

The following proof is essentially verbatim that of the corresponding result in the Ising spin setting (see Theorem 7.5 from Ref. 2). The only issue is that in this setting the proof of Holder continuity does not apply. A simple inspection, however, shows that one only needs the lower semicontinuity of the rate function which holds by definition. We provide the outline here for the convenience of the reader.

*β*,

*ξ*,

*h*) and

*u*∈ [−1, 1], the maps (

*β*,

*ξ*,

*h*) ↦

*I*(

*u*) are equicontinuous: there is a universal

*C*> 0 such that

*u*and

*ϵ*> 0, one then takes the

*ϵ*→ 0 limit).

*ξ*is even and is such that

*a*

_{2p}> 0 for all

*p*. In this case, the Parisi measure is the limiting distribution of the absolute value of the overlap (see Ref. 2, Sec. 6), from which it follows that if we let

*ζ*denote any weak limit point of the limiting law of the overlap distribution, we have that

*I*by definition of large deviation rate functions.

*ξ*is even. Let

*η*= ∑

_{p}2

^{−p}

*t*

^{p}. Then,

*ξ*

_{ϵ}=

*ξ*+

*ϵη*is as above. Thus, supp(

*μ*

_{ϵ}) ⊂ {

*I*

_{ϵ}= 0}, where

*μ*

_{ϵ}is the Parisi measure corresponding to

*ξ*

_{ϵ}. Recall now the elementary fact from Ref. 2, Lemma 7.6, that if

*ν*

_{ϵ}→

*ν*weak-

^{*}converge as probability measures on [0, 1], then for every

*q*∈ supp(

*ν*), there is a sequence of points

*q*

_{ϵ}∈ supp(

*μ*

_{ϵ}) with

*q*

_{ϵ}→

*q*. It is well-known

^{45}that

*μ*

_{ϵ}→

*μ*weak-

^{*}by a continuity argument. Thus, by lower-semicontinuity of

*I*and uniform continuity of (

*ξ*,

*β*,

*h*) ↦

*I*, if we let

*q*

_{ϵ}be the sequence corresponding to

*μ*

_{ϵ}→

*μ*, we have that

*Proof of Theorem 4.*

**RSB**and Theorem 3, there are at least two zeros. As the condition

**PREV**holds, one may apply Theorem 5.5 from Ref. 2 to obtain that around one of these points, we have the lower bound,

*q*

_{*}. (In fact, the proof implies the stronger, locally quadratic lower bound, $I(q)\u2265c\Lambda R(q*)(q\u2212q*)2>0$.) This implies that if we let the two zeros be

*q*

_{1}and

*q*

_{*}, then there is some

*q*satisfying the above bound which lies between

*q*

_{1}and

*q*

_{*}so that $H\u2265I(q*)>0$, i.e.,

**FEB**holds.

Finally to prove the desired bound regarding the dynamical transition, observe that as $\xi 0\u2033(0)=0$, the rule of signs (Theorem 1.4 of Ref. 45) and Corollary 1.3 of Ref. 45 imply the following lemma:

*Lemma 1.*

*For ξ*_{0} *with* $\xi 0\u2033(0)=0$ *and h* = 0, *we have that* 0 ∈ supp(*μ*) *is an isolated point of the support for all β* > 0.

*Proof of Theorem 5.*

*mutatis mutandis*. We present here a slightly different proof. For clarity, let us make explicit the dependence of the rate function in

*β*and denote it by

*I*

_{β}(

*x*). By Lemma 1 note that for

*β*>

*β*

_{s}, we have that there are at least two zeros of

*I*, namely,

*I*

_{β}(0) = 0 and there is some

*q*

_{1}(

*β*) > 0 such that

*I*

_{β}(

*q*

_{1}(

*β*)) = 0. Take any such sequence

*q*

_{1}(

*β*), send

*β*→

*β*

_{s}, and let $q*=lim\u0332q1(\beta )$. Observe that

*q*

_{1}is a zero of

*I*

_{β}, and the third follows by the bound (A1). [Note that implicitly in the statement of (A1) since

*I*

_{β}(

*q*

_{1}(

*β*)) = 0,

*I*

_{γ}(

*q*

_{1}(

*β*)) <

*∞*for all

*γ*.] Thus

*q*

_{*}is a zero of $I\beta s$. On the other hand, since 0 ∈ supp(

*μ*) by Lemma 1 and Λ

_{R}(0,

*μ*) = 1 > 0, it follows by (A2) that 0 is an isolated zero of

*q*. By the same reasoning, $H>0$. In particular, there is some

*q*with 0 <

*q*<

*q*

_{*}and

*I*(

*q*) > 0. Applying the intermediate value theorem to the map

*β*↦

*I*

_{β}(

*q*) −

*I*

_{β}(

*q*

_{*}) implies that in an open neighborhood of

*β*

_{s}, this difference is positive. Thus, $H>0$ in that neighborhood from which the result follows.

## REFERENCES

Note that in Ref. 2, this is stated with a sum in place of a max. As *I* is a rate function, this formulation is equivalent as at a maximum of *I*(*b*) − (*I*(*a*) + *I*(*c*)) with *a* ≤ *b* ≤ *c*, either a or c is a zero.

Note here that the definition of **FEB** here corresponds to that of **GFEB** in Ref. 2.

By weakly differentiable, we mean that its derivative in time is *W*^{1,1}.

In the sense of matrices, i.e., Q_{t} ≥ 0 as a matrix.

In Ref. 2, these conditions are called **GFEB**, **GRSB**, and **GPREV**, respectively. We drop the notation of G for notational brevity.