The first lattice correction to Bloch’s law

1 Introduction

The leading low-temperature magnetization deficit of a three-dimensional ferromagnet is determined by long-wavelength spin waves. At the next order, the lattice dispersion becomes visible. We prove that this first lattice correction gives the corresponding term in the spontaneous magnetization of the quantum Heisenberg ferromagnet, for every fixed positive half-integer spin.

1.1 The model and the result

Let S\in\{\tfrac12,1,\tfrac32,\ldots\} and \Lambda_N=\{-N,\ldots,N\}^3\subset\Z^3. At each site x let \boldsymbol S_x=(S_x^1,S_x^2,S_x^3) be the spin-S operators on \mathbb C^{2S+1}. With free boundary conditions, the Hamiltonian and total z-spin are

Every unordered lattice edge is counted once. For inverse temperature \beta>0 and field h\ge0, define

The pressure limit is the usual finite-range thermodynamic limit. Convexity and the bound |M_N|\le S|\Lambda_N| give its finite right derivative. Thus the volume limit precedes the zero-field limit in the definition of m_S.

The full ideal-magnon density uses the lattice dispersion \varepsilon_S(k)=2S\sum_{j=1}^3(1-\cos k_j):

The integral is finite in dimension three.

Expanding the lattice heat kernel gives the following explicit form. Here \zeta denotes the Riemann zeta function.

1.2 Historical context and inputs

The spin-wave description of a ferromagnet begins with Bloch’s prediction of a magnetization deficit proportional to T^{3/2}, where T=\beta^{-1} [2]. The bosonic representation of Holstein and Primakoff made the distinction between independent spin waves and their interactions explicit [9]. Dyson subsequently developed the interaction expansion [5, 6]. In that expansion, the terms of orders T^{5/2} and T^{7/2} in the magnetization already arise from the lattice dispersion of independent magnons [6, Section 5, Equation (71)]; the leading interaction contribution is of order T^4 [6, Section 8, Equation (122)]. Later effective-field-theory calculations retain this distinction and extend the expansion for cubic lattices [8]. The present result justifies the first lattice correction at fixed quantum spin, with magnetization defined by the zero-field right derivative of the thermodynamic pressure.

Rigorous thermodynamic estimates developed through a different sequence of arguments. Conlon and Solovej obtained a low-temperature free-energy bound with the predicted power but a nonoptimal coefficient [3]; Tóth improved this bound using a random-stirring representation [14]. Correggi, Giuliani, and Seiringer proved the leading spin-wave free-energy asymptotic in three dimensions for every fixed spin [4, Theorem 2.1]. Their result concerns the free energy at zero field; extracting the right derivative at zero field requires additional control. Benedikter obtained an interaction correction as a free-energy upper bound in the different regime S\to\infty with \beta S fixed [1].

The probabilistic framework comes from Tóth’s cycle representation for spin 1/2 [14] and Nachtergaele’s realization of spin S inside the symmetric tensor product of 2S spin-1/2 spaces [10, Section 2.2]. Within this framework, two companion manuscripts provide the direct inputs. Spontaneous magnetization in the quantum Heisenberg ferromagnet establishes low-temperature ordering and uniform estimates for loops conditioned to be up at prescribed points [13, Theorems 1.1 and 7.5]. Bloch’s Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions proves the leading magnetization asymptotic and develops the return-kernel and pressure arguments used below [12, Theorem 1.1 and Sections 2–7].

The heat-kernel bounds follow the energy-dissipation and composition argument of Nash [11, pp. 935–936], with the local geometric and flow estimates supplied by the companions.

1.3 The comparison behind the proof

The cycle representation replaces a spin-S site by 2S binary slots. Random exchanges join the slot trajectories into cycles on a time circle of length \beta. Requiring selected slot-time points to be up is called pinning. A single-slot down probability can be expressed through the return probability of a walk in the complement of the cycles that avoid the pins. We recall the precise identities in Section 2.

The return walk sees a periodic, moving set of obstacles. It loses an edge when the other endpoint is occupied by an obstacle, and is transported when that obstacle exchanges with a hole. The comparison must preserve both effects. For an isolated obstacle, their signed contribution to the free heat equation telescopes to the change of the profile along the obstacle trajectory, together with a cutoff error. The resulting bound is controlled by its displacement over a short block and smaller time-variation and cutoff terms. This is the cancellation that improves the error beyond the leading spin-wave scale.

We organize the proof around a finite-dimensional capacity criterion in Section 3. The criterion compares a hitting function with a trial potential and requires two estimates: a scalar residual pairing, and a bound on the residuals against Dirichlet energy. It applies to nonselfadjoint contraction kernels, so no reversibility of the exposed environment is needed. The free trial potential sums returns over all positive periods; the capacity comparison therefore retains the full Bose sum.

Section 4 strengthens the pin estimates to joint snapshot bounds and proves the path and heat-kernel estimates needed on short blocks. Section 5 constructs the free trial potential, and Section 6 proves the signed cancellation and the ordered Duhamel identities. Sections 7 and 8 establish the two residual estimates. Finally, Section 9 converts the uniform pinned comparison to the right derivative of the thermodynamic pressure and proves Corollary 1.2.

Throughout the proof S is fixed. Constants may depend on S; the local comparison estimates will be uniform in volume and in the allowed deterministic pin sets. All norms on finite slot spaces use counting measure.

2 Pinned cycles and the return walk

We first express a pinned spin deficit as a return probability for one particle. The resulting walk moves through a periodic set of holes. We state precisely the loop identities and pin estimates imported from [13, 12]; the estimates that improve their precision are proved in Section 4.

2.1 The picture and its pin law

Fix a finite free-boundary cube \Lambda=\Lambda_N\subset\Z^3 and write H_{S,\Lambda}=H_{S,N} and \ell=2S. Its slot set is V_\Lambda=\Lambda\times\{1,\ldots,\ell\}. A slot edge joins every pair of slots whose sites are nearest neighbors. The base picture consists of independent Poisson clocks of rate 1/2 on these undirected edges during [0,\beta), and independent uniform permutations of the \ell slots at each site at the seam 0. Following the swaps and then the seam permutations produces directed cycles on the time circle \R/\beta\Z. A slot-time point at an instantaneous operation is understood to lie just after that operation. Write \EE_{\mathrm b} for expectation under the base law. This is the exchange-cycle representation of Tóth [14], with the higher-spin symmetrization discussed by Nachtergaele [10, Section 2.2].

Let \partial_{\mathrm{in}}\Lambda be the sites of \Lambda having a neighbor outside it, and put

An allowed pin set is D=D_0\cup F, where F is a finite deterministic set of slot-time points. Pins require the corresponding cycles to be up. If n_D(\omega) is the number of cycles missing D, define

Under the associated color law, cycles meeting D are up and all other cycles are independently up or down with equal probabilities. These definitions also make sense for D=\varnothing. We denote probabilities under the color law by \PP_D^{\mathrm{col}}.

Hereafter all low-temperature bounds are uniform in the finite cube and in its allowed deterministic pin set. Their constants may depend on the fixed spin and on fixed geometric constants, but not on the number or placement of the extra pins.

2.2 Exposure and a fresh walk

For an allowed E and an unpinned point i, set P=E\cup\{i\}. Sample the picture under \PP_P and expose all cycles in \mathcal B_P, including their inter-site jumps and their seam mappings. At each time their occupied slots are called obstacles; the complementary slots are holes. Every pin is a hole at its prescribed time.

At fixed exposure, the fresh walk on holes has rate 1/2 on every edge between holes. When an obstacle swaps into a hole, that hole is transported in the opposite direction. A swap of two obstacles has no effect on the holes. At the seam the walk uses a fresh uniform bijection between the remaining slots at each site, respecting the exposed part of the seam permutation. The exposure is repeated periodically, while all transition randomness of the walk is resampled on each turn. This is a time-inhomogeneous Markov walk at fixed exposure; it is not obtained by following the unexposed part of a single conditioned picture.

Let q_i be its probability, starting from i, of returning to the same slot-time point at strictly positive elapsed time before hitting E. A return can occur only at an integer number of periods. We use r=\EE_Pq_i and d_E(i)=\PP_E^{\mathrm{col}}(i\text{ is down}).

The first assertion is the exact spin-S representation in [13, Proposition 2.2], with the present rate and Hamiltonian normalizations. The second combines the one-pin weight identity and fresh-turn identity in [12, Propositions 2.4 and 2.8]. The third combines [13, Proposition 3.1 and Lemma 4.4]; in particular, no spatial separation of the test points is assumed. The last assertion is [13, Lemma 2.4]. The statements include all-time boundary pins by the dense-boundary limit proved there.

2.3 Kernel conventions and energy

Kernels act forward on column vectors, with counting inner products and norms. Killing on a pin means a projection at its prescribed time; boundary killing is imposed throughout, including at interval endpoints. Killing on exit from a spatial set retains the exit rates on the generator’s diagonal. The adjoint of an interval reverses the order of all factors, inverts transports, transposes the seam averages, and reverses the order of killing projections. Thus the two sides of an instantaneous operation remain distinct slices. Each seam factor is included exactly once in a traversal.

Extend the hole graph by making every slot outside \Lambda available. Its edges join available slots over neighboring sites. For a vector extended by zero on the inner boundary and outside \Lambda, define

The exterior extension will allow flows to infinity; all test vectors will have finite support.

All these are finite-volume statements with piecewise smooth evolution. We disregard the null event of coincident Poisson marks or marks at a prescribed deterministic time. Deterministic operations that coincide are composed in their assigned order. The kernels are measurable from finite-time transition formulas, and hitting probabilities from increasing finite-horizon limits. Consequently we may average them under the actual picture law, including on events defined below.

3 A capacity comparison on one period

The probability of returning to a pin includes arbitrarily many turns of the fresh walk. We compare this probability directly, through a hitting problem on one cut. The local conclusion needed for magnetization is the following uniform comparison. Write B_r(x)=\{z\in\Z^3:|z-x|_\infty\le r\} for a site ball and set R_*=\lceil\beta^{5/3}\rceil.

The bound is uniform over the number and locations of all pins outside the displayed ball. This uniformity permits a field to be represented by additional random pins. The proof is completed in Section 9, after the required residual estimates have been established.

We first give a finite-dimensional criterion that identifies the two estimates needed for this comparison: a scalar residual at a trial potential and a residual bound in the energy seminorm. Both directions of the period occur because its transition matrix need not be symmetric. The relation between hitting potentials and energy also appears in nonreversible potential theory; see [7, Section 2, Equation (2.6)]. The comparison below follows from direct finite-dimensional identities.

Let H be a finite coordinate set, let K\subset H be killing coordinates, and fix i\in H\setminus K. Suppose that U is a nonnegative doubly substochastic matrix on \R^H, with zero rows and columns on K. It acts forward on columns. Complete each column to one by adding a cemetery state, and use U for the successive transitions of the resulting killed chain. Let q be its probability, starting at i, of returning to i after at least one step before killing. Let f_j be the probability of hitting i before killing from j, with f_i=1 and f=0 on K. Thus 0\le f\le1.

Write

All inner products and norms use counting measure. Since U is an \ell^2 contraction, e_z\ge0, and real inner products give e_z=\ip{z}{A^*z}. In the application, H is the set of holes at the time-zero cut and U is one traversal killed on E. Lemma 2.2 supplies exactly these contraction properties and their continuous-time energy interpretation.

Choose a real vector h with h_i=1 and h=0 on K, and a real number c. Define its two residuals by

Thus Ah=c\eta-\mu_+ and A^*h=c\eta-\mu_-.

The adjoint residual in (3.4) controls the distance between the hitting potential and the trial vector. The forward residual then controls the capacity itself. In particular, one scalar estimate and two energy estimates suffice; no expansion in the number of returns is involved.

The same criterion can be averaged over a random environment. Precisely, suppose that all preceding objects are measurable, that c\in[0,1] is deterministic, and that (3.6) holds on a measurable event \mathcal G with a uniform bound M. Then (3.7) and 0\le q\le1 give

We next construct an event on which diffusion and spatial flow estimates are available. After that construction, the all-turn free potential will provide the trial vector h.

4 Sparse obstacles and local propagation

The return comparison requires an exceptional set smaller than \beta^{-5/2} and spatial propagation sharper than a ballistic bound. We obtain both from a six-point sparsity estimate and a displacement estimate for isolated exposed strands. The former is an extension of the one-point argument in [12, Section 4]; the latter uses a direct recoloring of the exchange picture. We then fix a good exposure and derive the heat-kernel and flow estimates used in the comparison.

All site distances and balls in this section use the supremum norm. A site is blocked if none of its slots is a hole. In the extended hole graph, blocked sites occur only inside \Lambda.

4.1 Geometric inputs and the snapshot estimate

We first recall the geometric hypotheses under which the available slots support uniform diffusion and flows. Fix the constant M_*\ge48 used in [12, Section 3]; it is chosen there from [13, Lemma 5.5]. A time is regular at radius R_0 about x if:

1. 1.

   every B_{M_*}(z) with z\in B_{11R_0}(x) contains fewer than six blocked sites;

2. 2.

   for each blocked z with b=|z-x|_\infty>10R_0, its blocked component, with component steps of supremum length at most eight, has radius about z at most (1+b)^{1/300}.

We call the first condition local regularity.

The probability estimates are [13, Proposition 5.4 and Theorem 7.5], with their separate local and distant-component bounds, as applied in [12, Proposition 4.3]. To check the time-grid hypothesis of the pin theorem, include zero, the queried time, and every extra pin time in a finite grid and refine until every gap is at most p^{12}. Then \beta=5p^{-11/10}, and the theorem is uniform in the grid.

The Nash inequality is [13, Proposition 5.6]. Its proof uses only local regularity: alternatively fill all sufficiently remote sites with holes to obtain full regularity without changing the energy of a vector supported in B_{R_0}(x). The path assertion follows from [13, Lemma 5.5]. Every site in B_{2R_0+1}(x) has an unblocked nearest neighbor, since six blocked neighbors would violate local regularity. Along a nearest-neighbor site route replace blocked vertices by such neighbors and use the lemma’s bounded paths between successive surviving vertices. All detours have bounded size. Slots over the same site are connected through an unblocked neighbor. For R_0 above a fixed threshold the resulting paths stay in B_{3R_0}(x). The global flow envelope is [13, Proposition 6.2]; the inverse-square improvement is proved in [12, Lemma 3.6]. That proof uses only regularity at the stated radius and a fixed lower threshold, so the near envelope is available for arbitrary R_0 here.

We next strengthen the pin theorem to its natural diffusive density, for the finitely many points needed below. Write \lambda=\beta^{-3/2}.

4.2 Short blocks and isolated strands

The density \lambda in Proposition 4.2, in place of the coarser p in (4.1), makes six-obstacle events sufficiently rare on the larger spatial scales introduced next. The snapshot estimate will give at most five obstacles in a mesoscopic neighborhood. To compare an isolated moving obstacle with a smooth profile, we also need its displacement to be smaller than the block length. We prove that displacement bound before assembling the good event, so that no conditioning on that event enters its proof.

From now on i lies at time zero just after the seam, over site x, and P=E\cup\{i\}. Choose

We assume B_{50R_*}(x) is interior to \Lambda and contains no site above which E has a pin. All subsequent bounds are uniform under this assumption. We call an error rapidly small if, for every fixed j, its absolute value is at most C_j\beta^{-j} with this uniformity.

Partition each half-period into equal blocks of length between L and 2L. Subdivide the blocks into bins of length between one and three. Use the same partitions in reverse. The seam is assigned to the last forward block and bin and to the first reverse block and bin. An exposed strand in a block is the trajectory of one exposed slot at its starting slice in the chosen direction. This definition counts different strands of the same cycle separately.

In particular, (4.9) bounds a joint event, not the strand law conditioned on isolation. This is precisely what is needed when excluding rare isolated strands with large displacement.

4.3 Heat propagation at a fixed good exposure

From this point, all actual kernels kill on E, not at i. The estimates in this subsection are deterministic statements about the fresh walk at an exposure in \mathcal G. The earlier picture probabilities have already done their work in constructing that event.

4.4 Overlaps of flows

The final geometric estimate measures the energy cost of superposing flows from different source locations. It retains their separation, which will be essential when averaging the sparse source locations.

5 The ideal potential and its profiles

We now choose the trial vector for the capacity comparison. The free potential includes every positive number of periods, so its capacity already contains the full ideal-magnon density. A spatial cutoff makes it a finite-volume test vector without truncating those periods. We also obtain the spatial and temporal estimates needed to compare its evolution with the walk among holes.

Fix S, put \ell=2S, and use the scales in (4.8). Let E=D_0\cup F, where F is a finite deterministic set of slot-time pins, and let i=(x,a) be an unpinned post-seam slot at time zero. Throughout the comparison assume that B_{50R_*}(x) lies in the interior of the cube and that no point of E lies above this ball. The picture law is \PP_P, where P=E\cup\{i\}. At a fixed exposure let U be the forward one-period kernel killed on E, retaining its zero rows and columns on killed cut slots. Set A=I-U and use the energy e_y of Section 3.

5.1 The all-turn free potential

On \Z^3\times\{1,\ldots,\ell\}, let \mathscr L have rate 1/2 on every slot edge, and let T_s=\exp(s\mathscr L). Let J be the projection that averages the slots at each site. For a slot-uniform vector the site jump rate is \ell/2=S to each neighbor. Denote this site semigroup by P_s, and its kernel from the origin by p_s(z). Its Fourier multiplier is

If a vector has zero sum on the slots at every site, then \mathscr L acts on it as -3\ell I. Therefore

The free seam is J. It commutes with T_s, so the ideal full-period operator, in either direction, is

In particular the slot modes removed by the seam contribute nothing to a positive full turn.

With \eta=\ind_{\{i\}} on the full slot lattice, define the slot-uniform function

The estimates below show that the sum is finite and v(x)=O_S(\beta^{-3/2}). Positivity and the semigroup property give

Choose \chi:\R^3\to[0,1] smooth, equal to one when |z-x|_\infty\le R/2 and zero when |z-x|_\infty\ge R, with first and second derivatives bounded by C/R and C/R^2. Use its restriction to sites as a slot-uniform multiplier. The full trial vector is

and the same symbol denotes its restriction to the cut holes. It vanishes on every killed cut slot and satisfies h(i)=1. Moreover p_s(z)\le p_s(0), by the Fourier representation, so 0\le h\le1.

5.2 Forward and reverse profiles

Put g=c(\eta+v) on the full slot lattice. The forward and reverse profiles, before restriction to holes, are

Here s is elapsed time in the indicated direction. The forward traversal contains the seam at its end. The reverse traversal contains it at its beginning, with b_0 its incoming value and b_{0+} its outgoing value. Each traversal includes the seam exactly once. At every slice, the same symbols also denote the restrictions to the holes on the specified side of any instantaneous operation; “full profile values” will always refer to the formulas before restriction.

Multiplication by \chi commutes with J. Thus the full profiles undergo the ideal seam operation, and (5.4) gives the endpoint identities

The smooth part of either profile is the contribution of cv; the point part is the contribution of c\eta. This decomposition also applies to full profile values.

For r_z=|z-x|_\infty and d\ge1, introduce

For a slot function u, a bounded-distance difference at z means u(z,a)-u(z',a') with |z-z'|_\infty at most a fixed constant; the constants below may depend on that constant. This includes differences between two slots at the same site.

At the source site, the free potential therefore recovers the ideal density exactly. Fourier inversion and monotone convergence give

The integral is finite in dimension three, also as a consequence of the heat-kernel sum in (5.13).

The uncut potential need not be square summable. Its evolutions above are defined pointwise by the convergent positive series, while the trial vector and every restricted profile used in a finite cube have finite support. Equation (5.14) identifies v(x) with the target in Proposition 3.1. It remains to bound the change of return capacity when the ideal period is replaced by the walk among holes. Define \mu_+,\mu_- from this h and c by (3.2). Section 6 realizes these residuals as propagated local errors of the profiles.

6 Comparison sources and their cancellation

We now express the errors of the trial potential as sources in the actual heat equation. The key fact is a conservation identity: along an isolated obstacle, the signed source is an endpoint difference of the profile, up to its cutoff commutator. Its size is therefore controlled by the obstacle’s displacement, although the total variation of the source counts every jump. We establish this identity before using it to estimate either the scalar or the energy pairing.

Throughout this section we impose the interiority and absence of pins over B_{50R_*}(x) required in Section 5. Constants are uniform in the finite cube, the allowed pin set, and the injection slot. The two traversals and their profiles are those defined there: a_s is the forward profile and b_s is the reverse profile, each indexed by its own elapsed time s\in[0,\beta]. Profiles are evaluated on the full slot lattice before restriction to the current holes.

6.1 Sources on intervals and at operations

Fix one of the two directions and write \varphi_s for its profile. On an open interval containing no instantaneous operation, let \mathscr L_s^{\mathrm{act}} be the generator on the holes, with the prescribed killing. The continuous source is the vector

Here H_s denotes the hole set on the slice. At an instantaneous transition K:\R^{H_-}\to\R^{H_+}, the atomic source, on the outgoing slice, is

In reverse time these are the transposed maps, in reverse order. Thus a reverse atomic source lies before the operation when viewed in forward time.

Write s^+ and s^- for these collections of continuous densities and atomic vectors in the forward and reverse traversals. Integrating against a collection will mean a Lebesgue integral of its continuous part plus the sum over its atoms. Every source is supported above B_{R+2}(x). There are no pins of E or boundary slots in this region, so killing creates no additional source there.

There are two types of source. The first is the cutoff commutator, defined on the full lattice by

The uncut profile solves the free heat equation. Consequently \gamma_s is supported where R/3\le r_z\le2R, and

These assertions hold separately for the smooth part and the point part of either profile. Indeed the commutator of \mathscr L with \chi consists of a cutoff second difference times a profile value, and products of first differences. On this annulus, Lemma 5.1 gives values at most C/(\beta R) and first differences at most C/(\beta R^2), uniformly in elapsed time. The bounds on the cutoff derivatives prove (6.4). There is no cutoff atom at the seam, because the sitewise cutoff commutes with both full-profile seam averaging and restriction within a site.

The remaining sources come from obstacles. A removed edge from a hole j to an obstacle k contributes

If an obstacle moves from k to a hole j, the hole moves from j to k. The source on this arriving hole is

Exchanges between two obstacles create no source. At a seam the source is the difference between the actual average on the remaining hole slots and the full-profile average, restricted to the outgoing holes. It vanishes at sites with no obstacles.

Assign a removed-edge source to its obstacle and a transport source to the obstacle responsible for the transport. Assign seam sources at a site to its obstacle strands, in any fixed manner. If a strand is isolated, it is the only obstacle there and receives the entire seam source at that site. These assignments can be made either in a block or in a bin, with strands enumerated at its initial slice in the direction being used. On \mathcal G, tracing a strand back from a source site in B_{R+2}(x) uses at most CL steps in its block, by Proposition 4.4. Thus every strand contributing a nonzero source starts in B_{4R}(x), for large \beta. All subsequent strand sums are restricted to this ball.

6.2 Conservation along an isolated strand

We next turn this identity into bounds that retain both the signed mass and the total variation. Keeping the two quantities separate is essential: total variation controls propagation and localization, whereas the smaller signed mass controls long spatial scales.

Near elapsed time zero, W_d varies too quickly to use block cancellation. The bins of length between 1 and 3 provide the following uniform replacement. We write H_\beta=C(1+\log\beta)^C for a fixed power of a logarithm; the constants, and hence this power, may be enlarged at subsequent uses.

There are at most CR^3 possible starting slots in each bin and C\beta bins, while W+W_d\le C. Including the background, the total variation of either complete source collection on \mathcal G is therefore at most CH_\beta\beta R^3. In particular, multiplying a rapidly small propagation error by one or two such total variations still gives a rapidly small error.

6.3 Propagation and the two-source identity

Let T_{v,u} denote actual forward propagation between specified slices, including exactly the operations lying between them. Reverse propagation is the adjoint with the ordered endpoints. Propagate every source from its insertion slice to its terminal cut in its own direction. The resulting signed masses are precisely the residuals defined in Section 3:

For example, on a smooth interval the difference between the actual mass and the profile satisfies the actual heat equation with forcing (6.1). At an operation it is multiplied by the actual map and then receives (6.2). Telescoping from the initial profile h to the terminal profile h-c\eta proves (6.11).

For a scalar pairing it is useful to make this substitution twice. Read the reverse profile at forward physical time t by writing \bar b_t=b_{\beta-t}, always on the specified side of an operation. In particular it is evaluated on the outgoing slice when paired with a forward source.

7 The scalar error at the trial potential

The first error in the capacity comparison is the pairing of the propagated source with the trial potential itself. We estimate it using Lemma 6.4. Its first term benefits from signed cancellation on isolated strands. Its second term contains two sources, and heat-kernel decay and spatial localization make their joint contribution smaller. All expectations below use the actual pin law \PP_P.

7.1 A single source paired with the free profile

For forward time s put

The reverse profile \bar b_s has value bound C(G+P_{d'}). The basic weight integral is

To verify it, first sum the smooth product by radial shells:

For s\le\beta/2, Lemma 5.1 gives P_{d'}\le CG, and

Integration over this half-period costs at most C\beta^{-1}. For s\ge\beta/2, use W_d\le CW and \sum_zP_{d'}(z)\le C, so the remaining product costs at most C\sup W=C\beta^{-2} per unit time. This proves (7.2).

Consider first the blocks contained in [\beta^{1/4},\beta-\beta^{1/4}]. On each such block d,d'\ge\beta^{1/8} and their weights are comparable over the block and over distances CL. At a deterministic initial slice, Proposition 4.2 bounds the probability of an obstacle at a prescribed slot by C\lambda. If it is not isolated, a second distinct obstacle lies within c_2L, whence the same proposition and a union bound give

Here and below fixed slot multiplicities are included in constants.

For a nonisolated strand starting over z, the total variation bound in Lemma 6.2 bounds its scalar contribution by

For an isolated strand subtract from the testing profile \bar b its value at the strand’s initial site and time, at any one slot. The constant value pairs with the signed source, whose bound is CL^{3/4}(W+W_d)(z). The remaining variation of the testing profile is at most

Spatial variation follows from the difference bounds over distance CL; time variation over the block follows from the time derivative bounds. The last inequality uses W\le G/D_* and W_{d'}=P_{d'}/d'.

Take expectations, use (7.3), and sum the blocks. Their lengths are comparable to L, so their weight sums are bounded by L^{-1} times (7.2). The contribution of all these blocks is at most

The three powers of \beta are respectively -4+3\delta, -5/2-\delta/4, and -21/8+\delta. The middle term is the one for which signed cancellation is needed.

The remaining blocks lie within 2\beta^{1/4} of an endpoint, for large \beta, because L\ll\beta^{1/4}. Use the bin total variation bound from Lemma 6.3, without cancellation. Weights at times and sites within a bin change by at most H_\beta, so the expected contribution is bounded by H_\beta\lambda times the integral in (7.2) restricted to these endpoint intervals. On the first interval the preceding estimates give

The last interval costs at most C\beta^{-7/4}(1+\log\beta). Multiplication by H_\beta\lambda gives powers at most -23/8 and -13/4, both strictly below -5/2.

The cutoff background is estimated separately. On its annulus the reverse profile has size at most C/(\beta R), so

We have therefore bounded the first term of (6.12). It remains to control the interaction between its two source collections.

7.2 Snapshot moments for spatial boxes

The source locations at two different times are correlated. The following consequence of the snapshot estimate controls their pairing without an independence assumption.

In the applications, the weights are restricted to the deterministic anchor region B_{4R}(x). We bound their norms by the corresponding unrestricted envelopes. For these weights, denote the resulting bounds by

To check the first line, direct shell sums give \|W_d\|_2\le Cd^{-5/2} and \|W_d\|_1\le Cd^{-1}; the square root of the squared box sums is no greater than the total weight. For the second line, \|W\|_2\le C\beta^{-1}D_*^{-1/2}. It remains to bound the squared box sums by C\beta^{-2}F^2. Use the larger envelope C\beta^{-1}(1+r_z)^{-2}. The bounded number of cubes within distance CF of x each have weight sum at most CF/\beta. At distance comparable to nF, each cube has weight at most C\beta^{-1}F n^{-2} and there are O(n^2) cubes in such a shell. Summing their squares over n\ge1 gives C\beta^{-2}F^2\sum_{n\ge1}n^{-2}, as claimed.

The time sum of J over the bins of either traversal satisfies

Indeed the bin lengths are bounded above and below, so \sum(1+s_{\mathrm{bin}})^{-5/4}\le C and \sum(1+s_{\mathrm{bin}})^{-1/2}\le C\sqrt\beta. The latter gives C\beta^{-1}, which is smaller than the right side of (7.9).

7.3 Two sources separated in time

First remove interactions involving cutoff background. Counting mass and supremum contractions of the actual kernels imply that two background sources cost at most

For one background source, integrate its time variable and use its supremum bound. The resulting cost is at most C\beta b_* times the total variation of the other, obstacle, source collection. Lemmas 6.3 and 7.2, or simply the one-point snapshot bound, give

For the smooth weight, use \sum_{r_z\le4R}W(z)\le C\beta^{-1}R and O(\beta) bins. For the point weight use \sum_zW_d(z)\le C/d and sum in time. Thus all terms with background are o(\beta^{-5/2}).

We can now assume both sources are obstacle sources. Group their bin pairs by dyadic scales T\in\{1,2,4,\ldots\}, up to a constant times \beta, according to 1+t-s\asymp T. The scale T=1 also includes a bounded number of neighboring and identical bins. Equivalently one may group by bin indices; the bin length bounds preserve comparability away from this first scale. For a fixed bin there are O(T) partner bins at scale T.

First replace each intervening propagator by its version killed on exiting B_{R_*}(x). Every source starts in B_{R+2}(x), so Lemma 4.6 bounds the discarded mass by a rapidly small quantity, uniformly on \mathcal G. The source supports and total variations are bounded by fixed powers of \beta, by (6.4) and Lemma 6.3; hence the aggregate loss in the double pairing is still rapidly small. Lemma 4.5 now bounds each entry of the killed kernel by CT^{-3/2}. The displacement assertion in Lemma 4.6 further permits us, with another rapidly small aggregate error, to retain only anchor sites at distance at most

The C\log\beta shifts from the bin anchors to insertion sites are absorbed by L^{3/4}.

Apply Lemma 7.2 with boxes of side comparable to F(T). The two bin total variations contribute only H_\beta times their anchor weights. We treat the three possible weight products in turn.

Two smooth weights. There are O(\beta) first bins and O(T) partners per first bin. Their contribution at scale T is at most

Since F^2\le C(L^{3/2}+T^2L^{-1/2}), the dyadic sum is at most

Here \sum_TT^{-1/2}\le C and \sum_{T\le C\beta}T^{3/2}\le C\beta^{3/2}. The final term explains why a displacement bound smaller than T is needed in (7.11).

One smooth and one point weight. Sum the point weight over its own elapsed-time bins using (7.9); each such bin has O(T) smooth partners. The scale-T bound is

Inserting (7.8) and (7.11), and summing dyadically, gives at most

This time the growing sum is \sum_{T\le C\beta}T^{1/2}\le C\sqrt\beta. The calculation applies whichever direction supplies the point weight, because (7.9) holds in either elapsed-time order.

Two point weights. When t-s\le\beta/2, their elapsed profile times satisfy

At least one is therefore at least \beta/4, and its point weight is bounded by CW by Lemma 5.1. These pairs are already covered by (7.14). The remaining pairs have separation comparable to \beta. Use the entry bound C\beta^{-3/2} and the unrestricted two snapshot pairing from Lemma 7.2, with a single box containing the source support. Summing both time variables by (7.9) costs at most

This finishes the estimate of the second term in (6.12). To conclude, the smallest power gains in the entire scalar calculation are L^{-1/4}=\beta^{-\delta/4} in (7.5), \beta^{-\delta/4} in (7.6), and L^{-1/2}=\beta^{-\delta/2} in (7.13). Each dominates every fixed logarithmic power because \delta=1/200>0. All other displayed terms have larger power margins. Taking the expected absolute value in Lemma 6.4 proves Proposition 7.1.

8 Residuals in the dual energy norm

We now bound the two residuals from Lemma 6.4 against an arbitrary test vector, using the one-period energy of that vector. The scalar estimate of Section 7 is not sufficient for this purpose: the test vector need not resemble the free potential. Instead, we represent propagated sources as divergences of flows and pair those flows with the gradient of the propagated test vector. For a group of sources associated with one obstacle strand, we first route the signed mass to a nearby slot. Only the resulting net mass must then be sent to infinity. The cancellation in Lemma 6.2 reduces this second cost.

We work in one direction at a time, writing U_d=U for the forward direction and U_d=U^* for the reverse direction. Time in the following argument is elapsed time in this chosen traversal; all instantaneous operations retain the ordering fixed in Sections 2 and 6. Write T_{t,s} for its propagator from s to t, and extend this notation periodically when t>\beta.

For a terminal vector y, the adjoint suffix Y(t)=T_{\beta,t}^*y satisfies, by Lemma 2.2,

These bounds hold for signed vectors. In particular, they require neither a reversible one-period kernel nor time-reversal invariance of the exposure law.

At a fixed slice, a finite-energy flow J with divergence \nu gives

Here Y is extended by zero on the killed inner boundary and outside the cube. It has finite support, so summation by parts has no term at infinity, and boundary-crossing edges are already included in \D_t. Nonzero total mass is allowed: the flows in Lemma 4.7 carry that mass to infinity on the extended available-slot graph. This is the same flow-pairing principle as in [12, Section 3.3]; the block cancellation below provides the additional saving needed here.

8.1 Flows for the sources of one block

Divide the sources in the chosen direction into two collections. The first contains all cutoff sources, all obstacle sources from the smooth potential, and the point-profile obstacle sources whose elapsed times lie in [\beta/2,\beta]. The second contains the remaining point-profile obstacle sources, at elapsed times in [0,\beta/2). We begin with the first collection. Its obstacle sources satisfy the block bounds of Lemma 6.2 with envelope W, and its cutoff sources have size at most Cb_* per slot and unit time, where

is the bound in (6.3). Since \beta/2 is a block division, no block is truncated when the late point-profile sources are included.

Let I_1,\ldots,I_n be the consecutive blocks in this direction, with lengths between L and 2L. Thus n\le C\beta/L. For each block I_j, let \mathcal A_j be the obstacle slots at its starting slice whose sites lie in B_{4R}(x), and use the same index for the strand starting at such a slot. Denote its starting site by z_\alpha. Set

and

These are finite snapshot sums, with slot multiplicities retained.

8.2 Averaging over the next block

The flow bound controls a block at any fully regular slice of its successor. We now use these choices of slice to share a single Dirichlet energy budget among the blocks. The final block requires a separate step, because its successor lies beyond the terminal cut.

Let m_j be the terminal mass produced by the first collection of sources in I_j. For j<n, at a slice t\in I_{j+1} the block pairing is the propagated source mass paired with Y(t)=T_{\beta,t}^*y. Lemma 8.2 and (8.4) therefore apply directly. For the final block, put m=m_n and use the exact identity

The second term equals \ip{U_dm}{y}. It is represented by continuing the source mass through a copy of the period, with terminal test vector y at time 2\beta. At its averaging slice t\in\beta+I_1, the test vector is consequently T_{2\beta,t}^*y. The localized norm bound and (8.3) control the first term:

The localization error here is estimated in \ell^1, using \norm{y-U_d^*y}_\infty\le2C_0. Figure 2 shows the ordinary averaging windows and the window used for this continued final block.

For each j, average over the fully regular subset of its successor block. Proposition 4.4 gives this subset measure at least cL. Applying Cauchy–Schwarz in that time variable bounds its averaged localized pairing by

where Y_j(t)=T_{\beta,t}^*y for j<n, whereas Y_n(t)=T_{2\beta,t}^*y and I_{n+1}=\beta+I_1. The intervals I_2,\ldots,I_n are disjoint in the original period, and \beta+I_1 lies in the copied period. Each full period has the bound (8.3), so

Cauchy–Schwarz across the block indices in (8.14), followed by (8.13), gives

The sum of localization errors is still rapidly small because n\le C\beta/L.

The expectation of the first term in parentheses is bounded by C\beta/L^2 times the right side of (8.6). Substituting \lambda=\beta^{-3/2}, D_*=\sqrt\beta, and b_*=\beta^{-1}R^{-3}, this is

Likewise, (8.7) and the one-point snapshot bound give

Thus the block argument has converted cancellation of source mass into a dual energy bound. It remains to handle the early point-profile sources, for which cancellation at the scale L was not used.

8.3 Early point-profile sources

9 From the pinned comparison to magnetization

We first combine the two residual estimates to prove the uniform pinned comparison. We then express a positive field as a mixture of deterministic pin sets. Integrating the finite-volume magnetization bounds in the field will preserve the required order: volume tends to infinity before the field tends to zero.

9.1 Completion of the capacity comparison

9.2 Boundary pins and field pins

Write V_N=|\Lambda_N|, use \partial\Lambda_N=\partial_{\mathrm{in}}\Lambda_N for the inner boundary, and let q_{\Lambda_N} be the number of slot edges. For a picture \omega, let \mathcal C(\omega) be its cycles and a_C the number of time-zero slots on C. For colors \sigma_C\in\{-1,1\}, define

Let \chi_D(\omega,\sigma) indicate that every cycle meeting D is up. The spin-loop representation in Proposition 2.1 identifies the ordinary partition function with

Its boundary-pinned counterpart and pressure are

We use the surface estimate of [12, Proposition 7.1]. Its argument is short and records why all-times boundary pins do not change the pressure.

For t\ge0 put b_\beta(t)=1-e^{-\beta t}, and let \nu_b be the law that selects each time-zero slot independently with probability b=b_\beta(t). If A is the selected set and \operatorname{Up}(\omega,\sigma) is the set of up slots at that cut, then

Consequently the field law in (9.3) is the picture/color marginal of the joint law

where the color variable uses counting measure. Indeed the common factor extracted from its normalizer is e^{\beta q_{\Lambda_N}/4+\beta tSV_N}. Conditional on A, this is exactly the zero-field color law with the deterministic pins E=D_0\cup A.

The marginal law of A is generally not Bernoulli, because its normalization is weighted by the pinned partition function. Conditioning instead on (\omega,\sigma) makes selections on up slots independent Bernoulli choices of parameter b, and forbids selections on down slots. Thus every specified slot j satisfies

This is the only property of the posterior pin law needed below.

9.3 Thermodynamic and zero-field limits

9.4 The explicit lattice correction

The full ideal density has a two-term expansion with a remainder small enough to identify the coefficient in Corollary 1.2.